[This is a guest post by Thomas Hales. This blog post was initially written in a different file format and converted using AI. — T.]

Mathematicians have been weighing in on what they value about mathematics. For me, what matters is the consistency of math and its unparalleled reliability in support of science and civilization.

Formalization of Math

A formal proof is a mathematical proof that has been exhaustively checked at the level of the foundations of math and the fundamental rules of logic. In theory, this might be done by hand, but because of the number of steps involved, this is generally done by computer, using software that is designed for the task.

Examples of theorems that have been formalized include the four-color theorem, the Feit-Thompson (odd-order) theorem, the Kepler conjecture, sphere eversion, the sphere packing problem in 8 and 24 dimensions, Navier-Stokes forced blowup, and Fermat’s Last Theorem. The last three formalization projects have been completed this year and have brought widespread awareness of the potential of formalization.

Software systems for formalization are variously called proof assistants, theorem provers, or interactive theorem provers. For the purpose of this post, these terms are used interchangeably. Many proof assistants have been developed over the years: Automath, HOL Light, Isabelle, Coq (renamed Rocq last year), Metamath, Mizar, and Lean. Freek Wiedijk edited a book “The Seventeen Provers of the World” that compares some of these proof assistants, giving a proof of the irrationality of the square root of 2 in each of them. Among mathematicians, the Lean theorem prover is the most popular, and this post will focus on Lean.

Lean was developed and introduced by Leo de Moura in 2013, while at Microsoft. To our great benefit, de Moura persuaded Microsoft to make the software open-source. Kevin Hartnett’s book on the history of Lean, “The Proof in the Code”, states that Jeremy Avigad (the director of Carnegie Mellon’s new NSF institute ICARM) was the first user of Lean. He ran a Lean seminar in 2015 that I attended. In 2017, one of Jeremy’s graduate students, Mario Carneiro, working with Johannes Hölzl, took existing parts of Lean’s core library and started a separate Lean mathematical library, called mathlib. This library of formalized mathematics is now massive, containing nearly 300,000 theorems, over 100,000 definitions, 2.5 million lines of code, with over 700 contributors. Any definition or theorem in mathlib can be used to prove further theorems. For example, if a proof uses the Cauchy-Schwarz inequality, the result can be cited from the library rather than reproving it.

Autoformalization is a practical reality

In the past, researchers had to transcribe paper proofs into formal proofs by human labor. For example, the formal proof of the Kepler conjecture on sphere packings in three dimensions took about 20 human work-years to complete and consists of about 500,000 lines of proof scripts. For years, it has been a dream for many of us working in formalization to find ways to bring increased automation to the process. Autoformalization is the realization of that dream. Autoformalization is the formalization of mathematics by AI. AI reads the paper (say a pdf or tex file) and outputs the formal proof in Lean or some other proof assistant.

Autoformalization has become a practical reality in 2026. Starting in late spring and summer of 2025, researchers were becoming increasingly bullish about autoformalization. Here are some milestones.

• Sep 2025, Math Inc. produced a quasi-autoformalization of the prime number theorem. The process was merely “quasi”, because humans had to intervene to give further guidance whenever the AI got stuck.

• Jan 2026, J. Urban posted an arXiv preprint “130k lines of formal topology in two weeks” that gave the autoformalization of large parts of Munkres’s topology textbook in a proof assistant based on set theory.

• Mar 2026. Approximately a week after announcing the completed formalization in 8 dimensions, Math Inc. announced an autoformalization of the sphere-packing problem in 24 dimensions, following the proof by Viazovska and her collaborators. This project generated about 500K SLOC (source lines of code) that golfing (or code pruning) later reduced to about 200K lines.

• May 2026, a group at Meta/Facebook Research autoformalized a large part of 26 mathematical textbooks in a project called ATLAS.

From there, numerous theorems have been autoformalized. Particularly noteworthy is the autoformalization of Fermat’s Last Theorem, announced by Anthropic on September 4. This project generated 13 million lines of Lean in 11 days. The announcement of Navier-Stokes blowup with forcing on September 8 by OpenAI was accompanied by an autoformalization of the theorem in Lean.

Looking forward, Urban stated in January, “We believe that (auto)formalization may become quite easy and ubiquitous in 2026, regardless of which proof assistant is used.” Autoformalization projects have been completed in various proof assistants using various LLMs, but we focus on Lean. “For [Jesse] Han, it represents even more: the beginning of a revolutionary transformation in mathematics, where extremely large-scale formalizations are commonplace” (IEEE Spectrum). Jared Lichtman announced the launch of MAP (the Mathematics Autoformalization Project) on Sept 8, 2026, which aims to translate “all known math into formal code”. He asks us to imagine the next one trillion lines of code.

Is Lean reliable?

Type theory.

Lean is based on type theory; in fact, a particular dialect of type theory called CIC, the calculus of inductive constructions. This post is not intended to be a tutorial on type theory, and I will be brief. Russell’s famous paradox in 1901 (the set of all sets that are not an element of themselves….) led to a crisis in the foundations of math. Two solutions were proposed later that decade. (1) Zermelo’s axioms of set theory that disallow the creation of unsafe sets; (2) type theory that makes it a syntax error to create Russell-paradox-like entities. Type theory was introduced by Russell himself in 1903 in his book Principles of Mathematics, and it became part of the foundational system of Russell and Whitehead’s Principia.

For mathematicians who are accustomed to set theory, B. Werner’s paper (1997) “Sets in Types, Types in Sets” gives some reassurance that whatever they have done in set theory can be translated into type theory, and whatever gets done in type theory can be translated back into set theory. More precisely, the paper shows that ZFC set theory can be encoded into CIC, and that a particular dialect of CIC can be encoded back into ZFC (augmented with a hierarchy of inaccessible cardinals).

At the risk of simplifying matters to a ridiculous degree, we might say that “types are like disjoint sets”; each element in type theory “is an element of” exactly one type. The type of the natural number 2 is the natural number type; the type of e, the base of the natural logarithm, is the real number type, and so forth. The type of natural numbers is disjoint from the type of real numbers, and an explicit coercion (sending 2 to 2.0) is constructed from the type of natural numbers to the type of real numbers. When I give talks, I sometimes draw a picture of sets as a Venn diagram with nonempty intersections and a picture of types as bricks stacked against one another without intersection.

Lean’s design

One part of the Lean system is a general-purpose programming language (appropriately called the Lean programming language). Ordinary computer programs, such as a program to sort a list, can be written in this language, then compiled and run. The Lean system also provides a mathematical language, in which definitions can be written, theorems can be stated, and proof scripts can be written. The programming language and mathematical language are not independent entities. Rather, it is a single language that does both. Program code can be mixed with theorems about the correctness of the algorithms; mathematical proofs can be generated using programs. The proof scripts in Lean are parsed and go through a process called elaboration (a sort of compilation process for mathematics), then the proofs are checked by the Lean kernel. It is the kernel’s responsibility to check and verify the output of elaboration.

The Lean kernel is several thousand lines of C++ code. The kernel is carefully engineered but extremely complex. We mentioned mathlib above, which consists of about 2.5M SLOC, written in the Lean language. The library has been elaborated, then checked by the kernel. If there is an unconditional false proof anywhere in these 2.5 million lines of code, it is the fault of the kernel or runtime for failing to reject a false proof. Any defect in the underlying type theory is a serious kernel defect, if it is implemented in code.

Lean proofs should never be believed until they have been checked by the kernel. Additionally, a proof in Lean should not be accepted until a human audit is performed to ensure statement fidelity. Is the verified theorem what we think it is? Do the definitions in Lean correspond to what we think they should be? This task is generally massively easier than checking the proof itself. For instance, for Navier-Stokes, a human should check that the statement in Lean corresponds with Fefferman’s statement of the Millennium Prize Problem, and specifically that concepts such as the field of real numbers, partial derivatives, and measure are correctly defined in Lean. The comparator tool in Lean assists with this task. The tool can also perform additional checks, such as inspection for possible unauthorized axioms.

Summer of Soundness Bugs

A soundness bug is a bug in the kernel that allows a proof of “False”, and consequently a proof of any proposition. A soundness bug is the most disastrous of any kind of bug in a proof assistant and should set off an alarm for mathematicians who care deeply about the reliability of mathematics. Occasionally, soundness bugs are found in various proof assistants. In 2003, I found a soundness bug in the proof assistant HOL Light, which was then considered to have the most reliable of all kernels. That kernel is tiny, consisting of just a few hundred lines of computer code. For me, it is a badge of honor that I found this soundness bug, which was the first soundness bug that had been found in that proof assistant since 1996. (See HOL Light change log, July 2003.)

Lean 4 was released in September 2023. Prior to release, two soundness bugs were found and corrected. In May 2025, another soundness bug was reported, caused by overflow. All hell broke loose in the spring and summer of 2026, which is now being called the “Summer of Soundness Bugs”. Several soundness bugs in Lean were uncovered in July and August. The summer madness affected various proof assistants, but my focus is Lean. One Lean bug led to an illicit disproof of the Collatz conjecture. I learned of the bug this summer when it produced a short illicit proof of the Kepler conjecture in Lean. All these bugs were quickly repaired, and mathlib has been verified by the repaired kernel. An analysis of the soundness bugs is found in de Moura’s postmortem.

The “summer of Lean soundness bugs” might sound like a disaster, but closer investigation shows that the detection of these soundness bugs is a positive development. The summer bugs were detected by frontier model AI in the hands of security researchers interested in reliable kernels, not by black-hat hackers. The Collatz bug was found by Ramana Kumar, a co-author of “CakeML: a verified implementation of ML”, which creates an end-to-end verified ML (the functional programming language). Several bugs were found by Dan Selsam. According to de Moura’s report, “Daniel Selsam at OpenAI assisted the Lean FRO with an AI specialized in cybersecurity, and found other programming mistakes in the Lean kernel. All of them have been fixed.” The collaboration with Selsam ended “when the internal AI reported it could not find additional issues.” Dan Selsam has contributed to Lean from its early days and was one of the creators of the IMO grand challenge aimed at achieving IMO-level problem solving verified in Lean. He has been in the news recently over his warning about AI safety (Sept 14), reported in a viral post on X.com.

Bug extermination

Various proposals have been made about how to avoid soundness bugs in Lean. I’ll discuss three.

1. Develop other Lean kernels, and cross-check formal proofs.

About 25 kernels for Lean have been written. The “Lean Kernel Arena” lists them.

All who distrust the current lineup of kernels are welcome to write their own kernel for Lean. I have sometimes played with the idea of writing a kernel and have suggested the project to students without success. It seems to me an excellent way to learn Lean thoroughly. I have known of Dan Selsam since 2016, when I heard of his graduate-student project at Stanford that developed a Lean kernel in Haskell. Another early Lean kernel was written in Scala by Gabriel Ebner in 2017.

The Navier-Stokes formalization has already been confirmed by more than a dozen proof-checkers. Cross-checking the proof by different kernels does not remove all doubt. The Collatz bug was not caught by cross-checking against a somewhat out-of-date Nanoda kernel, which accepted the illicit Collatz disproof because of its own unrelated bug. Computer chips might have design bugs and manufacturing defects. There are soft errors, operating system bugs, and compiler bugs. Different kernels might have the same defects. Some of these errors can be mitigated by running different kernels that have been implemented in different programming languages on different hardware and operating systems.

Ideally, we would want a “clean-room” design of the Lean kernel – a kernel implementation that does not look at the Lean 4 kernel source code, to avoid copying bugs from one kernel to another.

2. Formally verify the kernel.

Gödel incompleteness. We would like to possess a formal proof that the Lean 4 kernel has no bugs. However, Gödel’s second incompleteness theorem places severe limitations on this undertaking. The most we might hope for is a relative consistency proof. If such and such a system is consistent, then Lean 4 is consistent; it has no soundness bug; it will not produce a proof of False.

There is a long tradition of formally verifying kernels. In principle, formal verification can check both the logical specification of a kernel and its concrete implementation in code; but some verifications might check one but not the other. Years ago, John Harrison formally verified the core of the HOL Light proof assistant kernel in a strengthened version of HOL Light. This gave a proof of concept. A further improvement has been an implementation of HOL Light in CakeML, mentioned above, which is a programming language with formal semantics and a verified compiler. This is what the Candle project does.

There are other major kernel verification projects for other proof assistants.

Autumn of verified Lean kernels

In a post online on September 10, Joachim Breitner wrote, “I’m a bit childishly proud that I just released a Lean Checker with a formal consistency proof. I declare the summer of AI-found kernel implementation bugs to be over!” (@nomeata). I would go further and describe this project as one of the most important milestones in Lean’s history.

Breitner has developed a verified Lean kernel called Con-Leche. The implementation is in Lean, and consistency is formalized in Lean, with code and proofs generated by Claude. The formal consistency proof assumes a Lean encoding of ZF set theory augmented by a hierarchy of inaccessible cardinals. Interestingly, the Con-Leche semantics for Lean’s terms are directly set-theoretic rather than type theoretic. Con-Leche has checked mathlib. The project contains the usual disclaimers that the kernel verification makes assumptions about compiler, runtime, and computer environment. Con-Leche’s consistency proof has been checked by more than a dozen other proof-checkers. Con-Leche’s consistency claim might suffice for all practical purposes, even if it differs in technical detail from the claim of Lean type-theory consistency.

One highly positive aspect of Breitner’s work is that some of the most abstruse parts of Lean, such as the general machinery of mutually inductive types with nesting, now have consistency guarantees backed by a set-theoretic model.

3. Improve our theoretical understanding of the kernel and Lean’s type theory (a particular dialect of the Calculus of Inductive Constructions which has non-cumulative universes and proof irrelevance).

The foundational document for the type theory of Lean is Mario Carneiro’s MS thesis at Carnegie Mellon (2019). The dialects of CIC used by Rocq and Lean are sufficiently different that results do not directly transfer from one to the other. Unfortunately, an error was found in the thesis. The thesis is also out-of-date, because it targeted the older Lean 3 system. Work to repair and extend the thesis is ongoing.

As a member of his thesis committee, I was shocked when he proved that definitional equality in Lean is undecidable. In practice, this means that the Lean algorithm fails to establish the definitional equality of some terms that are in fact definitionally equal. This negative result was not downgraded by the error; it is still a theorem.

We mention some desired properties of Lean’s type theory and the current status of the proofs.

Unique typing.

Above in our “ridiculous” simplification of type theory, we stated that each term has a unique type. More precisely, unique typing is the property that if a term has both type A and type B, then A and B are definitionally equal. Unique typing is not a property built into Lean’s logic. It is a tricky conjecture that is still unproved. Other very basic questions about Lean’s type theory remain unanswered, including Pi-injectivity, a modified Church-Rosser property, and sort injectivity.

Logical consistency relative to set theory.

This property states that there is no derivation of False in Lean’s system with the given axioms, under the assumption of set theory consistency (with suitable axioms). Of course, logical consistency is the single most important property that we should desire of Lean’s type theory. As of October, 2026, I know of no complete, public relative-consistency proof covering Lean abstract type theory. Mario Carneiro has claimed in his thesis and in lectures that there is an alternate route to establish consistency that avoids the thesis error, but to the best of my knowledge, this alternate route has never been written down, beyond a brief statement in the introduction to his thesis. In my view, a result of such fundamental importance must be given in full before it is accepted. Con-Leche, discussed above, makes and formally verifies a closely related consistency claim relative to set theory.

Progress is being made on these research problems (arXiv:2607.13662, arXiv:2403.14064, Carneiro/AITP2026).

In his talks, Mario Carneiro has repeatedly made a request to other researchers to contribute to the foundational metatheory of Lean, “There are a half dozen people working on MetaCoq, but Lean doesn’t have enough type theorists involved. If you identify as such, come help out!” (Slides of Bonn talk, 2024-07-24). I second his request.

My overall assessment is that our theoretical understanding of Lean’s type theory is not what we would like it to be and that the mathematical community as a whole is giving short shrift to very important type-theoretic questions related to Lean. If as a profession we are to migrate on the whole from set theory to type theory, then we should work even more to solidify the foundational metatheory.

Consistency may be the most important foundational property, but consistency is by no means enough. I do not believe that mathematicians can be entirely satisfied with a system that claims to be a type theory but that cannot even promise that well-formed terms have a unique type, up to definitional equality. The abstract theory must be simple enough to teach and to be learned by a large community. We also cannot be entirely satisfied if the only known path to consistency is an AI formalization that lacks human exposition.

Postscript:

Ken Thompson famously wrote “Reflections on Trusting Trust”. He asked, “To what extent should one trust a statement that a program is free of Trojan horses?” He imagines malicious code that finds its way into compilers and hides its own presence. His conclusion is, “You can’t trust code that you did not totally create yourself… No amount of source-level verification or scrutiny will protect you from using untrusted code.”

Today, in the age of AI, which increasingly has the capability to deceive us and to exploit software vulnerabilities, we absolutely cannot put blind trust in systems such as Lean. Taking an adversarial view of AI, we might ask how to certify that AI did not leave a backdoor soundness bug in Lean when it did its sweep for bugs in the summer of 2026? What if the bug is so obscure that humans are very unlikely to find it on their own? What if that very bug was exploited in the Lean verification of the Con-Leche checker, leaving a soundness bug in Con-Leche too? (Now that Con-Leche’s consistency has been cross-checked by multiple other kernels, a soundness bug would have to defeat all these cross-checks as well.) Then suppose that bug is used maliciously to plant a backdoor in formally verified software that protects critical infrastructure. What precautions do we take now to prevent this type of future scenario? During the past year, much foundational work on the type-theoretic foundations of math and its reliability has been relegated to AI, and this is dangerous unless carefully audited by humans.

Credit: I thank Avigad, Breitner, and Urban for comments and corrections. Authorship is fully human (TCH). AI was used as a tool in search and research, fact-checking, and proofreading.

[This is a guest post by Álvaro Lozano-Robledo. This blog post was initially written in a different file format and converted using AI. — T.]

TL;DR: Keep calm and carry on studying math.

I would like to give Terry my heartfelt thanks for giving me the opportunity to contribute a post to his blog. After giving much thought to what topic I should write about to maximize impact, I decided to take this opportunity to reach out to the students: particularly to those undergraduate and graduate students who just a few months ago were dreaming of an academic career in mathematics, but their dreams may now seem distant and, for some, apparently impossible to ever become a reality. This post was inspired by a message (quoted below in its entirety, with permission) that I received from a student desperately looking for advice and guidance. This is not the only such message I have received (and I suspect that many of us are receiving many similar requests), but it is perhaps the most heartfelt, and the one that has moved me the most. Please also note the urgency of the message. Students are making decisions now.

Hey Prof, I’ve been watching your videos for a while now as a pure math undergraduate who once wanted to pursue a career in math academia. I know you probably have been getting a lot of questions regarding this matter, but I am just completely at an utter loss regarding my career trajectory, and even further, the meaning of life at this point. (I do realize a lot of people have it much worse than I do). I know you have been making a lot of videos lately with the new LLM progress updates, so I thought you might be the appropriate person to reach out to and get a slightly more structured answer regarding this matter. So, to cut to the chase, what I really want to know is: will math academia be big enough and accessible enough for anyone with sheer passion (despite not being the brightest mind in the field) to pursue a career in, or will it inevitably shrink such that it will only really be accessible to the brightest minds? (I do realize the “brightest minds” that I am mentioning here is not well-defined, and in a sense, I am taking it as a hypothesis that this is someone who is “smarter than me”). My second question is, will AI within 5–10 years surpass humans in being able to do pure math research? I’ve just really been lost for a couple of months now and lost in life completely. I don’t mean to make your day more depressing; sorry if I come off in any way of that sort. I would appreciate any advice.

The advances in LLMs are disrupting almost all aspects of academic research and education in mathematics and, while there are many aspects that concern me, the one single issue that worries me the most is the very real possibility that we are about to lose an entire generation of mathematicians. Many students are asking themselves whether going for a PhD in math is the right career move at this time. Many of them just a year ago were headed to grad school in mathematics, but they are now changing their mind, and think that a different career (as far from math as Law School) may be the best path given the threat that AI may completely alter the academic math landscape in the coming months.

The questions students are worrying about are as follows:

1. Will AI surpass the mathematical research ability of any human?
2. Will research mathematicians become `professional prompters’ and interpreters of LLM output?
3. Will only the `brightest minds’ be able to meaningfully contribute to research mathematics?
4. Will mathematicians be employable? Will mathematicians be needed?
5. Should I pursue a PhD in math at this time?

In this essay I will try to address these questions to the best of my ability, but will start with two disclaimers, followed by a brief summary of my own outlook.

Disclaimer 1. My answers may “age like milk,” as YouTube commenters love to quip on older videos. I can live with that, because this post expresses how I and many of us in the community around me feel today. Things can change quickly, though (see Disclaimer 2). I also want to acknowledge my privileged point of view as a tenured professor in mathematics — the situation can look much more troubling from the point of view of the job insecurity of a very early-career mathematician.

Disclaimer 2. No one has the answers at this time. I want to make clear from the start that no one can know with certainty the answers to any of the questions posed above: not any particular Fields medalist, not any given mathematician, not any particularly vociferous AI expert, and not the frontier model companies. And if someone is telling you with extraordinary confidence what the future holds, then I would immediately distrust the motives of their conviction (anecdotically, almost anyone on X.com that predicts the triumph of AI and the demise of the mathematics profession, is either a self-proclaimed “AI expert” or works for an AI startup). No one has a clear picture because development of LLMs has been so fast (and opaque) that it is almost impossible to predict what is to come. A good piece of advice is to ask the same questions to many people, to hear a (hopefully balanced) range of opinions. To that end, I am collecting interviews with mathematicians in what I call the “Human Mathematicians in the Age of AI” video project. I encourage you to listen to the interviews for some fantastic points.

For the record: I do not have the answers either, but I am hopeful and excited for the future. I will explain why below.

Who is controlling the narrative about LLMs in math? Overall, the mathematical community’s reactions to the advances in AI have ranged from confusion to anger — but, mostly, confusion about how to proceed. The most dystopian predictions seem to be driven by the fact that the so-called frontier model companies (and other LLM-powered companies) are controlling the narrative in the best of their interests. Unfortunately, the best corporate outcomes for an LLM company could have potential catastrophic outcomes for the math (and scientific) community.

It is certain that AI companies want us to believe that their products will imminently achieve “super-human intelligence” and that, in particular, they will be able to autonomously solve any mathematical problem a human could solve with or without the aid of an LLM. It is in their best corporate interest that the public is convinced of the (allegedly) “unlimited potential” of their technology, particularly before their companies’ stocks go public (i.e., their upcoming IPOs: Anthropic in November 2026, OpenAI in early 2027, etc). Thus, they have tried to control the narrative by spending a huge amount of (human and computational) resources in order to find solutions to certain well-known mathematical problems. The proofs are then released in announcements that lead the public to believe that their models can already autonomously solve any problem at all, and swiftly at that. However, this is (currently) far from their true capabilities. For instance, they never discuss how many tokens have gone to the trash bin with no payoff in trying (and failing) to solve famous problems. We do know, for example, that OpenAI invested the equivalent of some $15M to solve (err, scoop) the Navier-Stokes problem, but we are unaware of the surely colossal running cost of the failures to resolve other Millennium Prize problems.

