VORLAGE

LEHRSTUHL

In the context of the proposed RTG we understand complexity as an intrinsic property that makes
it difficult to determine an appropriate mathematical representation of a real world problem, to assess the fundamental structures and properties of mathematical objects, and to algorithmically solve a given mathematical problem. By complexity reduction we refer to all approaches that help to overcome these difficulties in a systematic way and to achieve the aforementioned goals more efficiently.

For many mathematical tasks, approximation and dimension reduction are the most important tools to obtain
a simpler representation and computational speedups. We see complexity reduction in a more general way and
will also, e.g., investigate liftings to higher-dimensional spaces and consider the costs of data observation.
Our research goals are the development of cross-disciplinary mathematical theory and methods for complexity
reduction and the identification of relevant problem classes and effective exploitation of their structures.

We aim at a comprehensive teaching and research program based on geometric, algebraic, stochastic, and
analytic approaches, complemented by efficient numerical and computational implementations. In order to
ensure the success of our doctoral students, they will participate in a tailored structured study program. It will
contain training units in form of compact courses and weekly seminars, and encourage early integration into the
scientific community and networking. We expect that the RTG will also serve as a catalyst for a dissemination
of these successful practices within the Faculty of Mathematics and improve the gender situation.

Complexity reduction is a fundamental aspect of the scientific backgrounds of the principal investigators.
The combination of expertise from different areas of mathematics gives the RTG a unique profile, with high
chances for scientific breakthroughs. The RTG will be linked to two faculties, a Max Planck Institute, and
several national and international research activities in different scientific communities.

The students of the RTG will be trained to become proficient in a breadth of mathematical methods, and
thus be ready to cope with challenging tasks in particular in cross-disciplinary research teams. We expect an
impact both in terms of research successes, and in the education of the next generation of leading scientists in
academia and industry.

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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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Dieses Projekt untersucht sogenannte Schubertvarietäten und hat zum Ziel ein Kombinatorisches framework zu entwickeln um deren Tangentialräume zu verstehen und klassifizieren zu können.
Schubertvarietäten sind Untervarietäten von Fahnenvarietäten und spielen eine wichtige Rolle in der Darstellungstheorie.

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In diesem Projekt werden geometrische Methoden entwickelt um Dimensionen affiner Deligne-Lusztig Varietäten zu berechnen. Hierbei handelt es sich um Untervarietäten affiner Fahnenvarietäten.

Die Fragestellung stammt aus der arithmetischen Geometrie und wird hier mit neuen Methoden aus der geometrischen Gruppentheorie untersucht.

Das Projekt wird in Kooperation mit Elizabeth Milicevic (Haverford, USA) und Anne Thomas (Sydney, Australien) durchgeführt und durch ein ARC Discovery project gefördert.

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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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Kontraktionsränder sind Ränder metrischer Räume mit nichtpositiver Krümmung, sogenannte CAT(0) Räume, die invariant unter Quasi-Isometrie sind.
Daher eignen sie sich gut um das grobe Verhalten der metrischen Räume zu untersuchen.

Dieses Dissertationsprojekt hat zum Ziel für geeignete Klassen von CAT(0) Räume ebensolche Ränder zu berechnen.

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