Organizers: Laura Anderson, Emanuele Delucchi, and Thomas Zaslavsky.
In the last few years Chari's convex ear decomposition has been applied to several apparently unrelated order complexes of posets to derive new insight into their f-vectors. An unexpected consequence of these investigations is that their flag f-vectors are closely related to a largely unexplored interaction between descent sets and the weak order of finite Coxeter groups. The goal of the lecture is to explain how face enumeration on these posets naturally leads to a zoo of open problems involving descent sets and the weak order. In order to guarantee maximum accessibility I will concentrate on the special case of the symmetric group.
This talk will focus on the circuit axioms of matroid theory. I will start with a short introduction to matroids. I will then present a new strengthening of the circuit axioms that requires the elimination property only among modular pairs of circuits, and show how it can be cryptomorphically phrased in terms of Crapo's axioms for flats. This new point of view leads to a corresponding strengthening of the signed circuit axioms for oriented matroids.
A complex toric arrangement is a finite family of hypersurfaces in a complex torus (C*)n, every hypersurface being the kernel of a character (that is, a homomorphism into C*, which has the form (t1,...,tn) → t1a1···tnan, where (a1,...,an) is in Zn). I describe some properties of such arrangements, and I compare them with hyperplane arrangements.
The Tutte polynomial is an invariant which encodes a rich description of the topology and the combinatorics of a hyperplane arrangement, and satisfies a simple recurrence. I introduce the analogue of this polynomial for a toric arrangement. Furthermore, I show that the new polynomial counts integral points in the faces of a zonotope, and provides the graded dimension of a space of quasipolynomials introduced by Dahmen and Micchelli to study partition functions.