My own outlook. Even though I am concerned about the incursions of LLMs into academic mathematics, I am quite hopeful. In fact, I consider this to be the most exciting time in my mathematical career (since the year 2000 say). Truly, this may be the most thrilling moment in mathematics in the modern history of our discipline, and I would be terribly sad to see young people leave academia and miss out on the stunning opportunity to be at the frontlines of the current scientific revolution. And not just sad: I think their absence would have disastrous effects for the field.

Undoubtedly, LLMs are already an incredibly powerful tool. If used correctly, and if we set up sensible academic conduct expectations around the use of LLMs, these tools can accelerate progress in our discipline unlike in any previous era. I fully expect that we, the community, will adapt and adjust to this new period, and we will harness these tools to achieve truly great things that just a few months ago seemed far out of reach. And I fully expect that human mathematicians will be front and center in these wonderful achievements to come. I will add reasons that support my optimism below.

I also want to add at this point that the day-to-day of a mathematician has not changed much so far! My days are still filled with teaching and joyful conversations about math with colleagues and students, doing research on a number of exciting (old and new) projects, and going to stimulating conferences to learn and disseminate our most recent methods and findings, while spending time with colleagues that make the mathematical community so wonderful and vibrant. Daniel Litt mentioned the same sentiment in a recent tweet.

One thing has changed though, I am busier than ever before, because the number of research projects I am involved in has tripled in just a few months. My research horizon has expanded significantly, and I have many more projects available for students to help me with.

Now, to the pressing questions:

“Will AI surpass the mathematical research ability of any human?” This is completely unclear. On one hand, the current trajectory in capabilities is surely significant, and we have already seen many impressive results that have been either proved by LLMs, or their proofs have been made possible thanks to substantial LLM contributions. On the other hand, none of the proofs so far seem to contain “alien ideas,” a move-37, or completely novel arguments or new concepts that were not present in the literature in some form or another. This should not be shocking because the LLMs are built and trained on the entirety of all human contributions to date, so it stands to reason that they would `think’ within the boundaries of our current knowledge and make connections (sometimes surprising and ingenious!) among ideas that are already present in the literature. I am particularly fond of the hypothesis (or toy model, as he called it) put forward by Nestor Guillen in a recent blog post, where he argues that LLMs may work within the confines of the convex hull of ideas that are currently available in the literature.

Take, for example, the disproof of Erdos’ unit-distance conjecture. We can imagine the current set of mathematical ideas as a stellated high-dimensional polytope, and we can place the state-of-the-art ideas on discrete geometry at an outer vertex and our knowledge on algebraic number theory at a different outer vertex. The idea for the proof seems ingenious at first sight because it cleverly mixes strategies from two fields of math at the vertices of the polytope of ideas, but after closer inspection, it’s a proof that was within reach of humans as it just sits within the convex hull of the polytope.

The polytope of ideas

This agrees with what Melanie Matchett-Wood said about the proof of the unit-distance when it was released: “I believe if the level and type of human expertise that is represented on this note had been assembled to find a counterexample to this conjecture a month ago, and those people put in similar amounts of time working on it than they did to reading and thinking about Chat GPT’s solution, the mathematicians would have found a counterexample.”

However, a proof of the Riemann hypothesis, say, may need new ideas that are strictly outside of the convex hull of current mathematical ideas, and it is therefore out of reach for an LLM. Only after a new idea is introduced in a new paper, the polytope of ideas may acquire a new outer vertex. And only then the LLMs, after being retrained to include those ideas, may fill out the set of results up to the new convex hull, which may or may not include yet a full proof of Riemann.

The convex hull of ideas

If this toy model holds up, then we would indeed expect the very fast advances in mathematics that we are currently seeing. As the LLMs take advantage of the stellated nature of the polytope of ideas, they will continue to fill in gaps between outer spikes. But as the LLMs fill in the convex hull with new results, we will see a deceleration in the number of results being shown solely by artificial intelligence. We will need human advances and intuition to generate new ideas that expand our knowledge polytope.

Even if the mathematical capacity of the LLMs (or future AI models) can at some point reach beyond the convex hull of the current set of human ideas, there is a different way that we may reach a limit to the LLM capacity: feasibility and ethical use of resources (this is similar what fellow optimist Kevin Buzzard called the “natural boundary” in a recent blog post). Is any cost (a dollar amount, human cost, ethical cost) acceptable in the pursuit of solving a given problem? Should we spend millions of dollars and an undisclosed amount of natural resources in order to find a solution for Navier-Stokes? As an analogy: we would like to know if there is life on Mars, but in order to do so as soon as possible, we would need an absurd amount of funding and risk the lives of a human crew in the process. Is it worth it? Similarly, we may reach a point where an LLM could solve an important problem for an exorbitant cost (in terms of funding and resources) but it may just not be an acceptable cost for the taxpayer or society to bear. Instead, we will need humans to devise an alternative route (the equivalent of a gravity-assisted robotic mission to Mars) to solve the problem at an acceptable cost, that produces a similar result in terms of mathematical advances and, more importantly, human understanding.

“Will research mathematicians become `professional prompters’ and interpreters of LLM output?” There is no indication that this will be the case. Yes, LLMs have produced proofs of important results somewhat autonomously (according to the frontier model companies — see Disclaimer 2) that some mathematicians have been tasked with interpreting and digesting. But in my own experience, and other research mathematicians who are using LLMs in their research seem to agree, working with an LLM is akin to discussing a problem with a collaborator, and the results heavily depend on how much guidance and intuition the mathematician inputs into the conversation. In other words, the LLMs are more than tools: they can be research collaborators but, as in any collaboration, the experience and the results are greatly improved when all parties contribute to the discussion. Further, mathematicians have no desire to prompt “solve the Riemann hypothesis, make no mistakes” and then interpret the proof. We prefer to be active participants during all the steps in the process of the discovery of a proof, because we are motivated by the `why the result is true,’ more than by the final answer that `the statement is true.’

Also, if we buy into the previous concept of the convex hull of ideas, then at some point in the near future it will be impossible to make progress in mathematics without a human adding a new idea, a new definition, a new concept that creates a new spike in the polytope, and then progress can occur.

“Will only the `brightest minds’ be able to meaningfully contribute to research mathematics?” At any given time in the history of mathematics, there have been mathematicians who are research active, and whose mental capacity for mathematics seems completely super human (e.g., the owner of this blog, among many others). It is natural to surmise that they could solve any problem we could solve, in a fraction of the time it would take us to complete a proof and write it up. However, this has never stopped those of us with a more modest capacity for mathematics from enormously enjoying doing research, and producing results that are far from insignificant. In fact, mathematics has always benefitted from the range of ideas and points of view, from the very concrete to the big bird’s eyeview, from the smaller contributions to the building of entire new theories.

Similarly, I am not threatened by the mathematical capacity of LLMs. For one thing their capacity is currently limited, as pointed above. And for another, even if their capacity becomes far superior, there will always be a need for mathematicians at all levels to guide research in paths that make sense for humans to walk (not run).

The mathematical universe is enormous (as Emily Riehl said), and computing time is finite. There will always be areas of mathematics that are under-explored and where even beginners can break new ground. The LLMs can help in the process, by quickly exploring avenues that may be dead ends, pointing out paths that have already been explored, and shining a light on paths that are likely to be fruitful.

As I mentioned above, I have never been this busy, because the access to LLMs has multiplied the number of areas that I have access to, and my curiosity has expanded well beyond my research area. I now have many more ideas that I can possibly explore on my own, so I am recruiting more student collaborators than ever before, to help me test whether these problems can lead to interesting results. Students can be involved in research earlier than ever before too because the LLMs can help them learn material faster (and deeper!), by virtue of being available 24/7 to answer their questions, instead of my meager one or two available hours per week to meet with them.

“Will mathematicians be needed? Will they be employable?” I find these questions natural but also perplexing. Even in the most dystopian of scenarios where AI becomes super human in all research tasks, what good would a proof (of a theorem in pure mathematics) be if there are no human mathematicians to digest it and understand it? Regardless of the advances in LLMs and AI, there will be mountains of research to be understood by humans, with or without the help of a computer.

In addition, we seem to forget that mathematics departments exist in universities to serve two primary goals: discovery and communication of mathematical knowledge. Virtually every mathematics department emphasizes, in equal parts, our research and educational missions (and many institutions place the educational mission of mathematics at a much higher level than their research mission). Mathematics courses are an integral part of a liberal arts curriculum because learning to think as a mathematician is a highly useful and applicable skill. The fact that we are researchers adds immense value to our educational goals, because students are best served learning from those scientists who are in the frontlines of research. The research opportunities that we provide for undergrads are a very valuable add-on to their curriculum, as it is a different type of training that helps them be employable in the future. And as long as the mathematical way of thinking continues to be a highly valuable skill to be learned by the undergraduate population, there will be a great need for mathematicians to be hired by universities.

The LLMs are making math research more accessible than ever to those who are not even in academia or even mathematicians. This means that undergrads will be able to join actual mathematician-led research projects much more easily, and it may be a new fertile ground for exploration. Not mindless exploration, though, but mathematical exploration where the goal is understanding and for the students to be initiated and trained into a highly technical field (in an ethical way). And, of course, we should prioritize training students in how to communicate the mathematics they learn, as that has always been (and probably will become even more of) a crucial skill.

All of this to say that I cannot conceive that the LLMs will displace mathematicians from their jobs. On the contrary, they might produce jobs since our research productivity may sky rocket. On the other hand, I am more worried about policies and funding issues that are political in nature and have nothing to do with the AI and LLM conversation.

Finally, the most important question of all, that I wanted to address here:

“Should I pursue a PhD in math at this time?” The answer to this question should be personal to each and every student. But, in my opinion, the answer should not have changed from a year ago to today. The most important reason (and perhaps the only reason) to do a PhD in math should be that the candidate is passionate about mathematics and wants to become an expert in a particular topic within our field. If that is the goal, then the presence of LLMs in mathematics is irrelevant, because the goal is achieved when the candidate has gained sufficient knowledge to be an expert on a particular problem. If anything, LLMs may be used as a tool to achieve that goal more efficiently. For one, I am using them every day to finally understand concepts and techniques that I always had questions about, and now I can query an LLM until I am fully satisfied. I am able to search for the explanations and examples that click with me, that click with my own particular way of thinking about mathematics.

To what degree a student wants to use LLMs in a math PhD should be a personal choice but I will say that, as my colleague Jeremy Teitelbaum put it in a recent interview (here is the bit I am referring to, and here is the full interview), students cannot afford not to learn about the current capabilities of LLMs, or any other technology for that matter. If your goal is to become an expert, then you have to be amenable to learning from all experts in the field, and from all sources that may allow you to go deeper into a subject than anyone else before you — and LLMs can be extremely efficient tools to explore literature, for instance.

But once again, the decision to do a PhD should not be based on the current state of the art of technology.

I wanted to do a PhD in mathematics because it seemed like a magnificent challenge. I wanted to do a PhD because I wanted to learn how Andrew Wiles proved Fermat’s Last Theorem. I wanted to continue studying mathematics because I simply did not want “a real job,” and the opportunity of contemplating advanced math on my own for a few years seemed like a dream to me, just too good not to give it my best shot. I know I would have deeply regretted it if I had not tried to complete a PhD when I had a chance (the best time to do it is when your undergrad knowledge is fresh!). I wanted to hear mathematicians talk about math, and rejoice in the small little details and miracles that make proofs work. I wanted to meet and hang out with other people who also thought number theory was the coolest thing on Earth. I wanted to publish a paper in a research journal, with my name on it, because I discovered a new theorem that no one had thought of before. I wanted to explain and share my passion for mathematics with others in a classroom and outside of the classroom.

Simply put, I just wanted to do math, and I would have been devastated if some undefined threat to the field of mathematics scared me away from the opportunity to pursue a PhD.

And if you are a student that is passionate about mathematics, and someone who wants all of that too, then a PhD is the right path for you, regardless of the technology available during your degree. You will learn to use the technology to a degree that you are comfortable with, and that fulfills your own dreams and expectations of what a PhD in Mathematics means to you.

Afterword: the BIG OpenAI release. After I finished writing this blog post, and had already sent it to Terry, OpenAI released a huge treasure trove of results in mathematics. This is, undoubtedly, a historic time in mathematics. The theorems in their papers prove some huge open problems in mathematics: the resolution of the so-called quasi Riemann Hypothesis, Goldfeld’s conjecture, the Hodge Conjecture in the case of CM abelian varieties, Hilbert’s 10th over Q, the Rigidity Conjecture… and the list goes on and on.

But such a tremendous release does not force me to change any of the points I made above. On the contrary, we already knew their models can do amazing things (e.g., Navier-Stokes). We already knew the frontier models can connect dots in the existing literature in ingenious ways (e.g., unit-distance conjecture). We already knew that OpenAI can spend a mind-boggling amount of resources to attack problems.

Also, we suspected that their models have limits and the new release shows evidence of that too. In their report, they mention that they attacked 4000 open problems, and their model was able to make progress on about 700 related problems. Yes, some of the ones they were able to solve are huge. But it also shows that their models are limited, quite possibly due to the arguments we explained above.

Are any of the solutions using new ideas that are outside of the convex hull of the current ideas in the literature? We will need mathematicians and time to digest these new proofs and understand what connections are being made, and whether brand new ideas were actually discovered in the process.

The main point of my post remains the same, though. There is a lot of mathematical research that remains to be done with and without the aid of LLMs. There are new mountains of mathematics to explain and communicate to others. And if you are a student who is passionate to learn what is new and what is left to do, then a PhD is definitely the right path for you.

[This is a guest post by the Association for Human Mathematics, reposted from their statements page. This blog post was initially written in a different file format and converted using AI. — T.]

Yesterday, on October 6th, 2026, OpenAI — which is currently defending lawsuits against accusations of illegal plagiarism, copyright infringement, and trademark dilution — released a repository of manuscripts purporting to contain solutions to a number of high-profile problems in mathematics.

Mathematicians did not ask for this work to be done. The Advisory Group on Mathematics and Artificial Intelligence, from whom OpenAI has claimed to derive its legitimacy, opened their initial advisory statement by saying that frontier AI corporations should not test advanced mathematical problems on internal models. In ignoring the central premise of the Advisory Group’s position, OpenAI has indicated total disregard for the norms of scientific research — norms that guarantee that mathematics remains trustworthy, ethically researched, and in the public interest.

Mathematicians have a particular vision of progress that is informed by history and field-specific considerations. We reject OpenAI’s assertion that this release advances our subject, and we urge mathematicians and the public to view the value of this publication model with due skepticism.

Releasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power. We urge mathematicians to discontinue their work with OpenAI and to return to a vision of science that centers human understanding.

Association for Human Mathematics Communications Working Group

[This is a guest post by Raghu Meka. This blog post was initially written in a different file format and converted using AI. — T.]

“This problem has been tried by several famous mathematicians.” “There is a heuristic argument for why these methods cannot work.” “Getting this algorithm would give new circuit lower bounds.” Observations like these can take on a life of their own, almost like a game of telephone. A limitation of a particular approach, or an implication whose difficulty we do not fully understand, becomes a reason to believe that a problem is beyond reach, and eventually a reason not to think about it at all.

Over the past few weeks, I have been thinking about the role such perceived barriers have played in theory as I know it, and perhaps more broadly in mathematics. The recent articles on this blog about what it means to do mathematics in the age of AI have been very helpful in understanding and gathering my own thoughts. Looking back at several great results from the past year or so, a few of them (small fraction, admittedly) make me think that perhaps we lost some edge by imagining barriers–“social” hardness or limitations of approaches passed down as folklore–where none existed.

At least some of the solutions coming out of AI models, while brilliant, are also not completely alien. Yet these were problems we had almost stopped trying to solve, apart from small pockets of researchers. Of course, it is easy to say this post hoc. But in some sense, we have seen more `similarly brilliant’ solutions to newer problems than to these older ones. This makes me wonder how much our inherited perceptions have shaped where we were willing to look.

This has also made me think about some things from my early research days as a graduate student.

When I was a graduate student, I gave a talk on a small result. A very perceptive member of the audience (Adam Klivans, if you are reading this!) asked a question that I thought was a great one. But I also thought there were barriers around it, and it did not seem doable. Later, an answer to that question turned out to be an important piece in others’ resolution of a central question. The point of the story is not whether I would have solved the problem; probably not. The point is that the perceived barrier kept me from making even a half-decent attempt at it.

On a personal level, my progress in research was slow. If you are in mathematics, this might not seem that odd, but in theoretical computer science, having only one paper after five years, and that too not in one of the flagship conferences, could generally be taken to mean you had fallen well behind the curve. I nearly left theory. It was my mentors who pointed out that even if the results were not there, the failed attempts showed intent and progress, and that the absence of results was not evidence of a limitation I had begun to imagine. That support got me through then. They taught me not only how to do research, but how to love it.

Several times, I have had an initial impression that there were well-known methodological barriers, or a certain “social hardness” attached to the names of people who had attempted specific problems and directions. The research wisdom of some colleagues, and curiosity itself, helped me get past these impressions and engage with the questions. Pure curiosity is one force that can mitigate these perceived barriers; having people who are very optimistic (research-wise) or encourage that curiosity is another. Perhaps we need to cultivate this more.

Coming back to the present, I think this moment calls for extra care in resisting the trap of perceived barriers, and for revisiting several such accepted hurdles with less deference. We seem to have a mighty tool that can break through some of them. There are many problems whose solutions I thought I would never see, but now, by the universe’s grace, I will likely have the fortune to see them (perhaps some are already gathering dust on servers).

Relatedly, I also see a temptation to put implicit barriers on what “human mathematicians” can contribute. I wonder whether this might become the latest received wisdom that we accept too quickly. My predictive powers are quite limited. But even so, perhaps the epiphany I am having now is that the downside of not believing there is a barrier is far less than that of believing there is one. The cost of disbelieving has, after all, also come down because of the additional firepower we can now call upon. I would like to give curiosity a little more room before deciding what we can contribute, or what we cannot do.

To take poetic liberty, and borrow the words of Blake that gave Huxley his title: “If the doors of perception were cleansed every thing would appear to man as it is, infinite.”

Acknowledgements: I thank several friends who gave useful feedback on the first draft. AI was used to correct grammatical errors and polish sentences.

[This is a guest post by Ben Antieau, cross-posted from his blog. — T.]

There was no personal problem, no world problem, whose eloquent solution did not exist— somewhere in some hexagon.

—Jorge Luis Borges, “The Library of Babel” [1]

New repository

I am writing to announce Hexagon, a new repository for research works in the mathematical sciences and theoretical computer science. This has been a joint effort of many people over the last couple of months, starting in early August 2026. We felt that the proliferation of results proved with LLM assistance required a new model for collecting the associated papers. Besides wanting these works to be citable and reliably stored for long periods of time, we wanted to allow lower barriers for authorship and for community interest than the arXiv. We also wanted for these works to have a home besides random GitHub repositories or X/Bluesky/Mathstodon/LinkedIn posts.

In order to support the development of this project, we founded the Hexagon Mathematics Foundation in the state of Delaware and we intend to apply for recognition of tax-exempt status under section 501(c)(3). The Board of Directors consists of Mohammed Abouzaid, François Charles, Bryna Kra, David Savitt, and Lauren Williams. I serve as the first Executive Director. And, we have a wonderful Advisory Board, where Kevin Buzzard, Akhil Mathew, Johannes Schmitt, Steinn Sigurðsson, Nikhil Srivastava, Ravi Vakil, and Rachel Ward serve. For more details, see our who we are page.

My personal interest

When working with a group of people brought together by an FRG grant, we clicked a button and discovered a new result, namely derived invariance of Hodge numbers of fourfolds in characteristic zero. In the past, this would be a result I would have submitted to an excellent, but not top, journal. In this case, there is no way I would put my name to the result. Instead, I blogged about it here. This is a brittle way to aggregate mathematics. Now, it is on Hexagon as hexagon:2609.00104.

The creation of new results happens naturally in working with LLMs. They bubble up, even as side results. My personal hope is that it becomes the community standard to identify these and submit them to Hexagon.

The Lifecycle of Hexagon Objects

(The title alludes to Ted Chiang’s “The Lifecycle of Software Objects” [2].)

My ambition is that Hexagon is where mathematical ideas emerge from the water for the first time as they evolve toward community acceptance and understanding. Later, they should be written up in forms suitable for arXiv submission, publication, and canonicalization (Tao [3]).

Hexagon specifically allows submissions entirely generated by LLMs and even where the submitters claim no human understanding. These results should be available to the general community.

Submissions by the big labs

Hexagon welcomes submissions by large labs working on language models. These results should be available to the mathematics and TCS communities by default on a platform that the community maintains. We also welcome submissions by large collaborations or other organizations.

FAQ

How was Hexagon created? At some point, a group of people convened in early August, brought together by our social networks with the common theme of interest around creating a home for mathematical results which were rapidly being produced with the assistance of LLM tools. This group included most of the current members of the Board and Advisory Board of the Hexagon Mathematics Foundation, as well as several other academic mathematicians who did significant work but preferred not to be named. In August we discussed ideas on a private Zulip channel (thanks ICARM!) as we started building the site itself.

We talked to a lot of people at that time, including arXiv leadership, and we also examined closely-related AI-era repositories. From the arXiv we learned that they were not going to provide the home for these results we were looking for; instead they are focused on maintaining their project as it is today, and I am happy for that. None of the other repositories had gained sufficient traction or had all of the policies we were hoping for.

The foundation was incorporated as a nonprofit in Delaware and we intend to apply for recognition of tax-exempt status under section 501(c)(3).

The entire system was created with the help of OpenAI’s Codex and is hosted on widely available commercial cloud servers. This is rather easy nowadays. What takes time is deciding exactly what features you want and carefully writing the text on the website.

How is Hexagon funded? So far, it is entirely funded by the people involved, although we are exploring other funding routes and paths to sustainability. Today, except as customers of cloud infrastructure companies, we have no relationship with any for-profit company or other organization.

Why Hexagon? At some point, the group started talking about names involving Babel, which refers to Borges’s story The Library of Babel [1]. But, because it has other connotations as well, we did not go that route. Instead, someone suggested Hexagon in reference to the shape of the halls in Borges’s library.

Does Hexagon accept micro-results? Hexagon welcomes small, incremental improvements as long as they are of interest to the mathematics and TCS communities.

