SAAAJ (Seminar za analizo in algebro Alpe-Jadran)

Novo-ustanovljeni SAAAJ (Seminar za analizo in algebro Alpe-Jadran) bo imel drugo srečanje v soboto, 9.11.2019, v Ljubljani. Naslovi in povzetki sledijo. Vsi, ki se nam zelite pridruziti na kosilu, se javite pri Marjeti do cetrtka, 7.11. Vljudno vabljeni! Marjeta Kramar Fijavž in Igor Klep ------------------- 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 Arambasic 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.