Counting inversions and descents of random elements in finite Coxeter groups

Abstract:
In this talk I will report on Mahonian and Eulerian probability distributions given by inversions and descents in general finite Coxeter groups. I will provide uniform formulas for the mean values and variances in terms of Coxeter group data in both cases. I will also provide uniform formulas for the double-Eulerian probability distribution of the sum of descents and inverse descents. I will finally establish necessary and sufficient conditions for general sequences of Coxeter groups of increasing rank for which Mahonian and Eulerian probability distributions satisfy central and local limit theorems.

This talk is based on joint work with Thomas Kahle.

A Polyhedral Method for Sparse Systems with many Positive Solutions

Abstract:
We investigate a version of Viro's method for constructing polynomial systemswith many positive solutions, based on regular triangulations of the Newtonpolytope of the system. The number of positive solutions obtained with ourmethod is governed by the size of the largest positively decorable subcomplexof the triangulation. Here, positive decorability is a property that weintroduce and which is dual to being a subcomplex of some regulartriangulation. Using this duality, we produce large positively decorablesubcomplexes of the boundary complexes of cyclic polytopes. As a byproduct weget new lower bounds for themaximal number of positive solutions of polynomial systems with prescribednumbers of monomials and variables. We also study the asymptotics of thesenumbers and observe a log-concavity property.

This talk is based on joint work with Frédéric Bihan and Pierre-Jean Spaenlehauer.

Separation-type combinatorial invariants for triangulations of manifolds

Abstract:
In this talk, I propose a combinatorial invariant defined for simplicial complexes as an alternative way to obtain lower bounds on the size of triangulations of manifolds. The advantage of this approach lies within its simplicity: the invariants under invesitgation are defined by counting connected components and/or homological features of subcomplexes.

As an application I obtain a linear identity for triangulated 4-manifolds with a fixed f-vector, unifying previously known lower bounds on the number of faces needed to triangulate a) simply connected 4-manifolds, and b) 4-manifolds of type "boundary of a handlebody". Moreover, based on the same construction, I present a conjectured upper bound for 4-manifolds with prescribed vector of Betti-numbers.

This talk is based on joint work with Francisco Santos and Jonathan Spreer.