The Schedule

16th of December

10:15 - 10:30

Reception

André Carvalho and Lennart Obster

10:30 - 11:20

A lemma connected with Special Functions having two indices - II

Pedro Ribeiro

Abstract: In a previous independent talk, we have presented new integral formulas for a generalized Bessel function. Using these, we have indicated which kind of Eisenstein series have this Bessel function in their Fourier expansions.
In this talk, we embark on the study of the zeros of functions related to these Eisenstein series. Although the most part of them are not Dirichlet series in a classical sense, we shall see how our previous work fits in this broader framework.
If time allows, we will also discuss a conjecture of Steuding about the value-distribution of a family of series covered by our study.

Coffee Break 11:20 - 11:40

11:40 - 12:30

Towards a "dictionary" between commutative algebra and graph theory

Gonçalo Varejão

Abstract: Combinatorial commutative algebra is the field of mathematics dedicated to the study of algebraic structures, namely modules over polynomial rings, defined from some combinatorial object, and how their properties relate to the combinatorial properties of that object. In this seminar, we begin by showing the main classes of modules studied in this subject: the Stanley-Reisner ring; the edge ideal, and edge subring; the binomial edge ideal; the Eulerian ideal, which we highlight; among others. Afterwards, we focus on one of the most important invariants of these modules, the Castelnuovo-Mumford regularity. We define it through the construction of graded resolutions. Then, we present a result that motivates the current research on the regularity of powers of ideals, a topic of intensive investigation in the past decades. To conclude, we give the characterization of the regularity of the Eulerian ideal, and discuss basic ideas towards obtaining the regularity of its powers.

Lunch 12:30 - 14:00

14:00 - 14:50

Deformations of Lie algebras and Lie algebra morphisms

Sebastián Alfonso

Abstract: In deformation theory, we study a structure on a set, up to some notion of equivalence, by studying the space of possible structures around it, modulo this equivalence. Two main questions in deformation theory are:

1) When are all the nearby structures equivalent to the initial one? If this happens, the structure is called rigid.

2) Which properties are shared by nearby structures? Such properties will be called stable.

To answer these questions, we will use analytical and cohomological tools to encode the obstructions for rigidity or stability. We are interested in the questions mentioned above, because they give us more insight into our initial structure and set. For example, the ridigity of a structure can give us nice local decompositions (normal forms), and the stability of some property allow us to promote it to nearby structures. I'll introduce rigidity and stability problems of Lie algebras and of Lie algebra morphisms and show you which cohomologies control them.

Break 14:50 - 15:00

15:00 - 15:50

Computation of the commutation graph for the longest signed permutation

Diogo Soares

Abstract: Using the standard Coxeter presentation for the signed symmetric group on n+1 letters, two reduced expressions for a given signed permutation are in the same commutation class if one expression can be obtained from the other one by applying a finite sequence of commutations. The commutation classes of a given signed permutation can be seen as the vertices of a graph, called the commutation graph, where two classes are connected by an edge if there are elements in those classes that differ by a long braid relation. In this talk we are going to present some results obtained in my master thesis about the commutation graph for the longest signed permutation. Given the dimension and complexity of these structures, it was also developed an application in which we can visualize and interact with the graph for some values of n.

Break 15:50 - 16:00

16:00 - 16:50

Root systems and root data in the classification of Lie algebras and algebraic groups

Janet Flikkema

Abstract: A root system is a subset of an euclidean space satisfying certain geometrical axioms. Root systems play an important role in the classification and representation theory of semisimple Lie algebras. In fact, the classification of Lie algebras is where the concept of a root system was first introduced. A slightly more general notion is that of a root datum, which arises analogously in the classification of reductive algebraic groups. In this talk, we will introduce the basic definitions of root system and root datum and discuss some results, such as the classification of root systems by Dynkin diagrams. We will show how they play a role in the classification of Lie algebras and algebraic groups and, if time allows it, also their representation theory.

Coffee Break

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