Lehrstuhl Geometrie

We are principally interested in any nice problem, but our current main research interests may be subsumed to lie in Discrete and Computational Mathematics. Problems we have worked on fall into the following more specific categories:
* Algorithmic and Discrete Geometry
* Combinatorial Geometry, Polyhedral Combinatorics
* Geometry of Numbers, Packing- and Covering problems
* Quadratic Forms, Reduction theories
* Convex Geometry, Minkowski Spaces
* Algebraic Geometry, Compactifications of Moduli spaces
* Codes and Designs
* Algorithms and Complexity theory
* Computational Group Theory and Group Cohomology
* Symmetry in Optimization
* Linear and Convex Programming techniques
* Social Choice Theory
* Mathematics of Sailing
* Machine Learning Algorithms


DFG Project SCHU 1503/6-1 (2015-2018)

Geometry and Algorithms for Exploiting Polyhedral Symmetries

Main aim of this DFG-funded research project is to advance the use of symmetry in polyhedral computations. We intend to use the rich geometric structure of symmetric polyhedra, for improved algorithms in three main classes of computations.
I.   Polyhedral representation conversion using symmetry
II.  Symmetric integer linear and convex programming
III. Counting lattice points and exact volumes of symmetric polyhedra
There are multiple strong dependencies among these three topics and each one has its theoretical and algorithmic challenges as well as important applications.
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DFG Project SCHU 1503/7-1 (2017-2019)

Energy Minimizing Periodic Point Sets

Point configurations minimizing energy for a given pair potential function occur in diverse contexts of mathematics and its applications. In recent years the study of universally optimal point configurations has revealed some striking new phenomena. In this project we study periodic point configurations, that is, finite unions of translates of a lattice. Creating and using numerical tools for computational experiments, we expect to reveal new phenomena and collect evidence for the existence or non-existence of universally optimal periodic point configurations. The recently observed phenomenon of formally-dual periodic point sets is studied and, as far as possible, corresponding sets will be classified.
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Related Thesis:

  • Robert Schüler, Formal Duality, PhD thesis, Universität Rostock, 2019

DFG Project SCHU 1503/8-1 (2021-2024)

Perfect Copositive Matrices

In this project we develop a theory of perfect copositive matrices. These matrices can be used, for example, to discretize the convex cone of copositive matrices in a systematic way. This results in possible applications in copositive optimization, such as the algorithmic calculation of rational certificates for completely positive matrices.

A central component of our project — and important for algorithmic applications of the new theory — is the investigation of the neighborhood graph of perfect copositive matrices. This graph appears as an edge graph of a polyhedral surface, whose facets are determined by minimal vectors of copositive matrices. For the algorithmic treatment of the neighborhood graph it is therefore essential to compute minimal vectors of copositive matrices. For this important task new approaches are developed and tested. These new algorithms for determining minimal vectors themselves have the potential for interesting future applications.

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