# Seminar za analizu i algebru Alpe-Jadran

Poštovane kolegice, poštovani kolege,

Drugi sastanak **hrvatsko-slovenskog Seminara za analizu i algebru**

**Alpe-Jadran** bit će održan u subotu 9. studenog, na Fakultetu za

matematiku i fiziku u Ljubljani, prema sljedećem rasporedu:

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10.00 - 10.45

**Peter Šemrl**

**Title: Adjacency preserving maps**

**Abstract:**

In the last 20 years most of my research was closely connected to the

problem of describing the general form of adjacency preserving maps.

The notion of adjacency is very simple and the problem of finding the

general form of such maps can be easily understood by an undergraduate

student. But it turned out that this elementary linear algebra problem is

quite difficult and closely related to various parts of mathematics such as

geometry, operator theory, and mathematical physics. I will describe the

problem, some of the most interesting results, ideas, and connections

with other parts of mathematics and physics.

10.55 - 11.40

**Andrej Dujella
Title: Elliptic curves and Diophantine m-tuples**

**Abstract:**

In this talk, we will describe some connections between

Diophantine m-tuples and elliptic curves.

A rational Diophantine m-tuple is a set of m nonzero rationals

such that the product of any two of them increased by 1 is a perfect square.

The first rational Diophantine quadruple was found by Diophantus.

It is known that there are infinitely many Diophantine quadruples in integers

(the first example, the set {1,3,8,120}, was found by Fermat),

and He, Togbe and Ziegler proved recently that there are no

Diophantine quintuples in integers.

Euler proved that there are infinitely many rational Diophantine quintuples.

In 1999, Gibbs found the first example of a rational Diophantine sextuple.

It is still an open question whether there exist any rational

Diophantine septuple.

We will describe several constructions of infinite families of

rational Diophantine sextuples.

These constructions use properties of corresponding elliptic curves.

We will show how Diophantine m-tuples can be used in

construction of high-rank elliptic curves over Q

with given torsion group.

11.50 - 12.35

**Oliver Dragičević
Title: p-ellipticity**

**Abstract:** We introduce a condition on complex accretive matrices which

generalizes the notion of ellipticity. By presenting several examples,

we argue that the condition might be of interest for the L^p-theory of

elliptic PDE.

12.45 - 13.30

**Ljiljana Arambašić
Title: On orthogonalities in Hilbert $C^*$-modules**

**Abstract**: The notion of orthogonality in an arbitrary normed linear

space may be introduced in various ways. Let us mention only the

Birkhoff—James orthogonality and the Roberts orthogonality: if $x, y$

are elements of a normed linear space $X,$ then $x$ is orthogonal to

$y$ in the Birkhoff—James sense if $\|x+\lambda y\|\ge \|x\|$ for all

scalars $\lambda$, and $x$ and $y$ are Roberts orthogonal if

$\|x+\lambda y\|= \|x-\lambda y\|$ for all $\lambda$. A special class

of normed spaces are Hilbert $C^*$-modules where, besides B-J and

R-orthogonality, there is also orthogonality with respect to the

$C^*$-valued inner product. Since the role of scalars in Hilbert

$C^*$-modules is played by the elements of the underlying

$C^*$-algebra, it makes sense to introduce modular versions of BJ and

R-orthogonality and study their relations with existing

orthogonalities in a Hilbert $C^*$-module.

Pozivaju se članovi seminara i svi zainteresirani.

Pozdrav,

Ilja Gogić