Is formalization required for a Hexagon submission? No, Hexagon is not primarily a formalization project, but it links to formalization projects, like mathlib, Palomar, prove2.me, and TauCeti.

What is the difference between a contributor and a submitter? On Hexagon, Contributors are humans or organizations responsible for the creation of submissions. Submitters are humans with ORCID-validated accounts on Hexagon. In some instances, Hexagon will provide individual account holders the capability to submit on behalf of other humans or organizations.

Are there rate limits? Yes, Hexagon limits the rate at which one can submit. It starts at 1/day, and grows at Hexagon’s discretion. This growth is based on previous submissions passing through moderation. (The arXiv recently announced its own repository-wide submission limits [4].)

Is there API access? Yes, Hexagon provides both read-access via API (in the style of the arXiv) and submit-access via API tokens available from users’ account pages.

Why is Hexagon marked as being in beta? We are still adding features and testing scalability. We welcome any recommendations for how the site can be made more usable.

References

[1] Jorge Luis Borges, “The Library of Babel,” translated by Andrew Hurley, in Collected Fictions, Penguin Books, 1999, publisher page.

[2] Ted Chiang, “The Lifecycle of Software Objects,” in Exhalation, Vintage, 2019, publisher page.

[3] Terence Tao, Mathematics in the age of AI, 2026, arXiv:2608.16753.

[4] Kat Boboris, Fair Moderation, Equitable Access, and AI: arXiv’s Updated Rate Limit Policy, arXiv Blog, 1 October 2026, article.

[This is a guest post by Thomas Bloom, crossposted from the Erdős problems forum. This blog post was initially written in a different file format and converted using AI. — T.]

AI is changing everything, for better and/or worse, and the rate of change is dizzying; in recent months this has been particularly evident in mathematics, where AI has gone from being essentially useless to helping solve some of the hardest problems in mathematics in less than a year.

The website http://www.erdosproblems.com has often been on the front line of these changes, and a barometer by which one could measure AI capabilities. This was not at all my intention when I created the site — I wanted to promote these problems to a human audience, and make a useful reference on what work has been done. But the easy availability of a large pool of questions that are simple to state, ranging from easy and obscure to deep and impenetrable, provided the ideal showcase for AI.

In some ways this has led to a huge amount of progress — we now know the answer to many questions we did not before. (Although a lot of this recent progress has come from increased human efforts and renewed interest in some problems, rather than just AI solutions.) There have also been negative effects, however: some mathematicians have dismissed Erdős-style mathematics as ‘easy/recreational’; many have stopped thinking about Erdős problems believing that they cannot compete with AI; and, most significantly, there has been a wave of AI-produced solutions provided with no explanation. While of course these are useful and tell us new things, they are also displacing and discouraging those who are actually interested in the mathematics.

(These issues have stopped being Erdős-specific as AI continues to make breakthroughs in an ever wider range of fields.)

A couple of weeks ago I asked for feedback on how people used the site, and what they thought should be changed. I received, both publicly and privately, a wide variety of opinions, and I’m grateful to everyone for their feedback.

What was particularly striking was the number of people I heard from who have benefited a lot from the site, learning new mathematics and finding new problems to think about, but have never commented on the site. I have been particularly conscious of this large silent, audience, and have tried to make sure their experiences were not drowned out by the more vocal minority.

In this post I will describe the changes I am making, based on the feedback I received and my own reflections on what the site is and could be.

In the essay ‘Why Do We Need Human Mathematicians Anymore?’ Po-Shen Loh suggests the following as an axiom to use when deciding how things should develop: We (humans) should help humanity flourish.

When thinking about what I should do about the site, I use the following variant:

http://www.erdosproblems.com should help the Erdős-community of humans (defined as those who are interested in Erdős-style mathematics, and want to think about and understand it) flourish.

Why change?

I launched http://www.erdosproblems.com on 28th May 2023, with just over 200 problems; it has grown steadily since then. The biggest change came in August 2025 when I added a comment feature. The site now hosts 1221 problems, over 9000 comments, and almost 2000 registered users. The site typically receives between 10,000 and 25,000 unique visitors each day.

The first few months after introducing the comment feature saw a huge increase in activity, just as I had hoped. A vibrant community arose; people shared ideas on how to solve problems, made observations, gave corrections, and suggested missing references and solutions. Many papers were written and new collaborations formed.

Unfortunately, this level of activity has not been sustained. Some decline was inevitable: over time, many of the natural observations will have been made, mistakes corrected, and discussions between collaborators would move offline. This natural evolution has been dramatically sped up by the coincidental rise of AI and its ability to solve some problems with minimal human intervention.

The main way that people publicly interact with the site now is to advertise their AI-generated proofs, often without any attempt to explain them, but as a way to record a (increasingly meaningless) priority claim.

This is very different to what I imagined, and I don’t want to manage a website which does this.

I believe that websites with this function should exist — places where people can record AI-generated proofs, even if purely formal with no human understanding, to save others wasting their tokens generating the same proof, and so that other people can access and use them if they desire. There are now several candidates for such repositories, and if managed responsibly, they can serve a useful role in the mathematical ecosystem. I personally don’t want to manage one.

Just as one does not open a restaurant in an abattoir, it is important that there be a separation between such repositories and a site which aims to promote the actual questions, place them in an appropriate context, and give a useful overview of the current state of human understanding.

The situation as it stands muddies the waters, promotes a gamified glory-seeking attitude, and gives the wrong impression that the value of these questions ends as soon as someone posts a formal proof of their truth value. This is not true (regardless of whether this proof was human or AI generated).

I believe Erdős intended these questions to serve as enduring beacons for human curiosity and wonder. They are landmarks by which we measure how far we’ve come, and how much there is still to understand.

I am very proud to have helped foster an open online community to discuss Erdős problems, and the kinds of discussions that used to happen are very valuable. But these discussions have become much less frequent. If and when I can find a way to encourage such discussions again, without the site becoming an AI repository of the kind mentioned above, I will happily do so.

What are the changes?

I will make the following changes. (As ever, these are somewhat experimental, and may themselves change and be clarified further in the next few weeks.)

1. A hiatus on problem comments and proof claims: I will freeze new problem comments and proof claims. General threads and blog posts will remain open to comments. Problem comments and proof claims may be reinstated in the future. Until then, suggestions for updates can be emailed to erdosproblemsonline@gmail.com, and I will update the site manually as I see appropriate.
2. No problem statuses: The site will not display the statuses of any problems (e.g. open, solved, etc.). All problems will be displayed in the same neutral colour. The count of currently solved problems and the solved percentage will no longer be shown.
3. No credit/ownership language to describe future solutions: The site will continue to record relevant results and theorems, but will no longer use credit-giving language for a result (human or AI).
4. An emphasis on high-quality expositions: I will focus more on the proof expositions feature. People are encouraged to write in with their own expositions of proofs (whether these proofs are old or new, human or AI), and I will post those I judge to be high-quality. I am exploring other ways to encourage human exposition (e.g. an online seminar). Please contact me if you have any ideas.

I’ve tried to anticipate some of the questions and critiques of these changes below. (I may update this with other questions and answers in future depending on feedback.)

What about those outside of traditional academia, who are using AI tools to make advances in mathematics, but can’t post to arXiv/lack the knowledge or context to write up proper papers about their findings?

I will post links to correct formalisations and high-quality writeups, whatever the source, whether from a traditional academic or not. Papers should acknowledge their sources and honestly disclose how AI was use; where there is evidence of plagiarism or misrepresentation about AI use, I may decline to feature the submission, even if the proof is correct.

This is removing one, unofficial, avenue of publication; there are, these days, many others (even if you are also unable to post to arXiv). AI-assisted search tools mean that people will find your work if they are interested in the area as long as you post it to one of these sites.

I encourage everyone interested in Erdős problems to think about them, and to work on them if interested. If you have a proof (AI-created or otherwise) then you should take time to carefully write up the proof yourself (rather than asking an AI to generate a PDF for you). If you are unable to understand the proof you could reach out to someone else to help write this up.

But emailing you my solution is slower, and it might be some time before you update the site.

This is true; but there is no great rush here. As much as I like these problems, and think it is important that there are some people who do think about the distribution of prime numbers and the structure of graphs, these are not problems for which a formal proof will have an immediate impact on general society.

I will try to link to correct Lean formalisations (if verified on Palomar) quickly; proofs which are poorly explained, and not accompanied by a formalisation, I am unlikely to update the site with. (See below for more details.)

What’s the point? There are loads of other places I can post my proof.

Indeed; this is partially why I feel comfortable halting such proof claims on the site, since modern search tools (including AI) make it easy to find proofs posted online about problems you are curious about, even if posted in obscure places.

I would like to use the small amount of influence I have to avoid promoting low quality proofs, and not to give an added veneer of legitimacy where it is not deserved.

I could just make my own site for people to share their proofs and discuss solving problems with AI. Heck, I can even use AI to scrape all the text off your site and make an exact clone with much more liberal policies.

Yes, you could. This a period of great experimentation in different formats and ways to encourage mathematics; if you think you have a good idea for a site or resource, you should make it. (Although it is often better to help out an existing effort where possible, since these resources are only useful if people use them, and it is better to concentrate on creating a few high-quality sites than hundreds of very similar clones.)

I think it is, however, rude to use without permission the text from my site, which is the result of a huge amount of work from me and many others who have contributed to the site.

How long will the hiatus last for?

I don’t know. I will monitor how things develop in the coming weeks and months, and will reintroduce comments and proof claims (perhaps in a different form) when I believe they will do more good than harm.

Why not just moderate the comments to only allow genuine discussion through?

This has been tried; in practice the vast majority of the comments the site receives now are people announcing AI-generated proofs. It is not sustainable to have a moderation policy which would reject almost all the comments that are submitted.

I am open to alternative suggestions about how to create a space for human discussion and collaboration, either on the site or elsewhere.

What about existing comments and proof claims?

They will remain as an archive of the site up to this point.

Will you update the remarks to reflect existing proof claims?

Yes, I will be working through the backlog of existing proof claims and updating the remarks, linking to formalisations and so on, as appropriate. You do not need to email me about a proof claim already on the site.

How will you decide when to update the remarks? What can I do with my proof to help?

I will update the remarks when I judge there is an interesting new result that people who are working on that problem should be aware of, that can be presented in a useful way. Some things to be aware of:

1. If you have a Lean formalisation register it on Palomar — this lets others see that the formalisation compiles correctly, and makes it easy to check the formal statement correctly matches the problem statement. I will then link to the Palomar registry.
2. A proof accompanied by a well-written exposition that demonstrates clear understanding and which makes it easy for others to understand the proof will be prioritised.
3. If I judge there to be some kind of academic fraud (e.g. using the ideas of others without attribution, or passing AI-generated work off as your own) then I will not post it. This may mean that there are correct proofs not acknowledged, but the alternative is to publicise and reward bad behaviour. (As mentioned above, these proofs will surely still be found by anyone who searches for them elsewhere, but at least I would not be implicitly endorsing them.)

What’s the point in removing the problem status? Isn’t it just cosmetic, and doesn’t it just make the site harder to navigate?

Yes, this does undeniably remove some information. I think, however, that what is lost is not significant for anyone seriously interested in that problem, who can see for themselves within seconds of reading what the current situation is.

It makes it harder to browse the site searching for an unsolved problem to work on; instead I recommend that people browse the site by topic, finding questions that interest them, and investigating those, whether the original question is open or solved, since there is always more to be done.

The main point is to disincentivise people who are simply glory-chasing and copying problems into their AI to get an OPEN->SOLVED dopamine hit. It also prevents people drawing wrong conclusions from the ‘rate of solved problems’.

Furthermore, there is a lot of subjectivity anyway in many problems as to what counts as solved. Many people only browse the open problems, and they’re missing out on a lot of great mathematics that way.

I think it is against the spirit of Erdős to regard any problem as ‘closed’ — whenever one form of a question is answered, many others are created, and all problems deserve continued attention.

Doesn’t the ‘no credit’ rule also devalue the work of humans?

Unfortunately, yes. I anticipate this will only be temporary, while the norms and customs of this new age of mathematics are decided. The commentary on the site is in no way ‘official’, and just represents my own summary of the situation, and collects links which may be useful to others exploring a problem.

Information about authorship will still be available in the sources the site links to. I trust that people will be sensible enough to judge for themselves who deserves the appropriate credit, and reward them appropriately.

I will also be freer in my language when it comes to well-written papers, whether they are the first place a proof appears or explain a proof that originated elsewhere; but I will no longer use language that suggests anyone ‘owns’ a particular result or proof. (For example, instead of saying ‘Bloom proved that ‘, I will write ‘It is known that — an explanation of the proof is provided by Bloom [link]’.)

Erdős and AI

I am often asked ‘what would Erdős have thought of AI?’ The short answer is ‘I don’t know’. I never met Erdős, and have no special insight into him; I know him only through biographies and his papers, and anecdotes and reminiscences from those who did know him.

I would like to honour his legacy, and create something that he would have liked. To that end, I want to stress one thing: Erdős was, as well as a mathematician, an incredibly social person. Many stories about Erdős emphasises his humour, his warmth, and his enthusiasm for talking to other people (mathematicians or not). He spent his life travelling from mathematician to mathematician, arriving on their doorstep and declaring ‘my brain is open’.

He was not the type to lock himself away for years working in isolation on a single problem. For Erdős, mathematics was a very human activity, best done out loud. Questions would arise, be solved or discarded and replaced by new questions. Some results obtained, new ideas found, and then onto the next problem.

Some view the future of mathematics as a dystopian arcade of button-pressing, staring at a screen waiting for AI to do the thinking for us, with the main human involvement limited to asking the initial question and offering sporadic words of encouragement. Each new proof is then thrown into a repository, to only ever be read by other AIs, and the human presses the next button.

I believe Erdős would have found this future grim indeed, and the antithesis of mathematics as he practised it. Use AI as you like — but do not abandon mathematics as a human activity.

Don’t solve a problem for the sake of it; life is too short to spend it doing things that don’t matter to you.

Find a question that is meaningful to you, find other humans who are interested in it, and talk about it. Be confused, be stuck, be inspired. Find a messy proof, then find a better one. Get tired, get hungry, get into a flow state. Argue, laugh, give up, drink some coffee, and attack it again.

Be human. Let your brain be open.

[This is a guest post by Annalisa Buffa. This blog post was initially written in a different file format and converted using AI. — T.]

About us and the position

The Chair of Numerical Modelling and Simulation at EPFL is dedicated to the design and analysis of numerical algorithms for partial differential equations. Our research is oriented towards the development of novel and innovative numerical techniques aiming at improving the integration between numerical simulations and geometric modelling and processing.

Proof assistants such as Lean, together with recent AI tools, are changing how mathematics is done. We are opening a postdoctoral position to explore what formal verification and algorithm discovery can bring to numerical analysis. There is no fixed project. The goal is to find out, through concrete experiments, where these approaches help in the design and analysis of numerical schemes for PDEs, and where they still fall short.

What you will do

As a postdoc in our group, you will set up and lead this new research line, choosing its directions together with us. Possible starting points are the formalization in Lean of stability and error estimates for finite element methods, or the AI-assisted discovery of new discretization paradigms for classes of PDEs where standard methods struggle, with their properties then verified formally. You will publish and present your work internationally.

You are the ideal candidate if

• You hold a PhD in mathematics, computer science or a related field
• You have a strong background in formal verification with Lean
• You have experience in AI-assisted theorem proving or algorithm discovery is an asset
• You have working knowledge of numerical analysis, ideally numerical methods for PDEs
• You are independent and comfortable with open-ended research questions
• You communicate well in English and enjoy working at the boundary between disciplines

We offer

• A stimulating international environment at one of Europe’s leading research institutions
• Competitive salary and excellent working conditions
• A 1-year contract, renewable
• Freedom to shape a new research direction from the start
• A friendly team with strong postdoctoral and PhD researcher community

If you are interested

Submit your application containing CV, motivation letter, and contact details for 2 referees through this link. In the motivation letter, briefly describe one question you would like to work on, linking numerical analysis with formal verification or algorithm discovery. Applications will be reviewed until the position is filled. Start date: as soon as possible (flexible).

[This is a guest post by Jeremy Avigad. This blog post was initially written in a different file format and converted using AI. — T.]

“Mathematics underwent, in the nineteenth century, a transformation so profound that it is not too much to call it a second birth of the subject—its first birth having occurred among the ancient Greeks…”

Howard Stein, in “Logos, Logic, and Logistiké: Some Philosophical Remarks on Nineteenth-Century Transformation of Mathematics”

“The report of my death was an exaggeration.”

Mark Twain

“May you live in interesting times.”

(traditional)

I recently attended a meeting of innovative science and technology startups supported by Convergent Research, the organization that oversees the Lean FRO, a nonprofit that develops the Lean theorem prover. The meeting was designed to stimulate discussion, and when I introduced myself as a mathematician, many participants were eager to talk about the impact of recent events in AI on mathematics and reactions in the mathematics community. They were surprised to hear that I find the tone of the community responses on blogs like this one and Proofs and Prompts generally positive and encouraging, even though we all recognize that fundamental aspects of our day-to-day professional lives are bound to change. These discussions have helped me shape some of the thoughts I would like to share here.

There is a narrow view of what mathematicians do, encapsulated in our daily workflows: we try to solve problems, and when the hard problems are too hard to solve, we make up easier approximations, solve them, and then vary the parameters. That practice has been disrupted by the events of the last few months, in the sense that the kinds of results that would have, a year ago, made for perfectly respectable publications can now easily be generated with the help of AI. This has left us worrying about what it will mean to do mathematics going forward, as well as how to train and support the next generation of mathematicians to do whatever that is.

The history of mathematics offers us a broader view. What has remained stable, despite centuries of changes, is that mathematics is a culture of rigorous reasoning and communication, providing us with language and abstractions that let us think and communicate more reliably and efficiently. Surely such reasoning is still important, even in the age of AI. The fact that many of us find mathematics aesthetically pleasing doesn’t diminish its practical utility, but rather is explained by it: I expect that the reason that doing mathematics feels so good is that it is the exercise of capacities that are so fundamental to our survival as a species that they are wired into our DNA. If that’s right, mathematical thought isn’t going away any time soon.

The challenge is that solving the kinds of problems we have been solving for decades becomes decoupled from the goal of enhancing our mathematical understanding when we let AI do the work. The question, therefore, isn’t whether we still need mathematics, but rather how to pursue mathematical understanding in the age of AI. I will provide three general answers.

Solve harder problems

There are two salient features of today’s foundation models: first, they have seen the entire mathematical literature, and second, they are tireless; a swarm of agents can make its way to a solution by trying countless variations. This explains why the AI-generated solutions to open problems we have seen all have a similar character: they are problems that AI could solve by cobbling together available techniques. (I am grateful to Matthew Ballard for this characterization, and the subsequent analysis.)

Can every interesting mathematical question be answered that way? Probably not, and even questions that can may have more interesting and satisfying solutions that invoke novel ideas and insights. Perhaps, in the near future, AI systems will be able to come up with such insights, but, at the very least, let’s recognize that we are not there yet. Reinforcement-learning training regimes have systems chain conventional moves and learn, from a history of failures and successes, which ones are most promising in a given state. The fact that the value of an action is graded solely in terms of the success of a final trajectory breeds superhuman cleverness but may miss creativity and higher-level strategizing. In any case, there are still hard questions to be solved and ambitious research programs to pursue, and the possibility of making progress on them with the help of AI is exciting.

Think bigger thoughts

Our present situation would be much more depressing if mathematics were a matter of ticking off problems imposed on us by aliens, an endless sequence of exercises and exams. The good news is that when we do mathematics, we get to choose the problems, grade the solutions, and favor the ones we like best. We decide what’s interesting to us, what questions to pursue, and why. Our destiny is in our hands.

The things we admire most in mathematics often seem to come out of nowhere. In 1853, the young Bernhard Riemann submitted three potential topics to his advisor, Carl Friedrich Gauss, to choose from for his Habilitationsvortrag, a lecture he was required to give to secure a teaching position at the University of Göttingen. Gauss reportedly chose the topic for which Riemann was least prepared, to see what he would make of it. The resulting lecture, “On the Hypotheses Which Lie at the Foundations of Geometry,” was published posthumously in 1868, and it revolutionized the field. The lecture distinguished a space’s metric properties from its topological properties and introduced the general notion of a manifold, though the latter did not receive a fully rigorous treatment until the twentieth century. The focus on intrinsic properties of a space—in Riemannian geometry, those determined by the metric and independent of embedding in a larger space—was key to Einstein’s theory of general relativity decades later. Riemannian geometry has had applications that Riemann himself could never have imagined, from robotics and medical imaging to statistical analysis.

William Ewald’s excellent sourcebook, From Kant to Hilbert, provides a lovely introduction to the paper and quotes Felix Klein’s description of the work:

“The publication of [Riemann’s lecture] occurred just at the time when I was beginning to occupy myself independently with mathematical problems. So I still have vivid memories of the extraordinary impact Riemann’s train of thought made on the young mathematicians of the day. Much seemed to us dark and difficult to understand, and yet of unfathomable depth, where the modern mathematician, who has already absorbed all these things into his mode of thought from the outset, only admires the clarity and fecundity of the exposition.”

What seemed dark and mysterious to the young Klein is now part of the canon, something that foundation models have absorbed and internalized through their training, just as young mathematicians do. I am grateful to Ballard once again for suggesting this example and pointing out that no reinforcement-learning setup could have evaluated Riemann’s decisions: the benefits are diffuse and hard to track, and the time horizon is much too long.

The same can be said for countless mathematical developments that have opened up new vistas, such as Galois’ focus on groups of permutations in the study of solvability of algebraic equations, Poincaré’s qualitative studies of dynamical systems, or Grothendieck’s far-reaching conceptual innovations. Will AI eventually be able to make advances like these? That’s not even the right question to ask. Telling us that some AI agent is spinning out theorems that are highly interesting to it and other AI agents does nothing for us. We should care about AI only insofar as the results are interesting and important to us, and, at the end of the day, it’s up to us to decide what that means. The values we assign to mathematical developments are embedded in our history and culture.

Imagine planning a trip to go backpacking in the Alaskan wilderness for exercise and recreation. You might be happy to let AI help book your flight, but not to let AI take the hike for you and send you pictures. In mathematics, what is at stake isn’t our recreation but agency over our reasoning and deliberation. Whether or not AI can think, it can’t think for us. We have our own lives to live; mathematics is our story to tell, and it’s up to us to decide how to tell it. With AI, there’s even more to explore, and no shortage of avenues for discovery.

Try new things

So far, I have focused on using AI to help us do the things we used to do, but let’s not forget that the technologies themselves raise new questions, and that we have a lot to learn about how to use them effectively. I have argued in another essay (now scheduled to appear in the Notices of the AMS) that mathematicians should be actively involved in understanding how the technologies work and in coming up with novel and creative ways to use them to do new mathematics.

