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.
10.01.2023 um 14:15 Uhr in 32/109
Georg Loho (University of Twente)
17.01.2023 um 14:15 Uhr in 32/109
Marta Panizzut (TU Berlin)
24.01.2023 um 14:15 Uhr in 32/109
Irem Portakal (TU München)
31.01.2023 um 14:15 Uhr in 32/109
Nick Dewaele (KU Leuven)
07.02.2023 um 14:15 Uhr in 32/109
Sarah Eggleston (Uni Osnabrück)
The dimension of amoebas of linear spaces
The amoeba of an algebraic subvariety is the image of the subvariety under the logarithmic moment map. Following on work by Draisma-Rau-Yuen [DRY20], we study amoebas of linear subspaces of \C^n. We prove a conjecture by Rau regarding the relationship between the dimension of the amoeba and the matroid associated with the linear subspace. While [DRY20] showed that the dimension is given by a particular minimum over infinitely many subtori of (\C^*)^n, we show that finitely many subtori associated with partitions of the ground set of the matroid, [n], suffice.