The main goal of this Seminar Block Meeting is to increase the cohesion between the students of the IRTG, who are spread between different universities inside Germany. The students will be able to meet in a dedicated event, for some of them for the first time, and to get to know each other. 

In order to motivate the event there will be research talks given by some participants, in the same fashion as the regular IRTG online seminars. This will be the occasion for them to practice their communication during 50-minutes talks and to explain to the rest of the students which kind of research they are doing. Overall, the benefits for each participant will be to learn to connect with other mathematicians and to get introduced into research topics outside their expertise area(s). The event will be punctuated with different non-mathematical cohesion times during the three days.

• Diego A. Robayo Bargans (RPTU Kaiserslautern-Landau)

Title: Moduli of tropical covers and gonality

Abstract: Ranks of divisors on discrete (and metric!) graphs have been a general topic of interest in both combinatorics and tropical geometry. This talk focuses on the gonality of tropical curves, namely the case of rank 1 divisors on metric graphs. One way of producing rank 1 divisors on a specific curve is by tropical covers to trees from (tropical) modifications of the curve. In this talk I will give a short (and self contained) introduction to tropical curves, their covers, the respective moduli spaces and the gonality loci.

• Nicolas Faroß (Saarland University)

Title: Spatial partition quantum groups

Abstract: We begin with a short introduction to compact matrix quantum groups. These were first defined by Woronowicz and generalize classical groups of unitary matrices. Then we focus on some concrete examples which can be described by the combinatorics of so called spatial partitions.

• Jonas Hetz (Universität Stuttgart)

Title: Characters and character sheaves of finite groups of Lie Type

Abstract: An important task in the representation theory of finite groups is the determination of their character tables. As the classification of finite simple groups shows, the main difficulties in this context concern the finite groups of Lie type, which arise as an infinite series of finite groups associated to a certain algebraic group over a field of positive characteristic. In order to generically tackle the problem of determining the character tables of finite groups of Lie type, Lusztig developed the theory of character sheaves in the 1980s. In this framework, due to the work of Lusztig and Shoji, the problem is in principle reduced to determining certain roots of unity. We report on some recent progress in this area.

• Linda Hoyer (RWTH Aachen University)

Title: Orthogonal determinants of principal series characters of $GL_n(q)$

Abstract: For a finite group G, it is well-known that an ordinary irreducible character can be afforded by a real representation, if and only if the representation has a G-invariant nondegenerate symmetric bilinear form. If the degree of the character is even, there is an up to a square unique element d of the character field, such that the Gram determinant of any such bilinear form is equal to d, up to a square. We will call this the orthogonal determinant of the character.

For p an odd prime, q a power of p, n a natural number, we will regard the group G=GL_n(q), the general linear group over the field with q elements. We will recall the representation theory of these groups, and then move on to discussing methods how to calculate the orthogonal determinants of its characters. As it turns out, the case of the character not belonging to the principal series is easily handled, so the main focus of the talk will be about the principal series case.

• Fabian Mäurer (RPTU Kaiserslautern-Landau)

Title: On Fusion Categories and Their Centre

Abstract: I will give a quick introduction to the theory around fusion categories and talk about the question how to reasonably work with these structures in a computer. The main focus will lie on the computation of the categorial centre of a given fusion category.

• Max Mayer (RPTU Kaiserslautern-Landau)

Title: Computations in polycyclic groups

Abstract: Polycyclic groups are computational wise an interesting class of groups as a lot of usually hard or even undecidable questions can be answered algorithmically in these groups. In this talk we have a look at the subgroup algorithm which is central to many other algorithms. We discuss what the issues of the currently used algortihm are and how to improve it by the use of an LLL-reduced hermite normal form computation. Afterwards we have a look at an application to this algorithm namely the computation of the intersection of (two) subgroups.

• Stevell Muller (Saarland University)

Title: On Symplectic Birational Transformations of OG10-type Hyperkähler Manifolds

Abstract: Hyperkaehler manifolds, also known as irreducible holomorphic symplectic manifolds, are one of the three building blocks of an important class of complex manifolds. Their symplectic birational self maps have been widely studied: one could cite a celebrated paper of Mukai regarding the case of K3 surfaces.

In this talk, I plan to give a non-specialised overview on what motivates the study of symplectic birational self-maps of hyperkaehler manifolds. I explain how one can use a computer, in particular Oscar, to answer some questions requiring expensive computations. Without going too much into technical details, I use as a support for my talk a work in progress with my collaborator Lisa Marquand, about a classification of subgroups of symplectic birational self-maps for OG10-type hyperkaehler manifolds.

• Liam Rogel (RPTU Kaiserslautern-Landau)

Title: Categorifications of two-sided Kazhdan Lusztig cells using Soergel diagrammatics

Abstract: We introduce the diagrammatic Hecke Category by Elias and Williamson and show how it categorifies the Hecke-Algebra. We further motivate Lusztig's $H$-cell reduction by showing fusion ring properties on the Grothendieck ring and imitate the construction on the category level. Small examples of dihedral groups are discussed throughout the talk and we end up with complete fusion data needed to implement the asymptotic Hecke Category into the OSCAR package TensorCategories.jl.

• Victoria Schleis (University of Tübingen)

Title: Matroid quiver representations

Abstract: Grassmannians and flag varieties are important moduli spaces in algebraic geometry.  Their linear degenerations arise in representation theory as they describe quiver representations and their irreducible modules.  In joint work with Alessio Borzì, we introduce morphisms of valuated matroids and use them to study tropicalizations of quiver representations.

In this talk, I will give a self-contained introduction to linear tropical geometry, focused on defining linear maps corresponding to the morphisms we study. This notion allows us to view quiver representations in tropical geometry in a new light - not just as tropicalizations of algebraic objects, but as inherently combinatorial and tropical objects themselves.

The Ebernburg in Bad Münster am Stein can be easily reached by train (train station “Bad Münster am Stein”) or car from all main locations.