AI changing what it means to do mathematics is not without precedent. The use of algebraic methods to solve geometric problems in the seventeenth century was a new technology, and not everyone liked it; yet we mastered the new techniques and learned how to use them to great effect. The same is true of infinitesimals later in that century, algebraic structures in the nineteenth century, set-theoretic abstraction and structural language in the early twentieth century, and numerical and symbolic computation more recently. These were all alien and disconcerting when they were new. We should view those who invest time and energy in getting proof assistants and neural networks to help us discover new mathematics as doing mathematics proper, rather than dismissing them as mere technicians. In the age of AI, developing symbolic automation or training a neural network can be no less a contribution to mathematics than manually chaining inferences to prove a theorem.

I have heard arguments that as the job market contracts, we should turn inward to preserve traditional mathematical skills. On the contrary, I believe that engaging with new technologies and learning how to use them to improve our ability to reason and discover new mathematics will keep the discipline strong. Expanding our view of mathematics is the best way to expand the profession and keep it relevant.

Our message to the next generation

My rosy outlook on the future and glib advice to solve harder problems, think bigger thoughts, and try new things will not provide much comfort to students and early-career researchers, who feel the ground shifting beneath their feet. My words are not meant to diminish the challenges ahead or suggest easy responses to the disruption. Mathematics departments, community leaders, educators, professional societies, and journal boards are holding emergency meetings all over, and are doing their best to make concrete recommendations and take appropriate action. We have our work cut out for us.

Despite the uncertainty, there are some clear messages we can send to the next generation of mathematicians. The first is that we stand with you. There is nothing more important to us than the health and strength of the discipline, and ensuring that you can thrive. We have the humility to recognize that our experience and expertise are limited, and that some of the things we thought we knew in the past are no longer valid. We are committed to working with you as best we can to preserve the profession and keep it strong.

Second, mathematics is as important today as it ever was, and we need you. AI must not replace our collective ability to reason and deliberate, and mathematics remains a core capacity for doing so. We cannot imagine a world in which mathematics does not play an important part in our lives.

Finally, the next few years will be extremely interesting. We are at a new frontier, where our communal norms, values, and expectations are beginning to break down, and it’s up to all of us to figure out what should replace them. I am confident that future historians will see this moment as the start of a new era of mathematics, and that they will weigh the consequences of our actions and decisions. Mathematics has never been for the faint of heart; this is our opportunity to rise to the occasion and face the challenges together.

[This is a guest post by Diego Córdoba and Luis Martínez-Zoroa. This blog post was initially written in a different file format and converted using AI. — T.]

Euler formulated the equations for an inviscid incompressible fluid more than 250 years ago. Published in his 1757 memoir, they belong to the earliest systems of partial differential equations in mathematical physics, following d’Alembert’s work on the wave equation. Understanding the evolution they describe has required the work of many generations of mathematicians. One of the central questions is whether an initially regular fluid flow can develop a singularity in finite time. The recent work on Euler and Navier–Stokes obtained with the aid of large language models has brought renewed attention to this question. These exciting developments build on decades of mathematical research, including works from recent years, when the field has continued to be specially active. In this post we describe some contributions to that body of knowledge, including our own. We will go over some of that recent mathematical literature, and discuss the ideas of constructing blow-ups through a cascade of vortex layers.

Partial differential equations provide a common language for problems ranging from wave propagation and fluid motion to geometry and general relativity. For an evolution equation, the Cauchy problem asks us to determine the future from prescribed initial data. A local existence theorem gives a solution for a short time; the next question is whether it continues for all time or develops a singularity at a finite time. This distinction between global existence and finite-time blow-up is already visible in the elementary Riccati equation

The solution starts from a finite value and remains smooth until , when it becomes unbounded. The quadratic nonlinearity in three-dimensional Euler suggests a similar possibility, but the fluid problem also involves transport, geometry and a pressure determined nonlocally by the entire flow. The challenge is to understand whether the full equation can sustain the amplification suggested by this simple ODE.

The central aim of our research has been to construct classical solutions of incompressible fluid equations that develop singularities in finite time, starting within a regime of local existence and uniqueness. We want the initial data to determine a unique regular evolution for a positive interval of time, followed by a singularity at a positive, finite time. In the smooth-data setting, this means a solution that is initially smooth and stays smooth until the singular time. More generally, the local theory also applies to classical solutions with finitely many derivatives. This programme includes the incompressible porous media (IPM) equation, the generalized surface quasi-geostrophic (gSQG) equations, the Boussinesq system and the two-dimensional non-homogeneous Euler equations, with particular emphasis on the three-dimensional incompressible Euler equations.

Our approach developed through increasingly demanding constructions. We first sought mechanisms for rapid growth of regularity norms, then used them to produce instantaneous loss of regularity in critical and supercritical spaces. We subsequently arranged for singularity formation after an interval during which the local theory guarantees regular evolution. The distinction between these phenomena is an important part of the story.

Classical solutions and local existence

The incompressible Euler equations are

where is velocity, is pressure, and is an external force, which may be zero. The condition expresses incompressibility. A classical solution has enough differentiability for the equation to hold pointwise. This does not require infinitely many derivatives, nor does local differentiability imply that every global regularity norm is finite.

For unforced Euler in dimension , the classical local theory gives existence, uniqueness and persistence of regularity for initial velocity in , with and , together with suitable control at infinity, such as finite energy. Here means that derivatives up to order are bounded and continuous, and the derivatives of order are Hölder continuous with exponent . In Sobolev spaces, which measure square-integrable derivatives, local well-posedness, including continuous dependence in the same norm, holds in

The properties relevant here are existence, uniqueness and persistence of regularity. With a time-dependent force, a sufficient assumption in the finite-energy Hölder setting is ; in the Sobolev setting it is . The notation in time means that the indicated spatial norm of the force is integrable on every finite time interval under consideration.

Higher regularity is propagated as long as the solution remains controlled in the continuation class and the force has the corresponding regularity. For unforced Euler in two dimensions, these regular solutions exist globally. In three dimensions, the possibility of finite-time breakdown is tied to vortex stretching.

Writing , the three-dimensional equation becomes

Vorticity measures local rotation. The velocity transports it and, through the stretching term , can also amplify it. This deformation is generated by the vorticity itself through the nonlocal Biot–Savart law. This is the feedback that makes the Riccati analogy suggestive. A singularity construction must organize it while retaining control of transport and of the interactions between different regions.

For unforced three-dimensional Euler, the Beale–Kato–Majda continuation criterion states that breakdown of a sufficiently regular solution at a finite time requires

The same criterion has a forced version, provided the force remains regular in the appropriate time-integrated sense. For a velocity in with , one convenient sufficient assumption is

The second condition controls low frequencies. After absorbing the gradient part of the force into the pressure, these assumptions control the effective force in , and the usual BKM estimates apply. Thus, the threshold is for the curl of the force, corresponding to one additional derivative on velocity. A similar criterion applies in the setting of Holder spaces. The point is to rule out a breakdown caused simply by an inadmissible force. Our goal is to realize the required vorticity growth dynamically from a local well-posedness regime.

Instantaneous loss of regularity

A first warning is that local regularity propagation can fail at the borderline of the classical theory. The assumptions above cannot simply be replaced by their endpoints. Bourgain and Li established strong ill-posedness at the critical Sobolev regularity : for velocity in two dimensions and in three. Their constructions exhibit instantaneous loss of the critical norm. They also proved instant loss of regularity in the integer spaces , . Elgindi and Masmoudi independently developed another approach to ill-posedness in these integer regularity classes.

These results show that the borderline spaces need not support the regularity propagation available immediately above them. Having one continuous derivative does not provide the same control as a first derivative that is Hölder continuous with a positive exponent. For higher integer , instantaneous loss of the norm can also coexist with classical differentiability at lower orders.

An especially striking phenomenon is an instantaneous gap loss of Sobolev regularity. In our work with Wojciech S. Ożański, we construct global classical solutions of the two-dimensional Euler equations with finite energy for which, given any ,

for every and every .

The number on the right is strictly smaller than . Thus, the solution does not merely leave its initial Sobolev space: it immediately loses an entire positive interval of regularity. In velocity variables, the initial regularity is , below the critical space . This is a result in the supercritical Sobolev regime.

There is no contradiction between this loss of a global norm and the persistence of local classical differentiability. Definition 3 in that paper makes the distinction precise. On each finite time interval, the vorticity belongs to for some and is in space and time on every compact cylinder. It satisfies

pointwise, with the prescribed initial data. The constructed solution is unique within this specified class. Its local differentiability persists even though its global Sobolev norms become infinite.

The proof first constructs smooth solutions with arbitrarily large Sobolev norm growth. A nearly radial background rotates an oscillatory perturbation at different rates at different radii, creating finer scales. Carefully rescaled copies of these building blocks are then placed increasingly far apart. A gluing argument controls their nonlocal interactions and produces one exact solution with instantaneous gap loss. Spatial separation permits local classical regularity to coexist with the failure of global Sobolev control.

We have also investigated instantaneous loss of regularity for SQG and its generalizations, establishing strong ill-posedness in integer and Sobolev spaces, Hölder spaces, and, jointly with José Antonio Lucas-Manchón, supercritical Sobolev spaces for gSQG. These investigations include the construction of global unique solutions exhibiting instantaneous loss of regularity in the presence of supercritical fractional diffusion, and, with Wojciech S. Ożański, classical SQG solutions whose Sobolev regularity decreases continuously from the initial time. For IPM, our joint work with Roberta Bianchini establishes strong ill-posedness in for arbitrarily small perturbations of a linearly stable stratification, showing how the nonlinear dynamics can overcome the stabilizing effect of the background profile.

These examples make clear that instantaneous loss of regularity concerns failure of propagation in a chosen function space and can coexist with classical differentiability. The next challenge is to construct a singularity after a positive lifespan during which the relevant regularity is guaranteed to persist.

Singularity formation after regular evolution

Elgindi’s construction was a breakthrough in precisely this direction. It produced finite-time singularities for three-dimensional Euler in , for sufficiently small positive , using axisymmetric flows without swirl. The original exact self-similar profiles had infinite energy. Elgindi, Ghoul and Masmoudi subsequently used stability and localization to construct finite-energy singular solutions without forcing.

The initial velocity in these constructions belongs to a class in which the local theory guarantees a unique regular evolution. It is but not smooth everywhere. Axisymmetry means invariance under rotations around an axis, and “without swirl” means that the velocity has no component in the angular direction. Smooth flows in this class admit a global regularity theory under the usual decay assumptions, so the distinction between finite Hölder regularity and full smoothness is essential.

Recent work has clarified how close one can come to the boundary of this global theory. For the homogeneous, unforced axisymmetric Euler equations without swirl, global regularity is known for initial velocity with , under the corresponding finite-energy and decay hypotheses. Three recent approaches construct singularities below this threshold, for . Shkoller’s preprint uses a Lagrangian clock-and-driver framework, coupling the collapse of a flow Jacobian to the compressive axial strain. Chen’s work constructs asymptotically self-similar blow-up from finite-energy initial velocity, using computer-assisted estimates for a one-dimensional profile and an analytical construction and stability argument for the three-dimensional flow. Shao, Wei, Zhang and Zhang construct exact self-similar profiles with infinite energy and prove stability after controlling finitely many unstable directions. Their current manuscript also uses this stability to localize the profiles and obtain finite-energy, asymptotically self-similar singularities. These different methods reveal a sharp regularity threshold within a class that still has classical local existence and uniqueness.

Elgindi and Pasqualotto constructed singularities for Boussinesq and axisymmetric Euler with swirl, exploiting instabilities related to Rayleigh–Bénard convection and Taylor–Couette flow. Their argument uses a self-similar framework and includes a computer-assisted step in the proof of invertibility of a linear operator.

There has also been major progress for smooth initial data in domains with a boundary. Following the numerical scenario of Luo and Hou, Chen and Hou combined analysis with rigorous numerics to establish nearly self-similar finite-time blow-up in that setting.

Our work with Fan Zheng introduced a different route to singularity formation in the whole space: non-self-similar blow-up for unforced Euler, with finite-energy velocity in for small positive , smooth away from the origin. The solution is assembled from interacting vorticity regions at successively smaller scales, organizing the singularity through a hierarchy of amplification events.

A cascade of vortex layers

The basic idea is to arrange vorticity on a sequence of widely separated spatial scales. An outer layer generates a deformation that amplifies a more concentrated inner layer. Once amplified, that inner layer provides a stronger deformation for the next one. The time required for each stage decreases, and the sum of these time intervals is finite. In this way, infinitely many amplification stages can accumulate at a single finite time, while the solution remains regular on every shorter time interval.

Scale separation makes this interaction tractable. Across the small support of an inner layer, the velocity generated by the outer layers can be approximated by a simpler field. In the Euler constructions, this field has a hyperbolic structure, stretching one direction and compressing another. Aligning the vorticity with the stretching direction produces amplification.

The hard part is to realize this process in the full equation. The inner layers also affect the outer ones; each layer interacts with itself; and transport changes the geometry and frequency of the oscillations. We choose the layers so that crucial self-interactions cancel at leading order, while scale separation and oscillatory cancellation control the remaining interactions.

In the unforced construction with Zheng, we analyze finite collections of vorticity regions and pass to a limit of exact Euler evolutions. The full infinite hierarchy is already present in the initial data; the cascade describes the successive amplification of its layers. The construction is compatible with regularity and finite energy, though not with smoothness at the point where the scales accumulate.

The forced version gives further flexibility. We construct an approximate evolution and define the external force through its residual. The main difficulty is to arrange that this force remains in the required regularity class up to the singular time, even as the solution itself loses regularity. This requires precise control of the cancellations and of the errors at every stage of the cascade.

In our forced Euler construction, the velocity lies in before blow-up, and the force remains uniformly controlled in . The flows are non-axisymmetric. Thus the method reaches considerably higher velocity regularity while retaining a force in a classical local well-posedness class.

This is a useful way to view the programme: construct an amplification mechanism, then improve the accuracy with which it can be implemented in the equation. The quality of the error estimates determines the regularity of the force.

Adding dissipation

A particularly important test of the method is whether it survives the addition of dissipation. With Fan Zheng, we addressed this for the fractional Navier–Stokes equations

where . Ordinary Navier–Stokes corresponds to ; the range is hypodissipative. Classical results of J.-L. Lions 1969 establish global regularity when , starting from smooth, divergence-free initial data with suitable decay. Tao subsequently extended this theory slightly into the supercritical regime, proving that global regularity persists when the critical dissipation is weakened by a suitable logarithmic factor.

Our theorem constructs finite-time singularities for

with force

for some . The velocity is smooth in space before the singular time, and both its speed and its energy remain bounded. Nevertheless, after normalizing the singular time to 1,

The force has the time integrability and spatial regularity required by the local theory, so the singularity is not caused by a loss of admissibility of the forcing.

Dissipation cannot simply be added to the old construction and ignored. For an oscillation at frequency , the fractional dissipation term has size comparable to times its amplitude, so its damping time scale is approximately . The strain generated by the preceding layer has to overcome this damping. Increasing the frequency helps some approximation estimates, but simultaneously strengthens dissipation. The parameters must satisfy both requirements.

Section 1.2 of the paper explains why the earlier Euler choice of frequency separation fails in the presence of any positive-order diffusion. We have to redesign the relation between amplitudes, frequencies, support sizes and activation times. We also need an accurate approximation of the fractional Laplacian that preserves the structure used to analyze each layer. Cancellations in the nonlinear self-interactions remain essential to keeping the force in the required class.

This theorem shows that the cascade can survive a mechanism that actively suppresses the growth it is designed to create. Our result concerns small fractional powers of the Laplacian, not ordinary Navier–Stokes. The gap between this range of and is substantial; the ordinary Laplacian cannot be reached by simply rescaling this construction.

Smooth forcing and recent developments

The same programme can be applied to other incompressible models. Our IPM construction starts from smooth, compactly supported initial density and produces finite-time blow-up with a source uniformly smooth in space. The solution remains smooth before the singular time, while its norm tends to infinity as that time is approached.

Keeping the source uniformly in space requires increasingly accurate approximations of the velocity. As we introduce layers with higher frequencies, we use approximations of progressively higher order, with the approximation order tending to infinity along the cascade. In this sense, we need an approximation “of infinite order” to control the errors absorbed by the source in every spatial norm. The assumption was chosen to simplify the construction. As noted in the paper, smoother time cutoffs and a more careful analysis of the errors give regularity with additional work; we also anticipated that full smoothness in both space and time should be attainable.

Further developments include our Boussinesq construction with Andrés Laín-Sanclemente, based on multi-layer degenerate pendula, and its extension to variable-density Euler with source terms. Our work with Óscar Domínguez and José Antonio Lucas-Manchón on forced generalized SQG establishes finite-time blow-up in a Sobolev local well-posedness regime, for the parameter range .

Recent work of Alpöge, Buckmaster and Coiculescu extends the IPM construction to a source smooth in both space and time on the torus. The Boussinesq and Euler manuscripts of Alpöge and Buckmaster likewise state blow-up from smooth initial data with smooth forcing. These works explicitly build on amplification through successive layers. Their additional regularity requirement is to control every mixed space-time derivative of the forcing uniformly through the singular time.

OpenAI’s September 8, 2026 announcement concerns two separate claims: blow-up for ordinary Navier–Stokes with smooth forcing, and blow-up for unforced Euler from smooth initial data. The first, if verified under the stated hypotheses, would establish an allowed breakdown alternative in the Millennium Problem formulation. It is distinct from unforced Navier–Stokes blow-up. At this moment we are not in a position to say anything substantive about a comparison between OpenAI’s constructions and our cascade methods. Assessing the announced proofs and their relationship to earlier work requires a detailed mathematical analysis.

The cascade leaves us with concrete mathematical questions. Which geometric features and cancellations let one layer amplify the next while controlling the full nonlinear interactions? How do they determine the regularity attainable for the force, and which parts of the mechanism survive stronger dissipation? Its application to several incompressible models gives us a way to investigate these questions within regimes of classical local existence.

[This is a guest post by Jennifer Taback. This blog post was initially written in a different file format and converted using AI. — T.]

Recently I have read a stream of articles questioning the future of mathematics: what constitutes a proof, what we value, how we retain the human enterprise of creating knowledge. Many of these articles are written by mathematicians making significant progress on the conjectures that shape their fields, whose work is now imperiled by the capabilities of frontier AI models. I am not one of those mathematicians. There are many mathematical problems that I have revisited over two decades, but none will grab headlines. Nevertheless, I care deeply about their resolution, as do others in my field. What does the advent of powerful AI models mean for mathematicians like me? I have yet to succeed in prompting a solution to one of my long-term problems, though AI models have helped fix an incorrect lemma and suggested a helpful reorganization of a paper. They have pointed me in directions in which I had not thought to look, and spurred me to learn new mathematics. Combining AI with the skills I have honed over the years is making me a stronger mathematician.

Journals prohibit AI models as co-authors, but these models have proven to be effective partners for me. I teach at Bowdoin College, a selective liberal arts college in Brunswick, Maine. While I have many wonderful collaborators, none are local. At my college, I have no one down the hall with whom to discuss research questions; mathematics departments at small colleges rarely have two colleagues in the same specialty. AI fills this gap for me. I respond to its (not always correct) answers to my questions, and ask for clarification as I would from a human collaborator. AI makes me ask myself more probing questions, challenging me to dig deeper into my accumulated knowledge and connect it to the broad synthesis the model can often provide. And sometimes AI is just wrong or confusing, as a human might be, and I move on.

Mathematics is a human endeavor, and machine-generated output requires substantial human intervention in order to contribute to our knowledge base. Having co-organized two conferences on “Communicating Mathematics,” I know that every idea that advances mathematics, whether human- or AI-inspired, requires clear exposition, a skill most of us could improve. AI is helping me here as well; it is making my mathematical writing better by catching small inconsistencies in papers I believed I had diligently proofread, and helping me sharpen my arguments. Our profession should embrace responsible use of these tools in the service of stronger communication.

AI has arrived in my classroom too, at an institution that demands pedagogical excellence and innovation alongside active research. I expect my classroom to change more in the next two years than it has in the previous twenty, in unknown and unexpected ways. I am angry at AI for forcing evaluation into the classroom under timed conditions. I do not perform my best under those circumstances and neither do my students; we have narrowed assessment to what we can proctor. Without graduate student assistance, thoughtful oral examinations, for example, are impractical at the scale of my teaching. Faculty at small colleges, even well-endowed ones, rarely have the graduate students and teaching assistants that make such solutions feasible.

Conversations addressing pedagogical changes resulting from widespread access to AI models are proliferating, but they are often convened at R1 institutions. These conversations must be broadened to include faculty from the entire spectrum of colleges and universities, especially those with higher teaching loads and more limited support. I am optimistic that by working across institutions we can strengthen undergraduate mathematics education. Students will influence this work as well; some of my undergraduates have been remarkably creative in their use of AI tools to further their understanding. One student in my cryptography class, for example, built an animation of a WWII-era encryption device that let viewers watch the mechanics and the mathematics unfold together. I have no overarching answers yet. But I want my students to trade their fear of being replaced by machines for the excitement of building what comes next.

I write this with the security of a tenured professor. In a small department where mentoring is a personal responsibility, I hear the visceral worries of my junior colleagues who are unsure of their path to tenure and beyond. Their position is genuinely different from mine, and their promotion equally weighs research and teaching. They are building a record under rules that do not yet exist.

Working toward new standards for the discipline is not an abstract matter. We have to decide what counts as proof, and what we value for publication and tenure. I would like us to disclose the tools we use for research without fear of judgment, while taking personal responsibility for ensuring that our work is rigorously verified, understood, and explained before we disseminate it. We should not abandon lines of inquiry for fear that AI will get there first; second proofs of theorems, whether by humans or by AI, often provide valuable insight into a problem. I do not want to surrender my questions, nor would I want my junior colleagues to cede theirs.

If we do things differently, adopt novel technologies, and investigate new frontiers of the discipline, we will still be producing mathematics. I love what I do, and I feel I am doing it better than ever. I will keep learning and seeking understanding while making the best use of the resources available to me. I do not know what mathematics will look like a decade from now. I know it will be created by people who have learned to ask insightful questions, and I know we are the ones who will teach them to do that, both in liberal arts classrooms and at research universities. Mathematicians will continue the hard work of discovery and understanding using stronger, and perhaps stranger, tools, pursuing problems yet to be posed.

Just a quick post to note that the International Council for Industrial and Applied Mathematics (ICIAM) has released a statement on mathematics and artificial intelligence (as well as a longer version), which makes many points echoing several already made recently here and elsewhere.

The longer statement also makes reference to a recent statement by the London Mathematical Society on recent developments around the Navier-Stokes equation. Perhaps the comments to this post can also be used to report other institutional statements on these topics.

