# A(achen)-B(ochum)-C(ologne)-D(arstellungstheorie) Seminar

# Wintersemester 2019/20

### Organizer: Peter Littelmann, Xin Fang

## Speakers

Fabian Korthauer (Bochum), Sara Lamboglia (Frankfurt), Verity Mackscheidt (Aachen), Elizabeth Milićević (Haverford), Elizaveta Vishnyakova (Cologne)## Schedule

The seminar will be held on**4, February, 2020**in the Hoersaal (Main Lecture Hall, 2.03) in

Department of Mathematics

University of Cologne

Weyertal 86-90

50931, Cologne

### Schedule of talks:

11h-12h Elizaveta Vishnyakova, Simple Harish-Chandra modules over invariant subalgebras in a skew-group ring

12h-14h Lunch break

14h-14h15 Fabian Korthauer, Arithmetic representation growth of groups

14h30-15h30 Elizabeth Milićević, The Peterson Isomorphism: Moduli of Curves and Alcove Walks

15h30-16h Coffee break

16h-17h Sara Lamboglia, Tropical convex hulls of infinite sets

17h-17h15 Verity Mackscheidt, PBW deformations arising from algebraic groups

## Abstract

#### Elizaveta Vishnyakova

Simple Harish-Chandra modules over invariant subalgebras in a skew-group ringAbstract of the talk

#### Fabian Korthauer

Arithmetic representation growth of groupsIn 2014 Mozgovoy-Reineke proved that completely reducible representations of finitely generated free groups over the finite field F_q can be counted by evaluating polynomials with integer coefficients at q. This and some related results and ongoing generalizations of these will be presented in this talk.

#### Elizabeth Milićević

The Peterson Isomorphism: Moduli of Curves and Alcove WalksIn this talk, I will explain the combinatorial tool of folded alcove walks, in addition to surveying a wide range of applications in combinatorics, representation theory, and algebraic geometry. As a concrete example, I will describe a labeling of the points of the moduli space of genus zero curves in the complete complex flag variety using the combinatorial machinery of alcove walks. Following Peterson, this geometric labeling partially explains the "quantum equals affine" phenomenon which relates the quantum cohomology of this flag variety to the homology of the affine Grassmannian. This is joint work with Arun Ram.

#### Sara Lamboglia

Tropical convex hulls of infinite setsIn this talk I will present some recent results on the interplay between tropical and classical convexity. In particular, I will focus on the tropical convex hull of convex sets and polyhedral complexes and their explicit computation. This will lead to a lower bound on the degree of tropical curves. This is joint work with Cvetelina Hill and Faye Pasley Simon.

#### Verity Mackscheidt

PBW deformations arising from algebraic groupsStarting from a graded algebra given by generators and relations, one can deform the algebra by deforming the relations. In this talk, a special class of deformations, namely PBW deformations, will be presented. They are well known in some cases, e.g. for Lie algebras, but the theory contains interesting open questions in general. This talk presents some known results and generalises the context to PBW deformations arising from algebraic groups, which is ongoing work.

## Practical information

There is no registration needed.For further questions, please contact xfang@math.uni-koeln.de

Arbeitsgruppe Algebra und Zahlentheorie, Mathematisches Institut, Universität zu Köln