Goethe-Universität — Oberseminar Diskrete Mathematik, Geometrie und Optimierung

Diskrete Mathematik, Geometrie und Optimierung

Nov 17 2026

16:15

Raum 711 gr., Robert-Mayer-Straße 10

Jan Driesslein: Homotopiepfade bei der Bestimmung von Nash-Gleichgewichten

Diskrete Mathematik, Geometrie und Optimierung

Nov 3 2026

16:15

Raum 711 gr., Robert-Mayer-Str.. 10

Sudhir Ghorpade (IIT Bombay): Shellability and Homology related to Codes and Matroids

A little over a decade ago, Johnsen and Verdure introduced a fine set of
invariants, called Betti numbers, for a linear code, and more generally, a
matroid. These are obtained from graded minimal free resolutions of
Stanley-Reisner rings of simplicial complexes associated to the given linear
code (or matroid). This association is further facilitated by the fact that
simplicial complexes corresponding to codes, and more generally, matroids,
are shellable. The shellability also leads to the determination of homology
of these simplicial complexes. It turns out that the Betti numbers of a
linear code determine several of its important parameters such as
generalized Hamming weights and generalized weight enumerators. However,
computing Betti numbers is usually a hard problem. But it is tractable if
the free resolution is "pure". We will thus outline an intrinsic
characterization of purity of graded minimal free resolutions associated
with linear codes. Further, we will discuss a characterization of
(generalized) Reed-Muller and also projective Reed-Muller codes that admit a
pure resolution.

We then turn to the case of rank metric codes, which have been of some
current interest, and relevant q-analogues of the notions of matroids and
simplicial complexes, and their shellability and homology. We will outline
some recent results and questions in this direction.

This talk is mainly based on joint works with (i) Prasant Singh, (ii) Trygve
Johnsen, (iii) Rati Ludhani, (iv) Rakhi Pratihar and Tovohery
Randrianarisoa, and (v) Rakhi Pratihar, Tovohery Randrianarisoa, Hugues
Verdure and Glen Wilson.

Diskrete Mathematik, Geometrie und Optimierung

Okt 20 2026

15:00

Raum 711 gr., Robert-Mayer-Str. 10 (Besondere Uhrzeit! 15:00 s.t.)

Lorenzo Baldi: Totally real divisors on curves: metric bounds for algebraic invariants

Abstract: How to approximate the consecutive Minkowski sum of a smooth curve in the affine space? And in a compact torus? We give an answer based on convex geometry and zonoid calculus. We then apply this result to study totally real divisors on a real algebraic curve X, which are fundamental to study e.g. positive polynomials and projective embeddings. In this application, the torus is the Jacobian J of the algebraic curve X, and the smooth curve is the image of the real locus of X under the Abel-Jacobi map. We show in particular how the totally real divisor threshold, a natural algebraic invariant, is bounded by continuous metric properties of image of the Abel-Jacobi map. Based on a joint work with M. Kummer and D. Plaumann.

Diskrete Mathematik, Geometrie und Optimierung

Sep 29 2026

16:15

Raum 711 gr., Robert-Mayer-Str. 10

Paul Lindner: Polyedrische Geometrie von Rang-1-Spielen

Diskrete Mathematik, Geometrie und Optimierung

Sep 14 2026

- 16.09.2026

Frankfurt-Bockenheim | Physikalischer Verein

3. Jahrestagung des DFG-Schwerpunktprogramms “Combinatorial Synergies”

Diskrete Mathematik, Geometrie und Optimierung

Jul 14 2026

16:15

Raum 711 gr., Robert-Mayer-Str. 10

Jonas Glückmann: Nachbarschaftliche Polytope und Bimatrixspiele

Diskrete Mathematik, Geometrie und Optimierung

Jul 7 2026

16:15

Raum 711 gr., Robert-Mayer-Str. 10

Kolja Knauer: Geometry of Geometries

Universitat de Barcelona

Abstract: An abstract convex geometry describes the combinatorics of the convex subsets of a finite point set in Euclidean space. A complex of oriented matroids describes the combinatorics of a polyhedron sliced by hyperplanes in Euclidean space. It is untractable to decide whether a given abstract convex geometry or a given complex of oriented matroid can be realized by a point set or by a sliced polyhedron, respectively. We show that every abstract convex geometry can be realized by a sliced polyhedron, i.e., abstract convex geometries are realizable complexes of oriented matroids. We further discuss extensions to larger classes and bounds on the dimension of the realization.

Joint work with Victor Chepoi and Jérémie Chalopin

Diskrete Mathematik, Geometrie und Optimierung

Jun 30 2026

16:15

Raum 711 gr., Robert-Mayer-Str. 10

Dmitry Mineev: Combinatorics, Algebra, and Geometry of Locally Testable Codes

Bar-Ilan University, Tel Aviv Abstract: Sometimes one wants more from a linear error-correcting code than good rate and good distance. One may also want to check whether a message looks valid by examining only a tiny random portion of it. Properly formalized, this is called local testability. Many otherwise excellent codes, including ordinary random or expander LDPC codes, are not locally testable. This is bad news for naive constructions, but good news for mathematics: local testability seems to require a carefully curated balance of order and chaos, rather than pure randomness.

Diskrete Mathematik, Geometrie und Optimierung

Jun 23 2026

16:15

Raum 711 gr., Robert-Mayer-Str. 10

Dominik Rühmann: Nichtnegative Polynome, die nicht als SONC-Polynome darstellbar sind

Diskrete Mathematik, Geometrie und Optimierung

Jun 16 2026

16:15

Raum 711 gr., Robert-Mayer-Str. 10

Federico Castillo: Toric Nash blowups

Pontificia Universidad Católica de Chile

Abstract: Hironaka's celebrated resolution of singularities in characteristic 0 proceeds via blowups of carefully chosen subvarieties. There exists an alternative canonical process (requiring no choices) called Nash blowup, and a fundamental question is whether iterated Nash blowups resolve singularities. We show that this is not the case in dimensions greater than 3, and in all characteristics, by constructing counterexamples using toric varieties. In this context, the problem becomes a concrete polyhedral problem about rational cones and their Hilbert bases.
This is joint work with Daniel Duarte, Maximiliano Leyton, and Alvaro Liendo.

Oberseminar Diskrete Mathematik, Geometrie und Optimierung ​

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Oberseminar Diskrete Mathematik, Geometrie und Optimierung