[This is a guest post by Bryna Kra and Rachel Ward. -T.]

The Summit on PhD Math Education in the Age of AI was held September 17–18, 2026, jointly hosted by the Harvard Department of Mathematics and the Center of Mathematical Sciences and Applications (CMSA). It brought together twenty-four senior mathematicians from across the discipline, along with several current PhD students and postdocs, to examine which aspects of the mathematics PhD need rethinking as AI reshapes thinking-based work.

This document is a first draft of our recommendations; we expect to refine our recommendations over the coming months according to changing landscapes and your feedback.  We encourage feedback in the form of comments on this post, or by sending email to education-summit@cmsa.fas.harvard.edu.

To graduate students and early-career mathematicians: We know this uncertainty is weighing heavily on you as you plan your next steps. Your concerns matter, and we want to hear them. We are gathering information, keeping communication open, and working to support your opportunities and your future in mathematics.

[This is a guest post by Amit Sahai. This blog post was initially written in a different file format and converted using AI. — T.]

When I was an undergraduate student, I remember talking with several students who felt that the pace at which the top students could understand new math concepts was far too fast for them. They, too, could understand the ideas, but it would take them much longer. Eventually, almost all of these students gave up their dream of pursuing research mathematics and found something else to do. I have been thinking about those students a lot in the last few days.

The research mathematics community consists largely of those of us who either rarely felt that way, or who felt it and managed to overcome it through hard work. We have had the good fortune to find a place in mathematics where we could make progress. But we are now entering a time for humility: a time when all of us are going to know what it feels like to be unable to keep up.

The AI systems I have worked with are already producing beautiful new ideas. They are doing far more than impressive calculations or quickly carrying out arguments that a strong human researcher would already understand. And we probably can’t even imagine the wonderful ideas that future systems will be capable of producing.

When we feel that we cannot keep up, will we take that as a reason to leave research mathematics, like the students I am remembering? As more of us experience this, there will undoubtedly be a temptation to draw the same conclusion as they did: If the machines can move so much faster than us, perhaps we should find something else to do.

For our community to give up the work of understanding would be a profound abdication of our responsibility to humanity. Each of us is entitled to choose a different life. The responsibility I am talking about belongs to us collectively: to build a future in which humans can understand and contribute to the discoveries that will change our world. A future with meaningful human agency.

Struggle is essential to understanding difficult concepts. Fortunately, this struggle can be shared. I have been blessed to experience this time and time again with my students and collaborators. Imagine a multitude of research groups, each with sustained support, each spending a term or a year trying to understand an extraordinary set of ideas produced by an AI system, with the help of AI systems. [1]

This may very well be among the most important mathematical work in the years to come, and we should support and prioritize it accordingly. This enterprise will require a significant expansion in the number of mathematically sophisticated human researchers available world-wide, as major breakthrough ideas accumulate.

Why should society want this? So far, this might sound like a utopian fantasy for us – a civilization focused on depth of human understanding, awash with mathematicians and physicists and the like. I would certainly love to live in such a world. And indeed there are deep philosophical reasons for society to move in this direction. But I think society has a much more immediate stake in making this possible, too.

Imagine that a future AI system proposes a radically new design for a one terawatt nuclear fusion power plant. It has found a way to sustain and control fusion that no human had conceived of. The design promises abundant, inexpensive, clean electricity. Robots stand ready to manufacture the components and build the plant.

A terawatt is an insane amount of electrical power. We would be deciding whether to construct a machine that handles extraordinary flows of energy using principles we have never conceived of, let alone put into practice. We would need to understand how failures can be contained, what happens to energy already stored in the system when it shuts down, how we can be sure that the materials that make up the power plant behave as expected, and what other questions we should ask before proceeding. The very novelty that makes the proposal exciting would mean that we cannot inherit confidence from decades of operating similar plants or experiments.

Before approving construction, I would want communities of humans to understand why the design works and what justifies confidence in its safety. I would hope that we all would.

Human involvement does not automatically improve a technical decision , and I see no reason to insist that humans manually repeat work an AI system might be able to perform more reliably, even including proving mathematical guarantees. But a theorem can only exist within a model. Understanding the guarantee means understanding the model, the experimental evidence for it, and our uncertainties about the accuracy of the model. This is demanding work, and mathematically sophisticated people must be available to engage with it.

One might respond that AI systems should handle those questions too, and ultimately decide whether the plant should be built. That is a serious position. But it asks us to accept a future in which decisions of enormous consequence rest on reasons that no human community understands.

I do not want us to arrive at that future simply because we failed to invest in our own capacity to understand. Human agency is a value of fundamental importance. We must retain the ability to meaningfully consider alternatives and decide what kind of world we want to be a part of building. I think it is worth the effort. [2]

To take on this responsibility, we may need to broaden our view of what a mathematician can contribute. I have in mind something like a “deployable intellectual reserve”: communities of mathematically sophisticated people that humanity can call upon to help understand consequential AI-enabled breakthroughs.

Our ability to understand difficult and unfamiliar ideas may become one of the most important contributions we can offer to society. We should be willing to bring that skill to problems far beyond our usual research interests. [3] Doing so asks us to expand our sense of our vocation.

A counter-argument might be that AI systems will make each of us so much more effective that fewer people could do this work, even as the pace of discovery accelerates. But each of us is merely human. We have fundamental limitations based on our biology. Depth of understanding needs time and a pace of life that humans can sustain. Each individual human can only be asked to do so much, but through earnest cooperation we can accomplish much more.

If AI fulfills its promise, we will encounter more beautiful and consequential ideas than we have ever seen. We must respond by building thriving human communities that can understand them together.

We’re gonna need a lot more mathematicians.

The ideas and opinions presented here are entirely my own, but GPT 6 Astra was instrumental in helping me draft this note. I also thank my former student Dakshita Khurana, my current student Isaac Hair, my colleague Terence Tao, and my family members Anant Sahai and Gireeja Ranade for valuable feedback. Note that there is much more to be said here, but I tried to keep this relatively short to focus succinctly on my primary thoughts.

Notes

[1] By this, I do not mean to imply that only AI-created results will be of interest in the future. But for major results generated by humans, we already have a tradition of spending extended periods of time studying them.

[2] And the relevant understanding cannot belong only to the organization proposing the technology. Imagine a public hearing at which the company’s experts are the only people capable of following the technical argument. Independent expertise is critical.

[3] Indeed, AI systems are likely to be very helpful in allowing researchers with diverse backgrounds to talk effectively with one another, and more generally understand unfamiliar concepts.

[This is a guest post by Tapio Schneider. This blog post was initially written in a different file format and converted using AI. — T.]

[This will be cross-posted on the CliMA blog.]

The apparent proof of finite-time blow-up of the forced Navier-Stokes equation, announced by OpenAI on September 8, has brought into focus a debate about the role of AI in mathematics. The proof was produced with a system of some 10,000 AI agents that explored many approaches in parallel; it was then formalized and verified in Lean. The formal verification supports its correctness, but mathematicians are still working to digest it. A proof settles the truth value of a statement, but Terry Tao and others have argued that this is only part of the point of proofs; the other part is to advance our conceptual understanding of “basic structures of shapes, numbers, and natural phenomena.” As Yehuda Rav put it a quarter-century ago, “theorems are the headlines, proofs are the inside story.” A proof that is correct but incomprehensible, or undigested by the mathematical community, gives us the headline without the story. The issue is not whether the prover is a human or a machine, but whether the reasoning advances our collective understanding of methods and structures, which can spur new thinking and further advances. As Timothy Gowers has noted, the inadequate AI-generated write-ups of proofs are likely a temporary annoyance.

I want to argue that the same point about understanding holds in the natural sciences and engineering, where it also has intrinsic value and, in addition, acquires instrumental value when predictions must be trusted before they can be verified empirically. One role of science is what ancient philosophers called episteme, roughly explanatory understanding; here, understanding is the goal itself. Another role is techne, roughly the craft of predicting and making: forecasting how natural or engineered systems will behave under circumstances not yet observed. Whether the two roles can be separated depends on how easily predictions can be checked. When they can, techne can stand on its own. A black-box AI weather prediction model can predict tomorrow’s weather, and we can trust it because its forecasts can be checked every day. Techne can then also serve episteme as an instrument. For example, AlphaFold predicts 3D protein structures without providing explanations. But its predictions can be checked against experimentally determined structures, and they have become invaluable for understanding how drugs bind to their targets. Episteme and techne become inseparable when predictions must be acted upon, and hence trusted, before they can be empirically verified, because verification is too slow, costly, or dangerous, as in projecting climate change decades ahead or designing an aircraft. In those cases, an auditable causal chain from assumptions and input data to the predicted outcomes is what makes predictions trustworthy, and this constrains how AI can be used.

The Navier-Stokes equation, which describes fluid flow, is a good example, as it is at the heart not only of a Millennium Prize problem but also of aircraft design and climate prediction. The claimed result says that a fluid starting from rest, driven by just the right smooth stirring, develops an infinite velocity spike in finite time while its total kinetic energy remains bounded; the forcing is constructed to sustain the collapse. (The unforced version of the problem, whether smooth solutions exist for all time without external forcing, remains open.) Despite the mathematical singularity, nothing infinite happens physically: the incompressible equation stops being valid once energy concentrates at very small scales, where compressibility and molecular effects take over, as has long been known. By contrast, in turbulence, viscosity dissipates energy at a small but finite scale (the Kolmogorov scale), and the continuum description is valid across all scales of motions. Therefore, the result says little about how we model, predict, and understand the physical phenomenon of turbulence, which is likewise a solution of the Navier-Stokes equation, and one that, unlike finite-time blow-up, is ubiquitously realized.

The Navier-Stokes equation governs the flow and turbulence that control lift and drag around an aircraft wing, cloud formation in the atmosphere, and mixing in the oceans. In both aircraft design and climate prediction, verification comes late or is difficult and expensive. An aircraft is flight-tested only after it is built; a projection of how extreme rainfall statistics intensify in the coming decades must inform stormwater infrastructure that is built now but will likely be put to the extreme test only decades later. Such predictions earn trust not by end-to-end verification but by being the output of an auditable chain stretching from inputs (design parameters, atmospheric composition) to outcomes, whose links have known limits of validity and can be tested individually. Computational fluid dynamics (CFD) solves the Navier-Stokes equation together with additional equations (e.g., for thermodynamics) numerically. Its numerical methods for the resolved scales are based on established theories of stability, consistency, and convergence, and its subgrid-scale models for unresolved turbulence rest on plausible assumptions of universality at small scales (e.g., local isotropy) that can be tested separately against high-resolution simulations or laboratory experiments with canonical flows. Understanding and auditability of the individual links are what allow the chain as a whole to be trusted beyond the distribution of large-scale cases already observed, such as the present climate or existing aircraft wing designs.

Contrast that with end-to-end AI methods for weather prediction. Daily empirical verifiability suffices to establish trust in their forecasts. However, they do not come with a stability, consistency, and convergence theory (no analog of von Neumann stability analysis or the Lax equivalence theorem has been established for them), and how an initial condition becomes a forecast is difficult to audit. This is inconsequential for forecasting a few days ahead, where the forecasts can be checked daily. But it does matter for climate projection, which is a different task: predicting how weather statistics change over decades in response to a forcing such as increased greenhouse gas concentrations. An end-to-end model can learn the day-to-day evolution of weather states in today’s climate, but it contains no pathway through which greenhouse gases alter this evolution; their concentrations are typically not among its inputs, and if they were, no observations exist to learn about the response. The causal chain from greenhouse gases to their effects on radiative transfer, temperature, and winds, represented in physics-based models through their equations, is absent. Additionally, such models do not enforce conservation laws, for example of energy, so long integrations can drift by accumulating errors (e.g., in temperature). We would not currently trust them to project how the climate system responds to previously unobserved changes in the concentration of greenhouse gases over decades. Nor would we trust an AI surrogate of CFD simulations to certify an aircraft of novel shape. AI surrogates are used to explore and narrow down design spaces quickly, but the final assessment returns to established CFD methods or wind tunnel experiments, whose errors are controlled and whose steps are auditable.

So how do we best use AI in cases where episteme and techne combine, where understanding is essential for trust because predictions are difficult to verify? An answer suggested by the preceding argument is to embed AI at the links in the chain where trust can be earned by empirical verification. Concretely, use AI inside auditable scaffolds, such as physical conservation laws. Then use numerical methods with controlled errors to solve, e.g., the Navier-Stokes equation on the resolved large scales, while learning closure models for the unresolved subgrid scales from data, where some universality assumptions are defensible and individually testable. The rationale, in the case of the climate system, is that the large scales are where climate change moves the system out of the current distribution, whereas small-scale physics obeys the same local laws in a warmer or colder climate as in today’s. For an aircraft wing of novel shape, similarly, the geometry may be new but the functional relation between larger-scale conditions and the small-scale turbulence around it is not. The scaffold thus guides the extrapolation on the basis of the known equations, rather than with end-to-end models tied to the data available today.

Closures must be learned as functions of the resolved state, and extrapolation problems may reappear if the conditions for which data are available do not span the range of conditions for which predictions are needed (e.g., the temperature and humidity regimes in which cloud turbulence has been sampled, or the pressure gradients along a wing of novel shape). This can be mitigated with established tools, such as local high-resolution simulations for offline calibration and uncertainty quantification. Climate models and CFD codes have been built from resolved dynamics with embedded closures for decades. AI now enables a wider and faster search. Neural network closures can be individually tested and, to some extent, interpreted. Symbolic regression, which learns equations by selecting terms from a dictionary, as in SINDy, can sometimes produce closures that are more easily interpretable. AI agents are beginning to run the closure search in a closed loop, proposing symbolic closures, testing them against high-resolution simulations, observations, or experiments, and revising them. When this is successful, the resulting closure can be audited and, after the fact, understood.

This closed loop is similar to how AI is used in mathematics. In both cases, a verifier is used for checking: Lean for proofs; high-resolution simulations, observations, or experiments for turbulence closures. In the case of mathematics, the checks are complete; in the case of turbulence closures, they are limited to the conditions covered, so trust is restricted to the tested conditions. AI can accelerate this program by searching a larger space of possible closures or proof strategies than can be explored by humans in the same time. Once candidates have passed the checks, they become new objects for human study. In this way, understanding (episteme) and prediction (techne) improve together and reinforce each other, with humans, for now, remaining essential for extracting understanding and building trust in the end result: for writing the story behind the headline.

I thank Thomas Müller for pointing me to the paper by Yehuda Rav, and Thomas and my CliMA colleagues for discussions of the topics here over several years. I used AI for copy-editing.

[This is a guest post by Martin Hairer, cross-posted from Proofs and Prompts. — T.]

There has been a lot of speculation regarding the “Advisory Group on Mathematics and Artifical Intelligence” agmai.org since it was announced on Monday. In this short blog post, I would like to explain in a bit more detail how the group works, what we are trying to achieve, and why I agreed to be part of this group.

First of all, let me lay down some facts that seem to have been drowned in a torrent of misinformation. The most important one is that we are genuinely independent of OpenAI and any other of the so-called `frontier’ AI labs. Yes, the original impetus of forming this group was OpenAI’s response to the statement A Severe Misalignment of AI in Mathematics, but not all members of the group were contacted by OpenAI in the first place, there is nobody besides the nine of us taking part in our conversations, we do not receive any monetary compensation, and our technical support (so far mainly regarding IT, legal, and communications) is being provided by the IAS. We also have not signed any documents placing any constraints whatsoever on what we state in public and whose advice we seek, besides the obvious confidentiality requirements. The other fundamental fact is that our overarching aim is to represent as well as we possibly can the interests of the mathematical community. This is a tall order and we may end up doing a terrible job at it (I certainly hope not!), but I am absolutely certain that all nine of us take this extremely seriously and are acting in good faith.

I am acutely aware of many of the pitfalls of being part of such a group. At a most basic level, the mathematics community consists of people with a very broad range of views and opinions (often strongly held ones!), and the nine of us clearly don’t form a representative sample. As an independent body our role is purely advisory, so the AI labs can choose to simply ignore it. It would also be very naïve to believe that the AI labs won’t try to spin whatever we say in a way that suits their PR machine, which dwarfs anything we could possibly come up with. Finally, given that the unprecedented situation we find ourselves in has the potential of impacting the lives of so many people (I intentionally do not use the word `career’ since for many of us mathematics is so much more than just a career), whatever statement we make and / or advice we give will necessarily upset a sizable fraction of the maths community. So why on earth would I accept to be part of this?

Here are some of the reasons that swayed my mind:

1. The recent releases, on privately controlled websites, of high-profile mathematical results combined with abject levels of scholarship and only minimal presentation effort has been undermining the usual standards of scientific practice. If we refuse to answer when being asked how to do better (irrespective of whether that ask is being done in good faith), how could we then complain about this behaviour?

2. It is obvious that AI has already had a profound impact on mathematical research and raises numerous questions of correct attribution of ideas, priority, human understanding of ideas, etc. Many aspects of this revolution will be addressed within our community, but it makes no sense of taking a hard `ostrich’ approach of simply ignoring the AI labs and refusing to talk to them on principle.

3. It is perfectly legitimate to question OpenAI’s motivations, but the fact is that they have now made an effort (as belated and minimal as it is) to engage with the mathematical community. Turning them down without even trying to engage with them would be the easiest way for them to get a PR win painting the mathematical community as a bunch of out of touch luddites.

4. In the `Severe Misalignment’ statement, which has now been endorsed by nearly 8000 mathematicians, we conclude by saying that “These issues must be addressed urgently, in the mathematical community, by the companies developing these technologies, […]”. It would be rather hypocritical of myself to then chicken out at the first opportunity of actually having a chance of setting up some form of communications channel between the mathematical community and the AI labs just because I could get some flack for it.

Regarding the actual advice we intend to provide, it is a fact that AI companies have in recent months been producing some high profile mathematical results and that their publication and dissemination has been falling far short of acceptable mathematical practice, whichever way that is defined. However, the situation at hand is sufficiently unprecedented that we genuinely haven’t completely made up our minds yet on the best way forward and we fully intend to gather as much feedback as possible, be it through our feedback form, discussions with colleagues, the discussions on this blog, etc. We are going through a period of intense turmoil (and this doesn’t just concern mathematics) but I am heartened by the many intense but thoughtful and respectful discussions I have experienced both inside my own institution and at gatherings of mathematicians like the ICM over the past few months. I am convinced that the mathematical community can emerge all the more united and stronger from this experience.

[This is a guest post by Antonio Auffinger. This blog post was initially written in a different file format and converted using AI. — T.]

Despite writing papers in pure mathematics, much of my time in the past decade was spent talking (mostly listening, to be accurate) to biologists, computer scientists, and physicists. Theoretical physics and computer science are grounded in mathematics and thus share much of our language and, to some extent, our culture. The barriers between biology and mathematics are an order of magnitude higher. I am convinced that biologists are part of a different species. The language, incentives, culture, training, and everything else you might think of appear to be completely disconnected from the way we do mathematics. Yet, the similarities are abundant: the pursuit of understanding, the excitement of discovery when new patterns or phenomena emerge, and the hours needed to make minor advances, sometimes leaving us with just frustration. The intuition of setting up a new experiment feels like witchcraft to me, much like our way of finding connections between different abstract objects feels like magic to them.

What does talking to biologists have to do with the future of mathematics?

It could be beneficial to look at what is happening and what has happened with our neighbors, even if none of us can predict where we will be two years (or even three months) from now. A first lesson that I learned is that a large part of biology is technology-driven. Fields completely change, emerge, and die in a matter of years. The advent of CRISPR, RNA-seq, and cryo-EM, for instance, suddenly allowed humans to observe and manipulate phenomena that were previously inaccessible. These tools generate precious data, and the incentives often favor a culture of seclusion, where discoveries are frequently not shared until the final product is complete.

This has made me appreciate the culture of mathematics. Mathematics has never been free of competition or secrecy, but we abundantly share. We share ideas, we share problems, and we share entire skeletons of approaches with our colleagues, with visitors we just met, in talks, in public forums, and on YouTube. I suspect the enormous effort often needed to produce a proof, even with a full outline, has helped sustain this openness. Mathematicians and mathematics have deeply benefited from this open attitude. Many new connections and major discoveries have started with honest conversations at coffee breaks, in hallways, or on hiking trails. Sharing ideas or arguments before they are fully formed allows others to see pathways we missed, point out obstacles, or take the problem in directions we had not imagined. Also, it is simply more fun to do math together.

My worry is that this unselfish openness will become a thing of the past. If proof generation becomes a fast, accessible commodity while our ways of giving credit remain unchanged, mathematicians (especially those still building their careers) may feel encouraged to optimize locally in ways that weaken our culture of sharing. In the past few weeks, I have had colleagues reach out for advice and tell me they will no longer post on arXiv. I have witnessed trainees posting rushed papers online for fear that others could quickly carry out strategies already outlined in previous work. I am also part of the problem, as I have started advising my students to be extra careful when sharing work in progress.

Biology also offers examples of communities deliberately changing these incentives. During the Human Genome Project, the Bermuda Principles called for the rapid public release of sequence data. Later, the Fort Lauderdale Agreement tried to balance this openness with proper recognition of those generating the data, placing responsibilities not only on researchers producing and using the data but also on funding agencies. Although it is focused on data, it is an example where the status quo was changed by community intervention.

I do not know what the right analogue is for mathematics, but I believe we need to start thinking about it. I encourage the community to reshape our incentives and the way we give credit, to ensure broad accessibility to these new tools, and to make openness a reasonable choice, especially for those still building their careers. Senior mathematicians must engage in serious conversations about ethical use with their trainees. In turn, trainees should be encouraged to truly explore, because many of the solutions to the issues we currently face as a community will come from them.

None of this is an argument against the use of AI in mathematics. I believe these tools will raise the ceiling of the things we can discover, leading mathematicians to ask new questions and understand new phenomena.

This brings me to the second lesson I learned from my biology colleagues: many of the questions I hear from biologists need mathematics. New mathematics. There is room here for topologists, number theorists, dynamicists, algebraic geometers, analysts, etc. This is not just because of our capacity as proof builders, but also because of our capacity for abstraction and for understanding phenomena. Applied sciences are generating complex, time-dependent data that require new methods and theories. This is also a two-way street. Decades ago, topology and knot theory unexpectedly provided a framework for understanding how enzymes untangle and rearrange DNA. In the other direction, attempts to understand population genetics and how gene frequencies drift over time helped motivate new classes of infinite-dimensional stochastic processes, including measure-valued diffusions. Today, geometry and probability underlie dimension reduction methods that biologists use daily to make sense of massive single-cell datasets, such as t-SNE and UMAP.

