Program
All the talks will take place in the Lecture Hall of The Zuse Institute Berlin.
| Monday | Tuesday | Wednesday | |
|---|---|---|---|
| 09:00-09:15 | Paffenholz | Gaubert | |
| 09:15-10:15 | Röhrig | Theobald | |
| 10:15-11:00 | Coffee break | Coffee break | |
| 11:00-12:00 | Shaw | Jochemko | |
| 12:00-13:00 | Lunch | Closing remarks | |
| 13:00-14:00 | Lunch | ||
| 14:00-14:15 | Welcome | Lunch | |
| 14:15-14:30 | Sturmfels | Ziegler | |
| 14:30-15:30 | Stump | Casabella | |
| 15:30-16:00 | Coffee break | Coffee break | |
| 16:00-17:00 | Poster session | Gawrilow + more stories | |
| 17:00-18:00 | Poster session | ||
| 19:00- | Conference dinner |
Titles and abstracts
Laura Casabella - Regular subdivisions of hypersimplices
The hypersimplex is a polytope which appears naturally in many areas of mathematics, including algorithmic, discrete and tropical geometry. The regular subdivisions of a polytope are stratified by the secondary fan, which provides a combinatorial framework to understand them. We compute this fan for the hypersimplices (2,7) and (3,6) and find new families of rays of secondary fans of general hypersimplices. We explain the connections to the aforementioned branches of mathematics, such as the tropical Grassmannian, matroid theory, and finite metric spaces. This is joint work with Michael Joswig and Lars Kastner.
Katharina Jochemko - Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity
Generalized permutahedra form a combinatorially rich class of polytopes that appear naturally in various areas of mathematics. They include many interesting and significant classes of polytopes, in particular, matroid polytopes. We study functions on generalized permutahedra that behave linearly with respect to dilation and taking Minkowski sums. We present classification results and discuss how these can be applied to prove positivity of the linear coefficient of the Ehrhart polynomial of generalized permutahedra. This is based on joint work with Mohan Ravichandran.
Olivia Röhrig - Formal polytopes in lean
The mathlib project aims to compile a cohesive library of definitions and theorems from various areas of mathematics using the Lean proof assistant. It does not yet contain the language for talking about polytopes, which is what we are currently working on. In the talk I will explain the motivation for formalising mathematics and give an introduction to Lean and mathlib. I will then share the considerations that went into devising a theory of polytopes for mathlib as well as the current status of our formalization effort and the upcoming challenges.
This is joint work with Martin Winter.
Kris Shaw - Hitomezashi and patchworking
Hitomezashi is a Japanese stitching technique which produces beautiful and unexpected patterns and which has only recently been studied mathematically. In this talk, I will explain how hitomezashi patterns can be linked to real algebraic curves via Viro’s patchworking construction. I will also propose a generalisation of hitomezashi patterns to higher dimensions and explain how, with a bit of work, we can maintain a connection to real algebraic geometry in the case of surfaces. It can also be shown that in any dimension hitomezashi patterns are examples of positive parts of real tropical hypersurfaces.
Christian Stump - The cluster complex and tropical positivity
I will discuss why the cluster complex of a finite type cluster algebra is combinatorially isomorphic to the totally positive part of the corresponding cluster variety, as conjectured by Speyer-Williams. The talk will focus on presenting the involved objects in concrete examples, including intermediate structures such generalized associahedra and their type cones. The talk is based on joint work with Dennis Jahn and Robert Löwe.
Thorsten Theobald - A stable-set bound and maximal numbers of Nash equilibria in bimatrix games
Quint and Shubik (1997) conjectured that a non-degenerate $n \times n$ game has at most $2^n-1$ Nash equilibria in mixed strategies. The conjecture is true for $n \le 4$ but false for $n \ge 6$. We answer it positively for the remaining case $n=5$, which had been open since 1999. The problem can be translated to a combinatorial question about the vertices of a pair of simple $n$-polytopes with $2n$ facets. We introduce a novel obstruction based on the index of an equilibrium, which states that equilibrium vertices belong to two equal-sized disjoint stable sets of the graph of the polytope. This bound is verified directly using the known classification of the 159,375 combinatorial types of dual neighborly polytopes in dimension 5 with 10 facets. Non-neighborly polytopes are analyzed with additional combinatorial techniques where the bound is used for their disjoint facets.
Joint work with Constantin Ickstadt and Bernhard von Stengel.