Research

Profinite rigidity asks to determine a group by its finite quotients. This concept is classical in group theory and many results in this direction are known. Geometric group theory has picked up on this notion in recent years. We aim to study profinite rigidity for abstract reflection groups.

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Shadows in Coxeter groups are a well established tool which helps to characterize non-emptiness of double coset intersections in algebraic groups having these Coxeter groups as affine Weyl groups. These intersections in tern are relevant in the context of representation theory or in the study of non-emptiness and dimensions of certain varieties associated to the affine flag variety and affine Grassmannian. This project aims to find closed formulas for and a better algorithmic understanding of shadows.

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The conjugacy problem is one of Dehn's three classical problems in group theory. It asks to determine whether or not tow given elements in a group are conjugate. In this project we solve this problem and characterize the full conjugacy class of elements in split subgroups of the full isometry group of the n-dimensional real affine space.

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The aim of the project is to provide a uniform framework which allows us to treat Riemannian symmetric spaces of noncompact type and
Euclidean buildings on an equal footing. We will in particular consider
the question of the extension of automorphisms at infinity,
filling properties of S-arithmetic groups, and Kostant Convexity
from an unified viewpoint.

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This project aims to study reflection length in infinite, on-affine Coxeter groups. The goal is to find sequences of elements of growing reflection length, to describe the distribution of a fixed reflection length in hyperbolic space and to prove estimates of reflection length for a given S-length.

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In this project we introduce the galaxy of Coxeter groups - an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups - a result which is possibly of independent interest.

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Coxeter groups are abstract reflection groups
which can be classified roughly into three classes: spherical, affine,
and hyperbolic. The hyperbolic case is arguably the most flavorful and
many properties are clear in the spherical case, somewhat clear in the
affine case, and mysterious in the hyperbolic case. To tackle this
complexity, algebraic, combinatorial, and geometric methods need to be
combined.

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The conjugacy problem is one of Dehn's three classical problems in group theory. It asks to determine whether or not two given elements in a group are conjugate. In this project we solve this problem and characterize the full conjugacy class of elements in split subgroups of the full isometry group of the n-dimensional real affine space.

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In this project we use modern methods of geometric group theory to investigate connected components of clouds within in ICON weather model. We are currently developing a prototype to determine connected components in cloud data based on the triangular grid. First publications will be available, soon.This is a joint project with Aiko Voigt from KIT.

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This project investigates so-called Schubert varieties and aims to develop a combinatorial framework to understand and classify their tangent spaces.
Schubert varieties are subvarieties of flag varieties and play an important role in representation theory.
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In this project geometric methods are developed to calculate dimensions of affine Deligne-Lusztig varieties. These are sub-varieties of affine flag varieties.

The problem originates from arithmetic geometry and is investigated here using new methods from geometric group theory.

The project is carried out in cooperation with Elizabeth Milicevic (Haverford, USA) and Anne Thomas (Sydney, Australia) and is funded by an ARC Discovery project.
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The project is concerned with rigidity, compactifications and local-to-global principles in CAT(0) geometry. One aim is to give a uniform construction of compactifications of euclidean buildings, using Gromov's embedding into spaces of continuous functions.

The ultimate goal is to study the dynamics of discrete group actions on the building, using the compactification.

The project also intends to investigate LG-rigidity and non-rigidity for the 1-skeletons and chamber graphs of general Bruhat-Tits buildings.

Bruhat Tits buidlings are simplicial analogs of symmetric spaces and are a fundamental tool to study algebraic groups over non-archimedian local fields. Their combinatorial structure encodes a lot of information about flag varieties and Grassmannians.

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This project establishes a surprising connection between high-resolution climate modeling and geometric group theory.
We aim to address the need for fundamentally new strategies in analyzing the big-data output from next-generation climate models.The new German-community climate model ICON is a next-generation model (Zängl et al.,2015). Thanks to its triangular grid, ICON runs effectively on tens of thousands of CPUs and harvests advances in supercomputing.
In contrast to previous climate models this new model is based on a triangular grid. To provide fast computing algorithms one can thus no longer work with a cube-grid structure.

The main idea of this project is to use a technique from geometric group theory to translate the triangle structure into a parallel grid and back and thus to provide a methods to integrate existing fast algorithms into the new model.

This project won the "Best grant proposal award 2018" by the YIN@KIT

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Contraction edges are edges of metric spaces with non-positive curvature, so-called CAT(0) spaces, which are invariant under quasi-isometry.
Therefore, they are well suited to study the rough behavior of metric spaces.

This dissertation project aims to compute such edges for suitable classes of CAT(0) spaces.
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