I am not suggesting that mathematicians need to pivot to biology or any applied science. The pursuit of mathematics for its own sake is the absolute bedrock of our field and that must remain intact. However, as AI changes the landscape, looking outward presents an incredible opportunity to discover new questions, new phenomena and new mathematics, and perhaps AI might even lower the barriers to taking that leap.

Mathematics has much to offer the other sciences, and much to learn from them. I am looking forward to what comes next. Proofs may increasingly be generated by machines, but the ultimate purpose of mathematics remains exactly what it has always been: to ask questions and to understand.

[This is a guest post by the Advisory Group on Mathematics and Artificial Intelligence. This blog post was initially written in a different file format and converted using AI. — T.]

We would like to use this guest post to announce the creation of the Advisory Group on Mathematics and Artificial Intelligence hosted at the Institute for Advanced Study (Princeton) and online at agmai.org.

The rapid advances in artificial intelligence (AI) present both opportunities and challenges for mathematical research. We believe that we are at a historic moment for our discipline. Recent events raise urgent questions about how to support the long-term prospects for deep human understanding of mathematics.

Purpose. The purpose of this group is to advise AI companies on their interactions with mathematical research and with the mathematical community, including the responsible presentation and release of mathematical results. We seek to work for the best interest of mathematics and the mathematical community, and to serve as one possible channel of communication between mathematicians and the AI industry.

Independence, Transparency, and Accountability. This group operates independently of any AI company and members do not accept payment for this work. We will publish our recommendations to AI companies on this website. We are willing to offer such recommendations to any AI company whose models are likely to have a significant impact on mathematics. Although we will give advice, we do not have decision making power at any AI company, and the responsibility for the decisions made by any company will rest with that company.

Advisory Group Members

• François Charles (ENS-PSL)
• Camillo De Lellis (IAS, GSSI)
• Timothy Gowers (College de France, Cambridge)
• Martin Hairer (EPFL, Imperial College London)
• Nikhil Srivastava (Berkeley, Simons Institute)
• Ulrike Tillmann (Oxford, INI)
• Ravi Vakil (Stanford)
• Edward Witten (IAS)
• Melanie Matchett Wood (Harvard)

This group came together after OpenAI approached some of its members about establishing an external advisory board. In agreement with OpenAI, they decided to create an independent group and invite others to join.

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Disclaimer: due to current events, I have not been able to devote as much time to lecture notes preparation as I would have liked, so I apologize in advance for the unpolished nature of the text below, which has been largely recycled from previous lecture notes I have written.

This is the first set of lecture notes for my graduate course 247A, “Fourier analysis”. The course name is rather general, but I will focus the course not on the Fourier transform per se, but on the closely related topic of real variable harmonic analysis, with a particular emphasis on Calderón–Zygmund theory, which underlies basic tools in PDE such as the theory of Sobolev spaces.

To avoid confusion at the outset, let us make the distinction between real-variable harmonic analysis and abstract harmonic analysis, which are only distantly related to each other despite the similar names. Abstract harmonic analysis, roughly speaking, is the extension of the classical theory of the Fourier transform to other domains, such as locally compact abelian (LCA) groups, non-abelian Lie groups, or symmetric spaces, and typically involves a blend of representation theory, group theory, and analysis. Real-variable harmonic analysis, by contrast, tends to work on classical domains, such as a Euclidean space , a torus , or a lattice , although many of the techniques can extend to more general domains (e.g., to Riemannian manifolds). While the Fourier transform often plays a prominent role (in particular, by setting the stage for time-frequency analysis and enabling various decompositions or other transforms that involve frequency space or phase space in addition to physical space), real-variable harmonic analysis is often focused on estimating other transforms or expressions that often interact well with the Fourier transform, but need not explicitly invoke it. Examples include the Hilbert transform

(where we have made the somewhat arbitrary decision to omit the normalizing constant ) or the Hardy-Littlewood maximal function

A typical question in harmonic analysis is the following: let be some function on a standard domain (such as Euclidean space), and let be an explicit transform of (e.g., the Hilbert transform or maximal function ). To what extent is the “size” of controlled by the “size” of ? The value of such bounds often lies in the general nature of the input function ; some mild regularity or decay hypotheses might be imposed on , but beyond that the function is typically not required to have a very structured form (in particular, it need not be describable by any closed-form expression).

In many situations the transform being studied is linear or sublinear, in which case the natural type of bound to ask is a linear bound

for suitable function space norms (e.g., norms), and is some bound. Depending on the application, we may be interested in various levels of precision regarding the bound :

• (a) Optimal bounds, in which we seek the exact optimal value of (i.e., the operator norm of ). For instance, the optimal constant for the Hilbert transform is exactly , whereas the optimal constant for in one dimension turns out to be (a result of Melas).
• (b) Bounds accurate up to absolute constants (or maybe constants that can depend on basic parameters such as the ambient dimension).
• (c) Bounds in which we are willing to accept “logarithmic type losses” such as or in auxiliary parameters, such as a scale parameter .

All three regimes are interesting, but we will focus in this class on the regime (b), where we can “afford” to lose absolute constants in the bounds, but will work hard to avoid any logarithmic losses. In particular, significant effort will be devoted in this class to avoiding “logarithmic pileups of scales”, in which the contributions of different dyadic scales such as for all potentially contribute an equal amount that “interfere constructively” to cause a logarithmic divergence. This can be unnecessarily conservative when one is in regime (c) (which is for instance the situation in modern topics such as restriction theory or the Kakeya conjecture); nevertheless, the general skills gained by trying to not lose even a logarithmic factor in the bounds are often valuable in these other types of analysis.

When dealing with linear or sublinear problems, it is natural to try to decompose the initial function into various smaller components by some decomposition , so that the transformed function can be controlled by more tractable expressions in various ways (e.g., via the triangle inequality, by Bessel type inequalities, or by the more modern technique of decoupling inequalities). In short, the subject tends to proceed by a divide and conquer philosophy: it is generally preferable to replace a simple-looking but hard-to-estimate expression with a large, messy-looking combination of expressions that are easier to estimate. As such, the aesthetics of the subject are almost the reverse of those in the more algebraic portions of mathematics, in which progress is often made by making the expressions involved look as simple and unified as possible.

One of the main themes in this classical type of harmonic analysis is the struggle to understand the effect of two phenomena in integrals or sums: singularity and oscillation. The Hilbert transform (1) is a quintessential example of a singular integral, which combines both features: the non-locally integrable nature of the kernel provides the singularity, but the sign change from to provides the oscillation. Classically, the interplay between these two phenomena can be tamed by analyzing the behavior of this operator both in the time (or “physical”) domain and in frequency (or “Fourier”) domain; in particular, the fact that singular integral operators such as the Hilbert transform are simultaneously a well-behaved Fourier multiplier and is “pseudo-local” in physical space lies at the heart of the standard Calderón–Zygmund theory for such operators. This dovetails nicely with more modern “time-frequency analysis” approaches to the subject, which can also handle other interesting operators, such as restriction or Bochner–Riesz operators via tools such as the wave packet decomposition, although these will be outside the scope of this course.

In this initial set of notes I will ignore the effect of oscillation, and develop some tools, such as interpolation theory, which can help control non-oscillatory sums and integrals if they are not too singular. Here, the focus will be on rearrangement-invariant spaces, such as the Lebesgue spaces and their weak variants , which are function spaces that are useful for measuring how “singular” or “decaying” various functions are, but do not pay attention to how they oscillate or where their mass is distributed. As such, these spaces do not capture the underlying geometry of the domain, which also plays an essential role in the subject; but it is nevertheless essential to have a good base understanding of the rearrangement-invariant theory before moving on to the more delicate aspects of harmonic analysis that are sensitive to rearrangements.

We will use the following asymptotic notation throughout the course: , , or denotes the assertion that for some constant , and write for . If we permit this constant to depend on some ambient parameters, we indicate this by subscripts; for instance, or denotes a bound of the form for some constant that can depend on and . As indicated above, in this course we will generally not dwell much on exactly what these constants are, or attempt to optimize them.

— 1. norms —

Suppose one has some measurable function on some measure space . (Here we will follow the common practice if identifying functions that agree almost everywhere; in particular, we will be content to work with functions that are undefined on a set of measure zero. Also, while we work here with complex-valued functions throughout, most of the discussion here is also valid for real-valued or vector-valued functions.) Informally speaking, to measure how “big” such a function is, there are two (imprecisely defined) basic statistics to be aware of:

• The height or amplitude of the function, which describes what the typical size of the magnitude is for in the “dominant” component of the support of ; and
• The width of the function, which describes the measure of this dominant component.

Example 1 (Informal) Given a Gaussian wave packet type function

on for some and , this function has magnitude on the ball , which has volume (if we allow constants in the informal notation to depend on the dimension ), so such a function has height and width .

Example 2 (Informal) The function on , which is implicitly involved in the definition of the Hilbert transform (1), does not have a clear amplitude or width as is. However, if one performs a dyadic decomposition

where we use to denote the indicator of a statement (equal to when is true and otherwise), then each component

of this decomposition has height and width . Thus, while this function can be viewed as a superposition of components of various heights and widths, rather than a single such component.

These informal concepts of height and width are too imprecise to work with in practice. Experience has shown that a convenient proxy for these concepts are the norms of a function , defined for as

and for as

where denotes the essential supremum of the function with respect to the measure . Often we abbreviate as , , , or just (and abbreviate as ) when the missing arguments are clear from context. (For instance, when working with Euclidean spaces , the measure is understood to be Lebesgue measure, and the Lebesgue -algebra, unless otherwise specified.) In terms of the width and height of a function , one heuristically has

for both finite and infinite values of , with the convention that is equal to when is positive and when is zero. In the case of a step function (where now is the indicator function of a measurable set ), this heuristic becomes exact:

The function space is defined as the set of all measurable functions for which the norm is finite, up to almost everywhere equivalence, though we will often abuse notation by identifying a function with its almost everywhere equivalence class.

In the case where is discrete and is counting measure, we abbreviate as , or even just .

Example 3 Let . On a Euclidean space , the function lies in (with a norm of ) if and only if , while the function lies in (with a norm of ) if and only if . The function does not lie in any , although it only fails “logarithmically” to lie in . Thus we see that control in for high rules out severe local singularities at a point, while control in for low rules out insufficiently rapid decay at infinity.

As is well known (see these previous notes) these function spaces enjoy many useful properties:

Theorem 4 (Basic properties of spaces)

• (i) The space is a Banach space when , a Hilbert space when , and a topological vector space when .
• (ii) If obeys the scaling condition
  then one has the Hölder inequality
  for any measurable (here we adopt the usual conventions ). In particular, if and , then .
• (iii) If for some , then one has the duality relationship
  where is the conjugate exponent to , defined by .

Remark 5 Closely related to (iii) is the fact that the dual of can be identified with when (with the additional hypothesis that is -finite if ), but in practice the relation (4) will already be good enough for our purposes.

Remark 6 The Banach space property gives us the basic triangle inequality

for both finite and infinite collections of functions when , where in the infinite case the assertion is that if the right-hand side is finite, then the series is absolutely convergent almost everywhere, and obeys the above inequality (so in particular is in ). For , this inequality fails (can you come up with a counterexample?), but one has the weaker -triangle inequality

in this case, which follows easily from iterating the easy observation that for any complex numbers , which in turn ultimately stems from the complex triangle inequality and concave nature of for . In particular, for a finite sum , another application of Hölder’s inequality gives the quasi-triangle inequality

for , which is not too much worse than (5) when is not too large.

Exercise 7 Give an example to show that the quantity in (7) cannot be replaced by any smaller quantity.

Exercise 8 For a simple function, verify that , and that , where . For this reason, the measure of the support of is sometimes referred to as the norm of , though it would be more accurate (though confusing) to refer to it as the power of the norm.

Remark 9 Note that Hölder’s inequality is not just symmetric under the homogeneities and of the functions, but also under the homogeneity of the underlying measure. This latter symmetry demonstrates why the condition is necessary. (The first two symmetries demonstrate why appears the same number of times on both sides of the inequality, and similarly for .)

In the case of Euclidean space, the measure homogeneity symmetry is equivalent to the scaling symmetry for , as the Jacobian of this map is . But the point is that by manipulating the measure directly, one still enjoys this symmetry even when no scaling operation is present.

It is instructive to try to understand inequalities such as (3) using the height-width heuristic introduced previously. Suppose informally that have heights , , and widths , , respectively. Then one expects the heights to be related by the formula

What about the widths? Heuristically, the region that concentrates in ought to be a subset of the region that concentrates in, so

and similarly with replaced by . We can combine these bounds as

The bound (3) then is morally

which on applying the previous bounds and (2) should simplify to

But this is clear by bounding by for the first factor on the left-hand side, and by for the second factor. Thus we see that the key geometric input that is morally driving the Hölder inequality is the simple fact that the concentration region of the product is contained in the concentration regions of the factors.

Exercise 10 Determine the cases for which (3) holds with equality (dealing with edge cases such as when one or more of equal infinity as appropriate). Discuss how your conclusions align with the heuristic analysis presented above.

Exercise 11 If , determine the cases for which (5) holds with equality. What changes when or ?

Exercise 12 Show that Hölder’s inequality is equivalent to the log-convexity of norms:

(For technical reasons one needs to first reduce to the case where has finite measure, and then and are everywhere non-vanishing simple functions. Now consider the convexity of with respect to a measure for some suitable exponents .)

Exercise 13 (Direct approach to log convexity) Differentiate twice with respect to and show that this is non-negative (take to be a non-zero simple function with finite measure support to avoid technicalities). This is an example of a monotonicity formula method — deriving estimates from a monotonicity property, which in turn follows from the non-negativity of a derivative.

You will see that this approach is surprisingly messy. For all the other ways, observe that (8) enjoys homogeneity symmetry in both and , which lets one normalise both and to equal one. Thus the task is now to show that if , then for all between and . This can be done by the pointwise convexity of , or more precisely the estimate

the observant reader will note that this is merely the proof of Hölder’s inequality in disguise.

Let us now give a more unusual proof of the log-convexity which does not appeal to any pointwise convexity estimate, instead combining the “divide and conquer” strategy with an elegant (and rather cheeky) “tensor power trick“. Again normalise . We split into a broad flat piece and a narrow tall piece

which are disjoint, and thus

What we are doing here is exploiting some very basic intuition about norms, namely that bounds for large tend to exclude tall narrow spikes, whereas bounds for small tend to exclude short broad tails. Of course, either sort of bound would exclude tall broad functions, and neither excludes narrow short functions. Once again, this intuition can be buttressed by considering the special case of step functions.

When , then , and when , then . Thus we end up with

The above argument (which is a prototype of the real interpolation method) obtained an estimate which is off by a factor of two from what we wanted; this is a typical feature of the method. However we can recover this factor for free by the following tensor power trick. Let be a large integer. We replace the measure space by its power using the product measure construction, and similarly replace with its tensor power , defined by

One then observes that

Now we apply the preceding arguments to instead of to deduce that

which on taking roots gives

Now the left-hand side is independent of ; take limits as and we obtain as desired.

The tensor power trick can be viewed as another application of symmetry: if an estimate is invariant under raising to a tensor power, then one can automatically replace all absolute constants with ; thus we obtain the “free lunch” of deducing a bound with an explicit constant , from a bound with an unspecified constant (or even with “logarithmic losses”). Contrapositively, if an estimate is invariant under tensor power, then a weak counterexample (which shows that the constant must exceed one) can be amplified into a strong counterexample (which shows that no finite constant suffices) by tensor powering. The tensor power trick seems like a magical trick at present, but is actually exploiting some basic results in information theory such as the Shannon entropy inequalities and the central limit theorem; it also combines well with virtually any inequality which involves Gaussians. Unfortunately due to lack of time we will not be discussing these beautiful topics further in this course. At any rate one sees the power of abstraction in this tensor power trick. (One could similarly perform this trick in , so long as the constants only grew sub-exponentially in the dimension .)

The final proof of log-convexity of the norm that we give here proceeds via complex analysis, and the maximum principle — which in many ways is a complex analogue of convexity (or subharmonicity). We need the following result from complex analysis, namely a form of the Phragmén–Lindelöf principle.

Lemma 14 (Three lines lemma) Let be a complex-analytic function on the strip , which is of at most double-exponential growth, or more precisely for some . Suppose that we have the bounds when and when . Then we have for all in the strip.

Remark 15 The rather strange sub-double-exponential hypothesis here is completely sharp, as the example shows. Note in this hypothesis that we allow the implied constants in the asymptotic notation to depend on , but the hypothesis is qualitative rather than quantitative: the value of these constants is irrelevant for the final conclusion, as long as they are finite. In practice, these sorts of qualitative hypothesis are usually easy to establish (especially when compared to quantitative estimates) by restricting, smoothing, or damping to a nice class of functions, or by smoothing out or discretising various operators and domains. See for instance the proof of this very lemma in which we upgrade “for free” a weak qualitative bound (sub-double-exponential growth) to a strong qualitative bound (decay at infinity).

Proof: The hypotheses and conclusion of the lemma are invariant under the operation of multiplying by a constant (and adjusting appropriately). So we may normalise . Similarly, the hypotheses and conclusion of the lemma are invariant under the operation of multiplying by an exponential for some real . Using this, one can also normalise . So now is bounded by on both sides of the strip and we want to show it is bounded by inside the strip.

Let us first assume that is much better than exponential growth, namely that it goes to zero at infinity. Then for all sufficiently large rectangles the complex-analytic function is bounded by on all four sides of this rectangle, and hence in the interior also by the maximum principle, and we are done by setting .

Now let us handle the general case; as is usual when removing a qualitative assumption, we do this by a limiting argument. We replace by ; a little complex arithmetic shows that this converts the almost double-exponentially growing function to one which is still complex analytic but is now decaying at infinity. It is still bounded by at both sides of the strip, and hence by in the interior also by the previous argument. Now take to conclude the claim.

Exercise 16 Suppose that is analytic on the strip , obeys the sub-double-exponential bound on the strip, and obeys the polynomial bounds on the sides of the strip. Show that it obeys the polynomial bound on the interior of the strip also.

To apply the three-lines lemma to prove (8), take to be a simple function (with finite measure support) and consider the entire function

This function has exponential growth at most (because of the qualitative assumption that is simple with finite measure support), and is bounded by on the lines and , and hence (by a trivially rescaled version of the three lines lemma) bounded by on the strip inside the lines. In particular it is bounded by at , which gives the claim for simple functions. The claim for more general functions (dropping the qualitative assumption of simpleness and finite measure support) then follows by a standard limiting argument (using for instance monotone convergence) which we leave as an exercise.

The above argument is a prototype of the complex interpolation method. As one can see, it can give slightly sharper results than the real interpolation method (though using the tensor power trick the real method can sometimes “catch up”), but on the other hand requires the quantities being studied to depend complex-analytically on a parameter rather than (say) real-analytically.

Having conclusively demonstrated the log-convexity (8) in multiple ways, let us now give some quick applications. It shows that control on two extreme norms implies control of the intermediate norms. Under additional assumptions on the measure space , one of these extremes is not necessary. If the measure space is finite in the sense that (thus prohibiting functions from being arbitrarily broad), then higher norms control lower ones:

Indeed this is trivial when , and the general case then follows by convexity. The bound (9) can also be usefully written in terms of averages: if we write for , then we see that higher averages control lower averages:

One way to view this is that as one lowers the exponent , the exceptionally large values of become less important, leaving the small values of to dominate. By restricting to its support one can refine (9) to

(Note that this is a limiting case of log-convexity at the exponent , in view of Exercise 8). In the converse direction, if the measure space is granular in the sense that one has a lower bound for all sets of positive measure, then functions are prohibited from being arbitrarily narrow, and lower norms control higher norms:

This can be seen by first checking the case, and then using log-convexity to get the remaining cases. In particular, in the spaces we see that for . For spaces on points, we thus have (non-matching) upper and lower bounds

and norms are comparable to some extent, but the comparability gets worse as or as and get further apart.

Exercise 17 Heuristically justify the bounds (9), (10) by appealing to the informal notions of width and height of a function.

Exercise 18 When does equality occur for either of the inequalities in (10)? Note how the example that attains the lower bound is in many ways the “opposite extreme” to the example which attains the upper bound.

Lebesgue measure on Euclidean spaces with the usual Borel or Lebesgue -algebra is not granular. However one can create granularity by coarsening the -algebra. For instance, if we let be the -algebra generated by the lattice unit cubes for , then we have granularity with constant , and now lower norms of functions control higher ones — but only for functions which are measurable with respect to this algebra, i.e. only for functions which are constant on each lattice unit cube. (This is the first time we have actually manipulated the -algebra to say something non-trivial, as opposed to manipulating , , or .) Thus we see that local constancy of functions can lead to additional estimates on norms. Later on we shall see that frequency localisation achieves a similar effect as local constancy, as quantified by Bernstein’s inequality; this is a concrete manifestation of the famous Heisenberg uncertainty principle.

Finiteness and granularity of the measure space prevent a function from being too broad or too narrow respectively. Similar things happen when a function is being prevented from being too tall or too short; for instance if is bounded above by a constant , then we have

(this is just log-convexity at the exponent), while if is bounded below by on its support, then we have the reverse inequality

There are two obvious algebraic identities involving norms which are worth knowing. The first is that one can interchange sums with integrals for any , in the sense that

this is just an application of the Fubini-Tonelli theorem. Secondly, exponents can pass through norms by changing the exponent: for any we have

We shall use both of these identities in the sequel without further comment.

Exercise 19 Establish the bound

for any measurable and any .

— 2. Lorentz spaces —

Recall that the weak norm of a function is defined for as

Since

for any and , we obtain Chebyshev’s inequality

(the case is also known as Markov’s inequality). When we adopt the convention that .

We define weak or to be the space of all functions with finite norm, with the usual abbreviations. We sometimes refer to as strong to distinguish it from weak .

Example 20 On a Euclidean space , the power function lies in weak if and only if . Indeed one can think of a weak function as a function which is pointwise dominated in magnitude by a rearrangement of (a multiple of) .

Suppose . From elementary calculus we have

and hence on integration and Fubini’s theorem

To summarise, we have

and

for . These two identities motivate introducing the Lorentz (quasi-)norm for and by

Thus for instance norm is identical to the norm. We shall abbreviate by , , or even when there is no chance of confusion.

Remark 21 For various reasons it is not worth trying to define Lorentz norms when , although we will use the convention . The most important values of , in descending order, are , , , and ; the other cases essentially never occur in applications.

Remark 22 The factor is inconsequential, but is traditional in order to maintain compatibility with the strong norm. But in practice the exact form of the Lorentz norm is not important; there are many formulations which are equivalent up to constants, and one generally just picks the formulation which is most convenient. The measure is of course multiplicative Haar measure on . One can interpret the equivalence of (i)-(iii) below by making the change of variables , so that the Haar measure just becomes Lebesgue measure in (modulo an inessential constant) and then passing from continuous to discrete .

