08.11.2022 um 14:15 Uhr in 32/109
Simon Telen (MPI Leipzig/Centrum Wiskunde & Informatica, Amsterdam)
Toric geometry of entropic regularization
Entropic regularization is a method for large-scale linear programming. Geometrically, one traces intersections of the feasible polytope with scaled toric varieties, starting at the Birch point. We compare this to log-barrier methods, with reciprocal linear spaces, starting at the analytic center. We revisit entropic regularization for unbalanced optimal transport. We develop the use of optimal conic couplings and compute the degree of the associated toric variety. We also discuss generalizations for semidefinite programming.
15.11.2022 um 14:15 Uhr in 32/109
Elena Tielker (Universität Bielefeld)
Polytopes und Permutations
It is well-known that the numerator of the Ehrhart series, the so-called h*-polynomial, of a lattice polytope has nonnegative coefficients. Only in a few cases (e.g., for simplices) we are able to interpret the coefficients in the context of another counting problem. We present two further families of polytopes whose h*-polynomials can be interpreted in terms of permutation statistics. In this context we give new definitions of the major index and descent statistic on signed multiset permutations.
In this talk we give a brief introduction to Ehrhart theory and permutation statistics (and their generalisations), no previous knowledge is required.
22.11.2022 um 14:15 Uhr in 32/109
Alheydis Geiger (MPI MiS Leipzig)
A tropical count of real bitangents to plane quartic curves
A classical result by Plücker and Zeuthen states that a smooth complex quartic has exactly 28 bitangent lines, while a smooth real quartic has either 4, 8, 16 or 28 real bitangent lines.
A tropcial smooth quartic can have infinitely many bitangents, which are grouped into 7 equivalence classes. The shapes of these bitangent classes and their real lifting conditions were determined by Cueto and Markwig. Our investigation of the bitangent shapes allows to break the existence of tropical bitangents of quartics down to an analysis of the dual triangulation. Together with the results from Cueto and Markwig, this enabled us to implement a computational count of the numbers of real bitangents of quartics using polymake.
After a brief introduction of the tropical tools needed, we dive into the world of tropical bitangents finishing with a short demonstration of the polymake code that was developed during the project. This project is joint work with Marta Panizzut.
29.11.2022 um 14:15 Uhr in 32/109
Julia Lindberg (MPI Leipzig)
On the typical and atypical solutions to the Kuramoto equations
The Kuramoto model is a dynamical system that models the interaction of coupled oscillators. There has been much work to effectively bound the number of equilibria to the Kuramoto model for a given network. In this talk, I will relate the complex root count of the Kuramoto equations to the combinatorics of the underlying network by showing that the complex root count is generically equal to the normalized volume of the corresponding adjacency polytope of the network. I will then give explicit algebraic conditions under which this bound is strict and show that there are conditions where the Kuramoto equations have infinitely many equilibria. This is joint work with Tianran Chen and Evgeniia Korchevskaia.
06.12.2022 um 14:15 Uhr in 32/109
Gert Vercleyen (National University of Ireland, Maynooth)
Knots and topological quantum computing
In this seminar I hope to shed some light on the interplay between knot theory and topological quantum computation. This will be done in a more informal manner where the different concepts will be introduced without copious formulae or proofs so that people without a background in any of the fields can follow.
17.01.2023 um 14:15 Uhr in 32/109
Martina Juhnke-Kubitzke (Universität Osnabrück)
On cosmological polytopes
In this talk, I will consider cosmological polytopes, a class of lattice polytopes that can be constructed from a graph. Those polytopes were originally defined and studied by Arkani-Hamed, Benincasa, and Postnikov in their study of the wavefunction of the universe. The goal of this talk is to understand how geometric invariants, e.g., their volume and triangulations, of these polytopes are related to the combinatorics of the underlying graph. In particular, we will show that these polytopes have a regular unimodular triangulation by computing an explicit squarefree Gröbner basis for the corresponding toric ideals. We will describe the maximal cells in these triangulations by Feynman diagram like graphs. As a consequence, we are able to compute the normalized volume of the cosmological polytope for paths and cycles.
This is joint work (in progress) with Liam Solus and Lorenzo Venturello.
24.01.2023 um 14:15 Uhr in 32/109
Irem Portakal (TU München)
Rigid Gorenstein toric Fano varieties arising from directed graphs
A directed edge polytope AG is a lattice polytope arising from root system An and a finite directed graph G. If every directed edge of G belongs to a directed cycle in G, then AG is terminal and reflexive, that is, one can associate this polytope to a Gorenstein toric Fano variety XG with terminal singularities. It is shown by Totaro that a toric Fano variety which is smooth in codimension 2 and Q-factorial in codimension 3 is rigid. In this talk, we classify all directed graphs G such that XG is a toric Fano variety which is smooth in codimension 2 and Q-factorial in codimension 3. This is joint-work with Selvi Kara and Akiyoshi Tsuchiya.
31.01.2023 um 14:15 Uhr in 32/109
Nick Dewaele (KU Leuven)
A condition number for underdetermined systems
When solving the system of equations F(x) = y for x, the condition number provides a way of measuring the sensitivity of x to small changes in y. This tells us whether exact solutions to the system are also an approximate solutions to approximate systems, in an asymptotic sense. This talk starts with a gentle introduction to condition numbers of systems of equations. Furthermore, I argue that the concept of condition can be extended to underdetermined systems and that the way to compute the condition number is a straightforward generalisation of the computation for determined systems. I will illustrate how to interpret the condition number of simple matrix factorisation problems.