FB 6 Mathematik/Informatik

Institut für Mathematik


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SS 2019

16.04.2019 um 16:15 Uhr in 69/125:

Alexandros Grosdos Koutsoumpelias (Universität Osnabrück)

Combinatorial methods for algebraic problems in graphical models

Graphical models encode causal relations among random variables and find a range of applications in statistics. In the case of Gaussian random variables, the covariance matrix associated to the graph consists of rational functions in a polynomial ring. Goal of this talk is to shed light on algebraic questions related to the covariance matrix, such as finding its ideal or studying the fibers of the corresponding map. We present three combinatorial tools to do this, namely treks on graphs, graph decomposition and nested determinants.

28.05.2019 um 16:15 Uhr in 69/125:

Antonio Macchia (FU Berlin)

The minimal cellular resolution of the edge ideals of forests

I will present an explicit construction of a minimal cellular resolution for the edge ideals of forests, based on discrete Morse theory. In particular, the generators of the free modules are subsets of the generators of the modules in the Lyubeznik resolution. This procedure allows to simplify the computation of the graded Betti numbers and the projective dimension.
This is a joint work with Margherita Barile.

04.06.2019 um 16:15 Uhr in 69/125:

Milena Wrobel (MPI Leipzig)

Cox rings and T-varieties

Cox rings are a rich invariant of algebraic varieties with finitely generated divisor class group. Beside their theoretical meaning they play an important role for explicit methods and algorithmic approaches in algebraic geometry. In the talk we start with a discussion of the well known example class of toric varieties. Following this we will look at torus actions of higher complexity, i.e., the dimension of the general orbit is less than the dimension of the variety. I will discuss structural properties of Cox rings of these varieties and present new combinatorial methods and their applications in the classification of Fano varieties.

11.06.2019 um 16:15 Uhr in 69/125:

Janina Letz (University of Utah)

Local to global principles for generation time over commutative rings

In the derived category of modules over a commutative ring a complex G is said to generate a complex X if the latter can be obtained from the former by taking finitely many summands and cones. The number of cones needed in this process is the generation time of X. I will present some local to global type results for computing this invariant, and also discuss some applications.

25.06.2019 um 16:15 Uhr in 69/125:

Jose Alejandro Samper (MPI Leipzig)

Slicing matroid polytopes

We introduce the notion of a matroid threshold hypergraph: a set system obtained by slicing a matroid basis polytope and keeping the bases on one side. These hypergraphs can be then used to define a class of objects that slightly extends matroids and has a few new techincal advantages. For example, it is amenable to several inductive procedures that are out of reach to matroid theory. In this talk we will motivate the study of these families of hypergraphs, relate it to some old open questions and pose some new conjectures. To conclude we will show that the study of these objects helps us find a link between the geometry of the normal fan of the matroid polytope and shellability invariants/activities of independence complexes.

20.08.2019 um 16:15 Uhr in 69/125:

Vincent Delecroix (LaBRI Bordeaux)

Dirichlet domain computation for real hyperbolic groups

A real hyperbolic group of dimension d is a discrete subgroup of  SO(1,d) (the special orthogonal group of a quadratic form of signature (1,d) seen as the isometry group of the real hyperbolic space of dimension d). The main studied cases are d=2,3 which are the so-called Fuchsian and Kleinian groups.
In various situations, one has access to generators of a real hyperbolic group and would like to study its properties (e.g. find relations among the generators). A possible approach for this problem is the construction of a fundamental domain with respect to its action on the hyperbolic space. The Dirichlet fundamental domain is a systematic way to construct one.
After introducing the problem, I will provide several motivating examples. Then, I will discuss the algorithmic part which involves the incidence geometry of polyhedral cones and quadratic optimization (over number fields).