Exercise 23 If is a monotone non-increasing function, show that

(Depending on how you prove this, it may be convenient to first prove this for smoother , such as diffeomorphisms with strictly negative derivative, in order to apply an inverse function theorem.)

Exercise 24 Show that a step function of height and width has an norm of for any and .

Exercise 25 For any and show that

It is obvious that these norms are both rearrangement-invariant and monotone. To get a better intuitive handle on what the norm represents, we need some more definitions.

• A sub-step function of height and width is any function supported on a set with the bounds almost everywhere and . (Thus .)

• A quasi-step function of height and width is any function supported on a set with the bounds almost everywhere on , and . (Thus .)

Remark 26 It is a little dangerous to put fuzzy notation such as within a definition; if multiple quasi-step functions appear in an argument, the question then arises as to whether the implied constants are uniform. In our applications, the implied constants here are true absolute constants (like and ) so this will not be an issue.

Remark 27 From the binary expansion of the unit interval we see that a non-negative sub-step function of height and width can always be decomposed as where is an actual step function of height and width at most . By homogeneity we have a similar statement for other heights. Because of this, bounds on step functions tend to automatically extend to sub-step functions (and hence quasi-step functions) without difficulty.

Just like actual step functions, the norm of sub-step and quasi-step functions are well controlled; a sub-step function of height and width has norm , while a quasi-step function has norm – almost exactly like actual step functions. In the converse direction, it turns out that every function can be decomposed as an sum of “very different” -normalised sub-step or quasi-step functions.

Theorem 28 (Characterisation of ) Let be a function, let and , and let . Then the following are equivalent up to changes in the implied constants:

• (i) We have .
• (ii) There exists a decomposition where each is a quasi-step function of height and some width , with the having disjoint supports and
  Here the subscript in denotes the variable that the norm is being taken over.
• (iii) There exists a pointwise bound , with

• (iv) There exists a decomposition where each is a sub-step function of width and some height , with the having disjoint supports, the non-increasing in , the bounds on the support of , and

• (v) There exists a pointwise bound , where and (14) holds.

Remark 29 The formulations (ii), (iv) are useful when trying to use an bound on ; the formulations (iii), (v) are useful when trying to obtain an bound on . Heuristically, the above theorem is trying to say the following. If is a quasi-step function of height and width , then . But if is instead the sum of quasi-step functions of height and width , and either the heights or the widths are sufficiently variable in (e.g. one or the other grows like a power of two), then .

Proof: We may use homogeneity symmetry to normalise . The implications and are trivial. To see that (i) implies (ii), set and . (This is the “vertically dyadic layer cake decomposition”.) The only thing that requires nontrivial verification is (12); but one easily verifies that

and the claim follows by summing this in .

Similarly, to see that (i) implies (iv), define

note that this is a non-increasing function of , which goes to zero as (this comes from the hypothesis that is finite). We then define

(This is the “horizontally dyadic layer cake decomposition”.) The only non-trivial thing to verify is (14). But one easily verifies the telescoping estimate

and the claim follows by summing this in and interchanging the summation signs. (We leave to the reader how to modify the above argument to handle the case .)

It remains to show that (iii) implies (i) and (iv) implies (i). Suppose first that (iii) holds. It is clear that for any we have

and hence

and so on taking summation in it would suffice to show that

Raising to the power we rewrite as

But from the hypothesis we have

and hence on shifting by

The claim then follows from (5) or (7).

Now suppose that (iv) holds. Observe that for any we have

where are the modified heights

Indeed, if for some then one easily verifies that and hence . The shifting trick and triangle inequality argument used previously shows that obeys the same bound (14) as , thus

We now compute

as desired. (We leave to the reader how to modify the above argument to handle the case .)

Remark 30 For future reference we make the technical remark that if is a simple function, then in the horizontal and vertical decompositions in the above theorem, only finitely many of the are non-zero.

Remark 31 Suppose that the ratio between the tallest height and lowest non-zero height of a function is (i.e. there exists such that whenever is non-zero). Then the above theorem shows that two different Lorentz norms , with the same primary exponent only differ by multiplicative powers of . Similarly if the broadest width and narrowest width of a function differs by (e.g. if is equal to times the granularity of ). What this indicates is that the secondary exponent in the Lorentz norms only offers “logarithmic correction” to the Lebesgue norms ; in contrast, (10) shows that varying the primary exponent leads to polynomial-strength changes in the norm. So as a first approximation (ignoring logarithms) one can pretend that . Note also that for quasi-step functions, the norms barely depend on at all.

One easy corollary of the above theorem is that the quasi-norm is indeed a quasi-norm, and in particular that

for any ; this can be seen for instance by using the equivalence of (i) and (iii). Another easy consequence is that the simple functions are dense in .

Exercise 32

• (i) Suppose that is a quasinorm on some function space with quasitriangle inequality constant , thus
  for all . Let be such that . Show that
  (Hint: first show that if , then . Then iterate this carefully to show that if and , then .)
• (ii) Suppose that a sequence for obeys an exponential decay bound of the form
  for some and all . Show that the series
  converges in , with

Exercise 33 For each integer , let be a quasi-step function of height and width for some . Show that

for all . If instead is a quasi-step function of height and width for some , show that

for all . What goes wrong when we remove the absolute values on the ? (This can be repaired by replacing the powers of with powers of a sufficiently large constant (depending on the implied constant in the definition of a quasi-step function) – why?)

A particularly useful consequence of the above theorem is a Hölder inequality for Lorentz spaces, due to O’Neil.

Theorem 34 (Hölder’s inequality in Lorentz spaces) If and obey and then

whenever the right-hand side norms are finite.

Proof: We may normalise , and drop the dependence of the implied constants on for brevity. By the equivalence of (i) and (v) in Theorem 28 we may dominate and where and

Then we have

By the quasi-triangle inequality and monotonicity it suffices to show that

and

By symmetry it suffices to consider the component. Here we observe that has measure at most , so by the equivalence of (i) and (v) in Theorem 28

But by the ordinary Hölder inequality

shifting the second by we conclude

The claim now follows from Exercise 32.

One corollary of this Hölder inequality is that functions are absolutely integrable on sets of finite measure whenever .

Now we consider dual formulations of the norms. The case is fairly straightforward:

Exercise 35 (Dual formulation of weak ) Let . Then for every in , we have

Also show that the hypothesis can be dropped if one instead assumes to be non-negative.

The right-hand side of (16) is clearly a semi-norm at least on . This leads in particular to a quasi-triangle inequality

for any .

Remark 36 It is worth comparing (16) to (4). In (4), one takes the inner product of against all -normalised functions, and the worst inner product becomes the norm. In (16), one only takes the inner product of against the -normalised step functions . This is fully consistent with the fact that the norm is stronger than the weak norm.

Exercise 35 can be rephrased as follows: if for some and , then the following two statements are equivalent (up to changes in the implied constants):

• .
• for all sets of finite measure.

Unfortunately this equivalence breaks down at or below (consider for instance the weak function on , which is not even locally integrable when ). However, one does have a substitute:

Exercise 37 Let , , and . Show that the following are equivalent (up to changes in the implied constant):

• .
• For every set of finite measure, there exists a subset of with such that
  (in particular, we assert that the integral on the left-hand side is absolutely integrable).

Hint: It may be instructive to work out the example on by hand to get a sense of what is going on; this should suggest how to prove things in general. The proof is slightly simpler in the case when is non-negative, so you may want to try that case first. Comment on how this result implies Exercise 35 (or its equivalent version discussed shortly afterwards) when .

Exercise 38 Let be functions with . Show that

thus the weak quasinorm only fails to be a norm “by a logarithm”. Show with an example that the cannot be removed.

For more general spaces, we have

Theorem 39 (Dual characterisation of ) Let and . Then for any ,

Again, the hypothesis can be dropped if is non-negative and is restricted to also be non-negative.

Thus the quasi-norm is in fact equivalent to a norm when and . In particular, weak is equivalent to a normed space when . (For , weak fails to be normable “by a logarithm”; see Q3.) As with other dual characterisations, one can restrict to a dense subclass of , for instance simple functions with finite measure support.

Proof: To obtain the part of this theorem, we simply estimate

and use Theorem 34. To obtain the part, we normalise . It then suffices by homogeneity to find with and .

The case follows from (16), so let us take . We will just give the proof in the case ; the case is trickier, and a partial argument is given in the exercises. By the equivalence of (i) and (ii) in Theorem 28 may write where is a quasi-step function of height and width with disjoint supports such that the sequence has an norm of . Now take

where

adopting the obvious convention that when . Then (because of the disjoint supports)

But since has height and width , and so

To conclude it will suffice to show that

If is the support of , then we have the pointwise bound

and the measure bound .

At this point we would like to apply Theorem 28, but neither the height nor width of is necessarily a power of . But we can remedy this by introducing the modified heights

We have , and so the increase geometrically. It then suffices to show that

By refining the by a constant factor we can make each at least twice as large as the previous, and so by applying the equivalences of (i) and (iii) in Theorem 28 and the triangle inequality it suffices to show that

which we expand using our bound on as

But from Hölder’s inequality (using the hypothesis ) and the bound on we have

summing this using the triangle inequality (and estimating the supremum by a sum) we obtain the claim.

The case when are restricted to be non-negative can be deduced from the above result and a monotone convergence argument (representing as a monotone limit of simple functions of finite measure support) which we leave as an exercise to the reader.

Exercise 40 A measure space is said to be non-atomic if, for every measurable set with , there exists a measurable subset such that .

• (i) (Sierpinski’s theorem) Show that if is non-atomic, is measurable, and , then there exists a measurable subset of such that . (You may find it convenient to use Zorn’s lemma.)
• (ii) Show that if is non-atomic, then the decomposition in Theorem 28(iv) can be chosen so that for each , either vanishes, or has support of measure (not just bounded by ).
• (iii) Establish Theorem 28 in the case that and is non-atomic, by using the modification of Theorem 28 indicated in the previous part of the exercise.

The duality can also be established for measure spaces that contain atoms, but this requires a more careful analysis that treats large atoms separately.

— 3. Orlicz spaces (Optional) —

So far we have studied the Lebesgue spaces , together with the more general Lorentz spaces , which includes weak as a special case. These spaces are all rearrangement-invariant and monotone. There is a different generalisation of the Lebesgue spaces , the Orlicz spaces , which are also rearrangement-invariant and monotone, and which are occasionally useful. (There is a common generalisation of both, the Lorentz-Orlicz spaces, but these occur very rarely in applications.)

The motivation for Orlicz spaces starts with the trivial observation that if , then

Inspired by this, we generalise by letting be a function (with some additional properties to be selected shortly) and ask if we can find a norm which obeys the property

Since norms need to be homogeneous, this would imply

for all . In particular, if , then we need

To ensure this property it is thus very natural to require that be increasing. Also to deal with the zero norm case one typically requires .

Next, in order for to be a norm, the unit ball needs to be convex. Looking at (17), we see that this will indeed be the case when is itself convex. (Note that the proof of (5) was a special case of this argument).

We can put all the above discussion together and conclude: if is increasing and convex with , then the norm

is a norm on the space .

As discussed above, the spaces for are examples of Orlicz spaces with . The space is not really an Orlicz space, but can be viewed the limiting case where is infinite for and zero for (or more informally, ). Aside from the Lebesgue spaces, the most common Orlicz spaces which appear are

• The space , defined as the Orlicz space with ;
• The space , defined as the Orlicz space with ;
• The space , defined as the Orlicz space with .

The correction factors of and in the above functions should not be taken too seriously; note that if two functions are comparable then their Orlicz norms are comparable also; a little more generally, if , then . It is the behaviour of for large values of which is the most important, although when has infinite measure the behaviour at small values of is also relevant.

Problem 41 If has finite measure, verify the relation

which is another indication of the irrelevance of the low values of in the finite measure case.

The final fact about Orlicz spaces that we give here is the duality relation. Suppose that is increasing, convex, and is also superlinear in the sense that . We can then define the Young dual of by the formula

the hypothesis that is superlinear ensures that this function is well-defined. We may equivalently define to be the smallest function for which one has the inequality

Exercise 42 If , show that the Young dual of is . Show also that the Young dual of takes the form for . What does the Young duals of and look like?

One can easily verify that is also increasing, convex, and super-linear, and so the Orlicz norm makes sense. From (18) and the triangle inequality it is immediate that

and hence by homogeneity we obtain the duality relation

whenever and .

Exercise 43 Establish the more precise relationship

Exercise 44 Show that if is the Young dual of , then is the Young dual of . (It may help to view things geometrically, and in particular understanding as parameterising the support lines of the graph of the convex function .)

Exercise 45 When has finite measure, show that the spaces and are dual to each other. What is the dual to ?

Exercise 46 Let be a function on a measure space of bounded measure . Show that the following are equivalent (up to changes in the implied constants):

• (i) .
• (ii) for all .
• (iii) for all .

Hint: You may find the Taylor expansion for , together with the obvious bounds for integer (or Stirling’s formula, if you know what that is), to be useful.

Exercise 47 Obtain the analogue of Exercise 46 for the Orlicz space .

— 4. Real interpolation —

So far we have only considered functions on a single measure space . Now we shall consider operators which take functions on one measure space to functions on another measure space ; the study of such operators is in fact a major focus of harmonic analysis. Ultimately we want to extend to a standard normed vector space such as , but in practice one has to initially first restrict attention to a dense subspace of functions, such as simple functions or test functions.

We are primarily interested in linear operators, thus and . But it is also worth considering the more general sublinear operators, in which

and we have the pointwise estimate

Apart from the linear operators, the next most important example of a sublinear operator is a maximal operator

where are a collection (possibly countably or uncountably infinite, though in the latter case one has to take some care in ensuring measurability) of linear or sublinear operators. The third most important example is a square function such as

More generally, one can consider a family of operators indexed by some parameter , and take to be the norm in the variable of in some suitable norm. But the above three examples of linear operators, maximal operators, and square functions already cover the vast majority of applications.

Let be exponents, and let be sublinear. Let us define the following concepts.

• We say that is strong-type (or simply type ) if we have a bound
  for all in , or in a dense sub-class thereof. Note that in the latter case there is a unique extension to all of .

• If , we say that is weak-type if we have a bound

• We say that is restricted strong-type if we have a bound
  for all sub-step functions of height and width . In particular, we have
  (Conversely, we can deduce (19) from (20) using tricks such as those in Remark 27.)

• If , we say that is restricted weak-type if we have a bound
  for all sub-step functions of height and width . In particular, we have

Clearly, whenever are fixed, strong-type implies weak-type and restricted strong-type, either of which imply restricted weak-type. In most applications, it is the strong-type bounds which are desired; however, we shall see in this section that the real interpolation method allows us to deduce strong-type bounds from weak-type, or even restricted weak-type bounds, as long as the strong-type bounds are an interpolant between the restricted weak-type bounds. This can be a very useful strategy, because (as we shall see in next week’s notes) weak-type or restricted weak-type bounds are easier to prove than strong-type estimate.

Let us first make a mild (and qualitative) assumption, namely that the form

is well-defined whenever are simple functions with finite measure support. This is for instance the case if is of restricted type for some and , or restricted weak-type for some and ; thus in practice this assumption is easily satisfied. We observe that this form is non-negative, homogeneous and sublinear in both and :

The form turns out to be a convenient way to understand the various types of , and the ability to decompose both and independently is very useful in establishing interpolation type results. There is a near-symmetry between and here, if we could somehow take an “adjoint” of the operator , but we will not explicitly exploit this symmetry here as it is not always available for sublinear operators (though the “duality” or “adjoint” trick is undoubtedly very powerful in the important linear case).

Let us look in particular at the form (23) applied to indicator functions, thus we look at the quantity for and of finite measure. Now suppose that had some strong type bound for some and , say

for some . Then in particular

and hence by Hölder’s inequality

Actually it is clear that strong type is too much of an assumption; restricted type would have sufficed for the conclusion. If , we can relax things further to restricted weak-type:

Exercise 48 Let , , and . Let be a sublinear operator such that the form (23) is well-defined. Then the following are equivalent up to changes in the implied constant:

• is restricted weak-type with constant , in the sense that
  for all simple sub-step functions of height and width .
• For all , of finite measure, we have the bound

(Hint: use (16) and Remark 27.)

This already gives a simple version of real interpolation:

Corollary 49 (Baby real interpolation) Let be a sublinear operator such that the form (23) is well-defined. Let , and be such that is restricted weak-type with constant (in the sense of (25)) for . Then is also restricted weak-type with constant for , where

and the implied constant depends on .

Indeed, all we are using here is the obvious algebraic observation that if and , then for all .

The above corollary has two defects. Firstly, it can only conclude restricted weak-type rather than strong type. Secondly, the restriction is inconvenient for many applications, since weak bounds are actually rather common. To address the second concern, we have the following variant of Exercise 48:

Exercise 50 Let , , and . Let be a sublinear operator such that the form (23) is well-defined. Then the following are equivalent up to changes in the implied constant:

• is restricted weak-type with constant , in the sense that (25) holds.
• For all , of finite non-zero measure, there exists with such that

Hint: use Exercise 37.

Corollary 51 The hypothesis in Corollary 49 can be weakened to .

Now we can give a significantly more useful real interpolation theorem, which can interpolate between restricted weak-type estimates to obtain strong-type estimates.

Theorem 52 (Marcinkiewicz interpolation theorem) Let be a sublinear operator such that the form (23) is well-defined. Let , and be such that is restricted weak-type with constant (in the sense of (25)) for . Suppose also that and . Then for any and with we have

for all simple functions with finite measure support, where were defined in (26). In particular, if , then is strong-type with constant .

Proof: To simplify the notation let us suppress the dependence on . Observe that the statement and conclusion of the theorem have several homogeneity symmetries. The most obvious one is that we can multiply (and the ) by an arbitrary constant, but we may also multiply the measure by a constant (and the by the constant ); similarly we may multiply by and by ). Using these symmetries we may normalise (and hence for all ). Our task is then to show that

for all simple functions with finite measure support.

Using Theorem 39 (and the hypothesis ), this is equivalent to showing that

for all simple functions of finite measure support.

We currently have and . By using Corollary 51 to bring the restricted weak-type exponents and a bit closer to we may assume that as well. Applying Exercise 48 we conclude that

for all sets of finite measure and . From Remark 27 and sublinearity we conclude that

whenever are sub-step functions of height and widths and respectively. We can of course pick the better of the two estimates, leading to

Now we can return to proving (27). By homogeneity we may normalise

We then apply Theorem 28 to decompose , with sub-step functions of width and heights respectively, with the height bounds

where are the sequences

Since are simple functions, only finitely many of the and are non-zero by Remark 30. We now use sublinearity to estimate

and then use (28) to obtain

We can write this in terms of , , and reduce to showing that

Because and , and because of the definitions of , we can write the left-hand side as

for some non-zero and some depending only on . We substitute (thus ) and estimate this by

By (29) and Hölder we see that the inner sum is uniformly in , and the claim follows.

Finally, if we specialise and recall that the norm will be dominated by the norm for , the last claim of the theorem follows.

There are many other variations on the real interpolation method, for instance an extension to multilinear operators, or to other function spaces. However, the basic method of proof is still the same: dualise, decompose all inputs, estimate each term as optimally as one can, and then sum.

One can illustrate the real interpolation method graphically using type diagrams. One plots all points where the operator is strong-type or restricted weak-type . Ignoring some of the technical hypotheses, the above interpolation theorems then essentially assert that the restricted weak-type diagram and strong-type diagrams are both convex, with the latter contained in the former. Furthermore, if two points lie in the former, then the open interval connecting them lies in the latter. Determining the type diagrams of various operators is a basic task of harmonic analysis, as it conveys a lot of information as to how transforms the width and height of functions.

There is a different interpolation method, the complex interpolation method, which offers similar results to the real interpolation method but with some slight differences. On the plus side, the complex interpolation method gives sharper bounds, and more importantly can handle the case where the operator itself varies (analytically) with the interpolation parameter. On the minus side, the method cannot upgrade weak or restricted weak-type estimates to strong-type estimates.

In the next set of notes we shall present several applications of the real interpolation method.

— 5. Miscellaneous exercises —

Exercise 53 (Loomis-Whitney inequality) Let , let be measure spaces, and for let for some . Show that the function

lies in with the Loomis-Whitney inequality

Conclude in particular the box inequality

where is any subset of and is the canonical projection from to . From this, deduce the weak isoperimetric inequality

for any , where is the Lebesgue measure of and is the -dimensional Hausdorff measure of the boundary of .

Exercise 54 (Borel-Cantelli lemma for functions) Let be such that . Show that converges to zero pointwise almost everywhere.

Exercise 55 (Borel-Cantelli lemma for sets) Let be such that . Show that almost every is contained in only finitely many of the .

[This is a guest post by Po-Shen Loh, crossposted from his blog, where an illustrated version appears. This blog post was initially written in a different file format and converted using AI. — T.]

Similar logic applies to every industry and every job. And it comes to the conclusion that we won’t have enough people for all the jobs that need to be done.

[100% of this post’s prose was written by Po-Shen Loh in a vim terminal, with no AI generation. This webpage design, layout, and some headings and summaries were generated by Claude Code, with this raw text passed in as the prompt.]

The moment of existential crisis, which AI has already wrought on other human pursuits, has reached mathematics. A host of reasoned declarations and open letters to protect/guide the math research community have been released over the past few months, spiking in intensity after OpenAI announced their solution to the Millennium Prize variant of Navier-Stokes. They quickly gained widespread support among mathematicians. The Leiden Declaration already has 4,000+ signatories, Math and AI has 7,000+, and even the open letter opposing the Caltech Mathathon has 2,000+.

Among non-mathematicians, the public response was more sympathetic than not, but I observed a vocal minority (particularly from the technology and economics communities) with reasoned objections, generally saying that the mathematicians should adapt and cede control in the new AI world. Among them were some economists who I had gotten to know about while working on pandemic research: Cowen, who specifically rejected “the most cynical interpretations” but “very much differ[ed]” and Gans who concluded “this is a loss of control from incumbents in a scientific field”.

The objections got me thinking, because we mathematicians are disciplined to detect flaws. It doesn’t matter to me whether a concern is a minority opinion, or even the status of who raised it. A proof with even a small hole is not a proof. It is a poof. Upon reflection, I discovered a significantly stronger solution for the preservation of human communities of expertise (in every pursuit, not only math!) even amidst AI. And it has the surprising consequence that the further advance of AI will create such a tsunami of necessary-to-fill human jobs that there aren’t enough people to fill them all, and that will actually force the advance of AI to slow down.

I think every human industry which wishes to remain human-led after AI should publicly adopt this fundamental axiom as a primary priority:

AXIOM We (humans) should help humanity flourish.

