Séminaires : 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

Schedule

11:30-12:30:  Meral Tosun (Galatasaray University)

Title:  Geometric and combinatorial views on singularities

Abstract:  We focus on a special class of singularities that allow a polyhedral and

combinatorial description. We show how jet schemes capture fine local data

and lead to a resolution process in this class.

14:45-15:45: Dmitry Mineev ( Bar-Ilan University)

Title:  From tropical modifications of Bergman fans to correspondences and flag fans

Abstract:  Bergman fans are tropical counterparts of matroids. Strong maps between matroids admit a description as morphisms between Bergman fans. However, taking limits of even the simplest diagrams of matroids requires so-called weak maps, which do not translate to the tropical side. We suggest a fix, generalizing both weak maps of matroids and tropical morphisms, and construct a functor relating the two. We also introduce flag fans as a convenient tool for computations in this extended setting.

16:00-17:00: Christos Tatakis (University of Western Macedonia)

Title: The structure of complete intersection graphs and their planarity.

Abstract:  

Let G be a connected, undirected, finite and simple graph. We study the complete intersection property on the toric ideal $I_G$. In general, the toric ideal $I_G$ is complete intersection if and only if it can be generated by h binomials, where h=m-n+1 if G is a bipartite graph or h=m-n if G is not a bipartite graph, where by m we denote the number of the edges of G and by n the number of its vertices. The answer is known in the case of bipartite graphs, i.e. graphs with no odd cycles. In the last years, several useful partial results have been proved and they provide key properties of complete intersection toric ideals of graphs. 

We focus on the general case, where G is a random graph and we present a structural theorem which gives us necessary and sufficient conditions in which the toric ideal $I_G$ is complete intersection. Moreover, we characterize with sufficient and necessary conditions the complete intersection graphs which are planar. The talk is based on a joint work with Apostolos Thoma.