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

11:30-12:30:  Carsten Peterson (Sorbonne University)

Title:  A degenerate version of Brion's formula

Abstract: Brion's formula says that the continuous (resp. discrete) Fourier-Laplace transform of a polytope $P$ (resp. lattice points in a rational polytope) is equal to the sum of the continuous (resp. discrete) Fourier-Laplace transforms of the tangent cones of the vertices. However, whereas the former is an entire function, each latter function is merely meromorphic with singularities on the dual vectors $\xi$ which are constant on some positive-dimensional face of the polytope (resp. constant on the sublattice parallel to some positive-dimensional face). Because of this, one cannot ``plug into'' Brion's formula at such points.

We shall present a ``degenerate'' extension of Brion's formula for which one can still ``plug in'' at such troublesome points. Like Brion's formula it will be made up of terms each of which only depends on some local geometry of $P$. Our formula is particularly useful for understanding how the Fourier-Laplace transform varies over a family of polytopes with the same normal fan. In the generic case our formula reduces to the original Brion's formula, and in the maximally degenerate case ($\xi = 0$) it reduces to the volume of the polytope (resp. the Ehrhart quasi-polynomial).

14:45-15:45: Jacinta Torres (Jagiellonian University)
 

Title:  A new branching model in terms of flagged hives

Abstract: We prove a bijection between the branching models of Sundaram and Kwon. Along the way, we obtain a new branching model In terms of flagged hives polytopes. This is joint work with Sathish Kumar.

16:00-17:00: Evrydiki Nestoridi (Stony Brook University)

Title: Shuffling via transpositions

Abstract: In their seminal work, Diaconis and Shahshahani proved that shuffling a deck of $n$ cards sufficiently well via random transpositions takes $1/2 n log n$ steps. Their argument was algebraic and relied on the combinatorics of the symmetric group. In this talk, I will focus on two other shuffles, generalizing random transpositions and I will discuss the underlying combinatorics for understanding their mixing behavior and indeed proving cutoff. The talk will be based on joint works with A. Yan and S. Arfaee.