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.
- Achill Schürmann, Exploiting Symmetries in Polyhedral Computations, Fields Institute Communications, 69 (2013), 265-278
- Achill Schürmann, Exploiting Polyhedral Symmetries in Social Choice, Social Choice and Welfare, 40 (2013), 1097-1110
- Katrin Herr, Thomas Rehn and Achill Schürmann, Exploiting Symmetry in Integer Convex Optimization using Core Points, Operations Research Letters, 41 (2013), 298-304
- David Bremner, Mathieu Dutour Sikiric, Dimitii V. Pasechnik, Thomas Rehn and Achill Schürmann, Computing symmetry groups of polyhedra, LMS Journal of Computation and Mathematics, 17 (2014), 565-581.
- Katrin Herr, Thomas Rehn and Achill Schürmann, On Lattice-Free Orbit Polytopes, Discrete Comput. Geom., 53 (2015), 144-172.
- Frieder Ladisch and Erik Friese, Affine Symmetries of Orbit Polytopes, Adv. Math., 288 (2016), 386-425.
- Frieder Ladisch, Realizations of abstract regular polytopes from a representation theoretic view, Aequationes Math. 90 (2016), 1169-1193.
- Erik Friese, William V. Gehrlein, Dominique Lepelley and Achill Schürmann, The impact of dependence among voters' preferences with partial indifference, Quality & Quantity, 2017.
- Maren H. Ring and Achill Schürmann, Local formulas for Ehrhart coefficients from lattice tiles, Contributions to Algebra and Geometry, 61 (2020), 157–185 (2020)
- Thomas Rehn, Exploring core points for fun and profit — a study of lattice-free orbit polytopes, PhD thesis, Universität Rostock, 2013
- Frieder Ladisch, Representation Theory and Polytopes, Habilitation thesis, Universität Rostock, 2017
- Erik Friese, Generic Symmetries of Group Representations, PhD thesis, Universität Rostock, 2018
- Maren Helene Ring, Local formulas for Ehrhart coefficients from lattice tiles, PhD thesis, Universität Rostock, 2019
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.
- Henry Cohn, Abhinav Kumar and Achill Schürmann, Ground states and formal duality relations in the gaussian core model, Physical Review E 80 (2009).
- Renaud Coulangeon and Achill Schürmann, Energy minimization, periodic sets and spherical designs, Int. Math. Res. Not. IMRN (2012), 829-848.
- Henry Cohn, Abhinav Kumar, Christian Reiher and Achill Schürmann, Formal duality and generalizations of the Poisson summation formula, AMS Contemporary Mathematics, 625 (2014), 123-140.
- 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.
- Mathieu Dutour Sikirić, Achill Schürmann and Frank Vallentin, Rational factori- zations of completely positive matrices, Linear Algebra and its Applications, 523 (2017), 46–51.
- Mathieu Dutour Sikirić, Achill Schürmann and Frank Vallentin, A simplex algorithm for rational cp-factorization, Math. Program., online first, 21 pages, 2020.
- Valentin Dannenberg, Masterthesis (in German), Universität Rostock, November 2020.