Research

Research at the Institute of Algebra, Number Theory and Discrete Mathematics

A Weyl groupoid

Weyl groupoids are invariants of Nichols algebras.

A Weyl groupoid

Weyl groupoids are invariants of Nichols algebras.

Algebra

The focus area in algebra is the representation theory of finite groups and (usually finite-dimensional) algebras. The research topics often have a connection to algebraic combinatorics, algebraic geometry and also to number theory, whose methods are particularly helpful for arithmetic questions regarding dimensions. The representation theory of symmetric groups and related algebraic structures also lead to the study of (quasi-)symmetric functions and the corresponding Hopf algebras. In the representation theory of algebras, the focus is on cluster algebras, cluster categories and homological algebra; also here, combinatorial aspects are essential.

Discrete Mathematics

In discrete mathematics, a focus lies on reflection groups and arrangements of hyperplanes. Reflection groups and related structures also play an important role in algebra as invariants of certain Hopf algebras. These Hopf algebras were recently studied more intensively in mathematical physics. In the context of arrangements of hyperplanes, structures from algebraic and discrete geometry are studied in particular. These include (oriented) matroids, certain modules of derivations and also invariants from topology.

Number Theory

Our research in number theory is oriented towards arithmetic geometry, building a bridge to algebraic geometry. One focus is the existence and distribution of rational points on algebraic varieties. This is investigated using methods of algebraic and analytic number theory and also geometric methods. Here torsors and Cox rings play a central role.