Séminaires : Séminaire CAGe: Combinatorics, Arithmetic and Geometry

Les sujets sont ceux décrits par le titre :). Ils doivent être compris dans un sens large.

Notre objectif est de nous réunir avec une périodicité mensuelle.

https://sites.google.com/view/combarithmgeo/home?authuser=0

11:30-12:20:  Mikhail Shkolnikov (ICMS Sofia)

Title: Combinatorial and Geometric aspects of Tropical Optics

Abstract:  This talk is dedicated to an introduction and an overview of the current status of the theory of tropical wave fronts and their caustics seen from toric perspective. Particular emphasize will be made on the ongoing finite-time analysis of the structure of critical events in the tropical wave front propagation operating on convex polyhedra and the role of corresponding algebro-geometric singularities in the character of this canonical evolution as being developed in joint project with Grigory Mikhalkin. We will finish with discussing the fundamental open problem of understanding possible pre-final moments of such an evolution, which is fully resolved only in dimension 2, with final moments now being classified in all dimensions.

15:00-15:50:  Konstantinos Kartas (Université de Paris-Saclay)

Title: Taming perfectoid algebras

Abstract: The almost purity theorem is a foundational result about perfectoid algebras which, as the name suggests, holds only in an "almost" sense. We show that suitable completeness hypotheses on the algebras allow us to remove the "almost". More precisely,  spherically complete perfectoid algebras satisfy non-almost purity for Galois p-extensions and countably saturated perfectoid algebras satisfy non-almost purity for all finite étale extensions. Joint work with Franziska Jahnke.

16:00-16:50:  Yiyu Wang (Ohio State University)

Title: Grothendieck weight on permutohedral varieties and matroids

Abstract: Minkowski weights provide a combinatorial model for the Chow ring of smooth toric varieties. In this talk, I will introduce their K-theoretic analogue, the Grothendieck weights, and describe them explicitly for permutohedral varieties. Our results include a balancing condition and a product rule for Grothendieck weights. These results give a concrete way to work in the K-ring of the permutohedral variety. 

I will then discuss applications to matroids, where many naturally occurring K-classes admit simple descriptions in terms of Grothendieck weights. In particular, I will explain how the motivic Chern class of a hyperplane arrangement complement in its wonderful model only depends on its underlying matroid.

If time permits, I will briefly mention some further applications of Grothendieck weights to positivity questions in matroid K-theory.