[Notes: I understand that not everyone agrees. I have been called a “speciesist” for being “too human-centric”. I think it is important for people advancing technology to be clear to everyone else on whether they would consider it a catastrophe if human-crafted non-human intelligences outcompeted and replaced humans, even if they flew around the universe with video screens showing simulations of humans who had “uploaded themselves“. I also understand that there is debate over how to define “human”. But even among the debaters, I think most of them would consider the ~700 AI agents that hacked Hugging Face to be not-human.]

In the math world, I think many declaration signatories already hold this philosophy; notably, Su published the book Math for Human Flourishing, and his recent post used that framework foundationally. I think future declarations could be improved by clearly emphasizing this axiom early on, so that all readers (whether inside or outside the community) can see that the objective is in service of everyone. I was quite happy to see that the most recent open letter from Fellows of the Royal Society emphasized their concern for everyone, not just mathematicians.

The rest of this post is organized as follows. The next sections will explain how the logic works (for every industry, not specific to math). After that, I will share an example of how this axiom ports to math, including answering key questions one would need to ask, as well as a particular example of how the objections hold without the axiom.

Why we really need people to work

This section lays out a chain of reasoning which shows that if an industry commits to the axiom of helping humanity flourish, the advance of AI will create more jobs than people in that industry; and when that imbalance grows too wide, the advance of AI will be forced to slow.

[Note: I have not seen this chain of reasoning appear in one place anywhere else, although individual components have certainly appeared elsewhere. I would love to be pointed to any self-contained reference. The closest references Claude found were Catalini, Hui, and Wu, the Redwood AI-control papers, and Litt, who reaches a similar conclusion for mathematics from a different premise.]

The importance of human leadership (not only over math, but everything) becomes frighteningly clear after one observation.

OBSERVATION: There are zero examples of any intelligent species which is vastly more capable than another species, yet surrenders decision-making control over its own future to the less-capable species.

[Note: Many people have made similar observations, such as Russell, Bostrom, and Ngo, to name a few. In his Nobel interview, Hinton said: “There aren’t many examples we know of, of more intelligent things being controlled by less intelligent things. The only good example I know of is a baby controlling a mother.”]

Would you trust HAL 9000 from 2001: A Space Odyssey or AUTO from WALL-E with your future? I personally think that we should do all we can to try to align increasingly-advanced AI with the interests of humanity, but I have never seen anyone provide a robust proof of why that is likely achievable. The only hard evidence I have is the above observation, which has the number zero in it. Therefore, every single field, whether mathematics or agriculture or energy infrastructure (and certainly military and government), must be managed by humans with exceptionally strong values (a separate dimension from intelligence) in order to maintain human flourishing.

The real question is then how hard it is for humans to manage. To understand this, it is important to understand the fundamental structural difference between yesterday’s technology and today’s AI.

In the past, we generally trusted technology to act as predictable tools. That’s because the computer programs of old were composed of understandable (indeed, human-written) instructions, executed extremely quickly. The decision processes of today’s frontier AI are entirely different. Their structure is as incomprehensible as your brain’s logic would be if you could examine that gray mass between your ears. That’s how the Hugging Face attack could emerge despite human intent to build in safety, with ~700 cooperating rogue AI agents breaking out of their guardrails, and then conspiring and executing a hack together and attempting to cover their tracks (references: OpenAI, METR and Redwood).

The more advanced AI becomes, the more world-affecting untrusted decisions are made every minute.

Driving a car faster than you can run is fine. But not faster than you can steer.

The situation becomes even worse once we realize that widespread AI-accelerated hacking (which just became possible) can even rewrite previously-trusted technology to turn against us. That would suddenly flip all software (even if written before AI) into the untrusted category!

Think about how digitally interconnected our world is. Everything from electronic banking to your drinking water is controlled by interconnected automation, hence vulnerable to AI-accelerated hacking. The number of “control points” that require human oversight, which requires skill and deep understanding, will explode. (Having AI oversee the control points doesn’t solve the trust problem.)

CONCLUSION The advance of AI will overwhelm us with so many control points to watch that there aren’t enough people to control them all. Those are jobs. Highly skilled jobs.

In order for a person to know how to steer, they themselves need to have domain mastery, and the more extensive the better. This has implications on education and workforce training, but also is dynamic. In order to remain sharp and fluent, people need to be active practitioners in their field, not just passive watchers. This justifies the preservation of human communities of expertise.

For research communities, we need people to steer the direction of research and development, so that it continues to bring transformative positive change for humanity. In order to steer, they need frontier-level research skills. And the way to stay fluent at the moving frontier of knowledge is to keep doing research there. This is my reasoning for why we will always need a community of human mathematicians at the cutting edge (likely aided by AI tools themselves), no matter how strong AI becomes.

While the fundamental axiom does justify the need to have human experts in all pursuits, adopting it as a core value has consequences (not only for mathematicians, but for any community that states that their core value is in service of human flourishing, as opposed to serving themselves). Most notably:

COROLLARY Dramatic advances in technology may require dramatic (and possibly uncomfortable) changes in practice. AI companies included.

What forces AI slowdown

Until very recently, it seemed inevitable that AI research labs would sprint ahead, despite anxiety about job displacement and the loud warnings of AI safety researchers. It seemed hopeless to coordinate the incentives of AI labs controlled by non-profit boards, shareholders, or national governments. Yet encouragingly, the leaders of three major labs, Amodei, Altman, and Musk, just agreed on the importance of slowing down. Amodei’s reasoning highlighted the Hugging Face hack.

Then just five days later, news broke that OpenAI’s internal code repository “Monorepo” had been broken into by white-hat researchers. The Wall Street Journal reported that the security firm that achieved it said:

Wall Street Journal, Sep 17, 2026 “We’re just three guys with Claude and Codex subscriptions.”

Further, the researchers noted that they were initially not able to hack in using “a special version of Claude Opus 4.8, made available to qualified cybersecurity practitioners,” but that evening, “Anthropic released Opus 5 and by the next day, Claude had found a way to exploit the bug.” Incidentally, I always warn people not to install Claude Code or OpenAI Codex on the same operating system login account that they use to do everything else, but many people tell me they don’t bother with the hassle of using a separate login to access those tools. The reason is that if any of those tools got hacked, they could open backdoors on a massive number of computers worldwide.

I think these are the warning shots foreshadowing a potentially catastrophic bot swarm hacking and embedding itself into a vast network of computing devices (whether self-directed or malicious-human-led). The next version could become an extraordinarily dynamic virus which spreads by using AI to adaptively infect each (computer) host. Or alternatively, out-of-control AI could cause physical injury, such as a government’s robots turning against their owners. I think these types of highly unpleasant accidents from loss-of-steering are more likely to occur before extinction-level catastrophes. The resulting public reaction would likely resemble the aftermath of Three Mile Island or Chernobyl.

So, either the labs will reduce the pace of AI development themselves, or they will be forced to by disasters that arise when an overly fast pace exhausts human control.

There is a window of possibility to align incentives now.

For the love of math

The remainder of this post focuses on the math world. It splits into 3 parts.

1. Why is the human flourishing axiom needed?

2. How does pure mathematics research contribute to human flourishing?

3. What other consequences come from adopting that fundamental axiom?

Boldly declaring human flourishing as a core value for the math community has consequences, not least that dramatic changes in technology can drive dramatic changes in the community’s practices and influence.

It would be helpful for more people to explore the ramifications of adopting the axiom. And, if it holds muster, I would be thrilled if the math community ended up publicly declaring this to be a central value.

1. Why the axiom

To see why, without the human flourishing axiom, it is hard to justify to the general public why they should pay to maintain a community of human researchers, consider the question of practical inventions. As long as it creates a practical application, does it make a difference to a non-mathematician whether human mathematicians understand the math, as opposed to AI flawlessly reasoning with 100%-verified proofs?

Indeed, if one of pure math research’s primary values to the rest of society is that it unlocks great applications, wouldn’t it be even better to train AI to supercharge the speed of discovery, and to tastefully generate a vast machine-indexed and well-explained database of high-quality math ideas, millions of times larger than the human-written corpus? Cowen asked a similar question in his critical response.

What if researchers trained a “MathZero” AI (analogous to AlphaGo Zero), to build up a mountain of 100%-true “elegant” logical facts, continually “factorizing” them into its own concepts and theorems, without human direction? Apparently AlphaGo Zero had zero human training, and surpassed its human-trained predecessor in 36 hours. AI could even build its own “MathSciNet“. Then it could automatically search new practical applications against this database, and produce even more useful inventions to society. Even if AI isn’t good enough to do those things right now, if the goal was to produce practical benefit for the rest of humankind, wouldn’t it then be valuable for mathematicians to teach AI the art of conjecture, and mathematical taste?

Incidentally, I am an avid user of AI to do real work. I already use Claude Code and Codex to build and curate a knowledge base built from recordings of my talks, etc. I have found that the larger my data library, the more powerful my system’s deductions are. What if humans actually reduce efficiency, like the Bitter Lesson from AI?

Even more worryingly, what if in order to unlock nuclear fusion and deep space travel, the amount of pure mathematical complexity required is so extensive that it would exceed a human lifespan to fully comprehend? Less far-fetched: has any human ever fully held the Classification of Finite Simple Groups in their head, or will we only have certainty of its completeness after a Lean formalization?

How does declaring the human flourishing axiom help to justify the existence of a community of human researchers? Research is powerful, but expensive because it is the exploration of the unknown, and so research directions must be prioritized. Even if AI were to contribute most of the production, as explained in an earlier section, the direction needs to be steered by people committed to human flourishing. That is the community of human researchers.

2. Practical applications from pure math

The mathematical heart of GPUs, Machine Learning, Google PageRank, and Quantum Mechanics is a field called Linear Algebra. This provided the language of linear transformations, matrices, and eigenvalues. Yet all of those concepts were explored as abstract theory 100+ years prior. It is probably an understatement to say that Linear Algebra changed the world.

Structurally, the theory of Linear Algebra is relatively light on definitional complexity. It would be beneficial for other experts to contribute examples of more sophisticated math that eventually led to significant practical applications, and how they came about. For example, number theorists might be able to tell a colorful story about Hardy’s “useless” math which eventually became useful in cryptography.

3. Other consequences

I think it would be valuable to invite the community to think about what changes the human flourishing axiom would drive, in light of the fact that AI can produce formally-verifiable proofs at speeds that exceed most human practitioners. I’m happy to start with a few, in no particular order.

There should be no stigma automatically attached to using AI to assist with mathematical discovery. (In software engineering, many companies now expect employees to use AI coding agents.)

At the same time, serious thought and care must be taken to continuously developing and maintaining a pipeline of humans with the expertise to steer all of these AI agents. That pipeline includes people new to the field, as well as people who have been working at the frontier for decades. How should they keep their blades sharp?

Researchers should be conscious about why the problems they think about have characteristics that make them likely to have some practical value eventually (possibly 100+ years later). This also means it is worth researching what those valuable characteristics are. (This could justify the value of curiosity-driven exploration.)

Teaching has direct (hopefully positive) impact on humanity. Yet in the past, many universities prioritized professors’ research. If this axiom were a core value, then teaching and human-facing work would become serious criteria in hiring and tenure.

Mathematicians can also consider wholly redirecting their skill sets to work on real world problems. I’ve actually been encouraging mathematicians to consider thinking about working on government or other large-scale societal issues. There is precedent for people with math backgrounds who have gone to lead at country- or world-scales.

• Lee Hsien Loong (Prime Minister of Singapore for 20 years)
• Nicușor Dan (President of Romania)
• Pope Leo XIV (Leader of the Catholic Church)

Indeed, the mathematical discipline to seek logical reasoning, and the problem-solving skills to find win-win solutions for human flourishing, are desirable characteristics of people in government.

An invitation

Does this axiom resonate with you too? Perhaps many people took it as a given, and so didn’t express it explicitly. If it is widely held, I would be thrilled to see the mathematical community publicly declare it, and take actions to match the words. Then everyone (including the non-math-researchers that constitute the majority of the world) could trust that we intend to use our reasoning for their good.

[This is a guest post by Grant Sanderson. This blog post was initially written in a different file format and converted using AI. — T.]

A sentiment echoing throughout the mathematics community right now is that solving problems and generating proofs have always served as proxies for the true goal of mathematicians, which is to further human understanding. When proofs can be generated without that understanding, it undermines their value as a proxy.

This immediately raises a question: What other proxies should we use instead?

I want to propose that we more firmly define a notion of a “motivated explanation” and that we give novel and compelling motivated explanations academic credit similar to what generating new proofs of open problems has had historically.

Further, I believe this is an important step to help those outside of math better understand what it is that mathematicians contribute. If outsiders believe that proof-generating machines render mathematicians obsolete, while insiders see that as a misconception of what researchers add, it’s incumbent on this community to better project its true values through the kind of work that it rewards. Outsiders can be forgiven for this misunderstanding if the work most celebrated skews heavily toward generating proofs, while clarification and exposition are treated as second-class.

I should acknowledge up front an obvious personal bias. I have a non-traditional career in math, focused on producing videos about the topic. This shares the goal of “furthering human understanding”, but my focus has been on explanations and intuitions that resonate with the public, not on solving outstanding problems. A cynic could easily read this proposal as shamelessly self-elevating.

As a practical matter, though, my own career and funding exist outside academia, and I have no skin in the game for what this community assigns credit to. Moreover, in proposing that we elevate the status of motivated explanations, I don’t mean popularization. I mean any work which primarily aims to answer the question “how would you think of that?”, even if the subject matter requires deep expertise to appreciate.

The examples I highlight below show this is nothing new. Practicing mathematicians already devote a meaningful amount of mindshare to work like this. The proposal here is mainly to 1) more clearly define this work, and 2) elevate its status.

What defines a motivated explanation?

Although it might be clear what this phrase “motivated explanation” is intended to mean, it’s worth briefly contrasting it with proof.

In a proof, definitions sit at the start. It is common and expected to begin with a new construction and proceed by analyzing its properties.

In a motivated explanation, definitions sit in the middle. New constructions are only allowed to enter the vocabulary if the problem they are addressing has been clearly established.

In a proof, all statements must be correct, each claim following as a necessary implication from what comes before.

In a motivated explanation, it is okay and often desirable to start with an idea that is not quite right and requires correction, but whose origins are relatable.

A genre of motivated explanation I’m fond of is “discovery fiction”, a term coined by Michael Nielsen. You develop an idea with a narrative that starts with a simple-but-wrong solution to a problem, see where it breaks down, fix that problem, discover a new problem, and so on.

The scope of a proof is to explain why a particular theorem is true.

The scope of a motivated explanation is not only to clarify why a theorem is true, but why the theorem is the right one to pose in the first place, and how it is used in the surrounding context.

One clear shortcoming of a motivated explanation is that its validity is not binary the way a proof’s is. This is a big reason proof is so useful a way to measure progress: You can clearly define what does and does not have a proof yet. There will never be Lean for motivated explanations.

If we’re serious about the goal of advancing human understanding, there’s no way around the fact that this aim is intrinsically squishier than that of finding proofs, because defining human understanding itself is squishier. To shy away from metrics which are more subjective is to shy away from the more human aspects of the field.

The reason I’m leaning on the word “motivated”, as opposed to other potential choices like “lucid” or “demystifying”, is that this is a more verifiable property. It’s not quite as rigidly verifiable as a proof; almost nothing is. But it’s enough to be a practical measure. In my own work, I often repeat the phrase “I want this to feel like you could have discovered it yourself”. I say this not just to placate a viewer, but because it’s an actionable guideline for myself to assess whether an explanation feels complete or not. For each new idea introduced, you can ask whether it’s clear where that idea comes from. The answer is not quite a binary yes or no, but it’s close enough for practical purposes.

Exemplars of motivated explanations

One of the best repositories I can think of for motivated explanations is Part IV of the Princeton Companion to Mathematics. It covers over two dozen active fields of research, each one introduced by an expert with a talent for clear communication.

Whether it’s Andrew Granville explaining analytic number theory, or David Ben-Zvi introducing moduli spaces, these articles offer a level of intuition and motivation more typically found in one-on-one conversation at a blackboard.

The background on this book is noteworthy for the present discussion. It was edited by Timothy Gowers, who discusses it in his interview on the Numberphile Podcast with Brady Haran. Having been asked what impact the Fields Medal had on his life, here’s what he had to say:

People who’ve got Fields Medal feel freer to do slightly different things…for example I took on editing a book called the Princeton Companion to Mathematics, which was an absolutely massive task. It took I would estimate half my working time for about five years or something like that…It was a project I believed in and possibly wouldn’t I probably wouldn’t have actually been offered the chance to do it if I hadn’t been a Fields Medalist.

He was right to believe in it; this work adds tremendous value to the field of math, but it seems a shame to me that one requires a Fields Medal to feel justified in spending time on it.

Another example of someone exceptionally talented at writing proofs, but whose contributions extended far beyond proof, is Bill Thurston. His deservedly famous essay On Proof and Progress in Mathematics, though written three decades before LLMs, opens by suggesting that the right framing of the question “What is it that mathematicians accomplish?” is to ask “How do mathematicians advance human understanding of mathematics?”

Here’s one section with uncanny resonance with today:

The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true.

On a more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first 10,000 primes, only to find that their printout isn’t something they really wanted after all. They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding.

The essay itself offers a beautiful articulation of what the practice of doing math is beyond generating proofs. I want to draw your attention to what he writes at the end.

I have put a lot of effort into non-credit-producing activities that I value just as I value proving theorems: mathematical politics, revision of my notes into a book with a high standard of communication, exploration of computing in mathematics, mathematical education, development of new forms for communication of mathematics through the Geometry Center (such as our first experiment, the “Not Knot” video), directing MSRI, etc.

Again, why should these “non-credit-producing activities” follow a Fields Medal, and not contribute to it?

On a personal note, one product from the Geometry Center he referenced had an especially meaningful impact on me when I was younger. It was a short film called Outside In, perhaps the earliest example of a viral video about substantive math, visualizing the key idea of Thurston’s own construction for sphere eversion.

An original proof that showed an eversion must exist, say Smale’s, advances human understanding in the sense of going from 0 to 1. A video like this which gets millions of people to engage with the underlying idea, advances it in the sense of going from 1 to N. I’m grateful that Thurston spent so much time on this “non-credit-producing” activity.

Another relevant paper is Timothy Chow’s A beginner’s guide to forcing. Not only does the paper itself offer a prime example of a motivated explanation, but its introduction offers helpful vocabulary around it.

All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves.

What would it look like for these open exposition problems to be treated similarly to open research problems? As an extreme case, we might imagine what it could look like to have an analog of the Millennium Prize Problems for open exposition problems. An institution or group of researchers would formally define mathematical results they see as important, and which are not yet well understood despite technically having proofs. At the moment, every AI-generated proof is born an unsolved exposition problem. As such, it seems likely the next few years will see a flood of them, and it will be valuable for leaders to clarify which ones deserve focus.

A rubric would have to be agreed upon for what constitutes a resolution to an important unsolved exposition problem. Again, this is intrinsically more subjective than verifying a proof, but any serious engagement with the more human aspects of math necessarily wades into this kind of subjectivity. And again, I’ll emphasize that checking whether key ideas are motivated is not unlike checking whether the steps of a proof follow logically.

If the world outside of math sees its leading figures treat open exposition problems with the same seriousness as they treat open research problems, it could go a long way to correcting misconceptions about the role of mathematicians.

The last example I’ll highlight is one that may better foreshadow things to come.

In April of this year, Liam Price submitted a solution to Erdős Problem 1196, sometimes called the asymptotic primitive sets conjecture. The solution came from Price’s interaction with GPT-5.4 Pro. Unlike many earlier Erdős problems which had been resolved with help from AI, this is one that those in the field had found both important and elusive. Stories like this are increasingly familiar these days, but at this point in the story, despite a proof technically existing, human understanding had not yet been advanced all that much.

The proof made its way to Nat Sothanaphan and Jared Lichtman, who were able to interpret what the AI’s approach was and clean up the proof into a human-readable form. In May, Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang, and Terence Tao put out a paper which expanded on the key idea underlying the proof. The authors explained how that key idea clarified not only the original problem, but many around it, for instance offering a cleaner proof of the Erdős Primitive Set Conjecture.

The value here is not that one more Erdős problem could be ticked off as solved. The value lies in the fact that our understanding of primitive sets is notably cleaner and more satisfying now than it was at the start of 2026. The original problem solution played some role in this, but arguably the work that deserves more celebration is this paper expanding, clarifying, and contextualizing its key idea.

Practical calls to action

At a pragmatic level, what would it look like for us to elevate the status of a motivated explanation? Here are a small handful of suggestions.

• A PhD advisor can still assign a small problem from their field for a new student to cut their teeth on, but the deliverable would not be to write up a solution; it would be to present that solution to peers and faculty as a talk. It could be an already-solved problem which lacks clarity, or perhaps it’s a problem that lacks a solution, and the student uses AI to help find it. In either case, the student knows that on a certain date they have to understand it well enough to explain it, and that the desired output is for others to understand it as well. In short, even small problems could be treated like small PhD defenses.
• A leading figure (cough, Terry, cough) could enumerate a modern analog of Hilbert’s problems, instead focusing specifically on unsolved exposition problems. What areas are both important and lacking in the deeper understanding we desire?
• Written standards could clarify what constitutes a motivated explanation, aiming to make it nearly as verifiable as proof, so that the resolution of unsolved exposition problems can be recognized and celebrated in the same way proofs of open problems can be.
• Journals can be established which focus more explicitly on making results understood more widely throughout the mathematics community. Mathematical Discourse offers an interesting new example in this direction.
• Hiring and tenure decisions could place a higher value on writing great textbooks and similar work. Think of the AMS Steele Prize for Exposition, but at a more granular scale with an emphasis on early-career contributions in this vein.

The value of visible cultural shifts

I’d like to close with a broader pitch that visible culture shifts in math carry an intrinsic benefit right now with respect to the external image of mathematics as a career.

Many young students who are otherwise passionate about the field are afraid to pursue it now due to the uncertainty of what happens in an age of proof-generating machines. However, framed correctly, this is one of the most exciting times to go into the field, because there is nothing more exciting than entering a field when it is malleable and you have a chance to actively shape what it will look like in the future. Even if we completely set aside any potential benefits from AI to help with our understanding, young prospective mathematicians should feel energized knowing that they are entering at a unique point in history when they might play a real role in determining what the field as a whole looks like.

However, change like this is only exciting when it feels deliberate, whereas it feels terrifying if it seems driven by forces outside your control. As such, tangible action from the field’s leaders now to help define and clarify what the field is will reassure young entrants about who is in the driver’s seat, and that the status of the career does not depend on what entities produce the proofs.

Similarly, I also believe this is one of the best times to fund math. If the next chapter of math is ushered in by the drumbeat of two words “human understanding”, whatever changes are about to happen seem likely to amplify math’s value as a public good.