Abstracts for Oberseminar Representation Theory

The common neighborhood of a set of nodes in a simple graph is a
Galois connection with associated closure operation on the power set
of the node set. An application of this concept to root systems
yields a new Galois connection on the (conjugacy classes of) parabolic
subgroups of a finite Coxeter group which we have used to refine
Howlett's description of the normalizers of its parabolic subgroups.
This talk is based on joint work G. Roehrle and J.M. Douglass
(arXiv:2509.15850).

This talk is based on joint work with Omar Gómez (Bielefeld) and Marius Nielsen (NTNU). Hopfological algebra is a variant of classical homological algebra introduced by Khovanov and Qi, motivated by potential applications in the categorification of quantum invariants of 3-manifolds. In this talk, I will explain an infinity-categorical approach to the theory that leads, in particular, to refined foundations as well as to "hopfological analogues" of classical invariants such as Hochschild (co)homology.

Bott-Samelson spaces are spaces with an action of a torus, such that their equivariant cohomology recovers Soergel bimodules and their Hochschild homology. These spaces, originally appearing as resolutions of Schubert varieties, have been used in several ways to give geometric constructions of Khovanov-Rozansky HOMFLY-PT link homology. They are also the starting point for generalizations of Soergel bimodules to other cohomology theories such as K-theory. In this talk, I will explore some implications of this torus-equivariant perspective.

A matroid is a fundamental and actively studied object in combinatorics. Matroids generalize linear independence in vector spaces as well as many aspects of graph theory. After a short introduction to matroids, I will present parts of a new OSCAR module for matroids through several examples. I will focus on computing the moduli space of a matroid, which is the space of all arrangements of hyperplanes with that matroid as their intersection lattice.