Title: "I am not Lou Billera"
Abstract: I will present the slides from a talk Lou Billera gave at the Stanley birthday conference last summer regarding the history of f-vectors.
Title: r-Stable Hypersimplices
Abstract: The n,k-hypersimplices are a well-studied collection of polytopes. Inside each n,k-hypersimplex we can define a finite nesting of subpolytopes that we call the r-stable n,k-hypersimplices. In this talk, we will define the r-stable hypersimplices and then see that they share a nice geometric relationship via a well-known regular unimodular triangulation of the n,k-hypersimplex in which they live. Using this relationship, we will then identify some geometric and combinatorial properties of the r-stable hypersimplices. In doing so, we will see that a number of the properties of the n,k-hypersimplex also hold for the r-stable hypersimplices within.
Title: Hopf Lefschetz theorem for posets
Abstract: The Hopf-Lefschetz theorem is a classical fixed point result from topology relating the Euler characteristic and the traces of certain matrices. In this talk we will prove a generalization of this theorem to order preserving maps on posets due to Baclawski and Björner. Additionally, we will prove a number of sufficient conditions on a poset P guaranteeing that all order preserving maps on P have a fixed point.
Title: Groupoids with weak orders
Abstract: We discuss certain groupoids equipped with partial orders satisfying properties which abstract those of weak orders of Coxeter groups (the resulting structures are not "partially ordered groupoids" in the usual sense). In particular, we describe braid presentations of the underlying groupoids and some of the very strong closure properties of these structures under natural categorical constructions. Applied even to familiar examples such as the symmetric groups, these constructions produce interesting new structures.
Title: The complex of not 2-connected graphs
Abstract: With every graph property that is monotone, we can associate for every positive integer n an abstract simplicial complex. The vertices of this complex are the edges of the complete graph on n nodes, and the faces are the sets of edges having this graph property. We will present and outline the proof of a result by Babson, Bjorner, Linusson, Shareshian and Welker of the homotopy type of this simplicial complex for not 2-connected graphs.
Title: A poset view of the major index
Abstract: We introduce the Major MacMahon map from non-commutative polynomials in the variables a and b to polynomials in q, and show how this map commutes with the pyramid and bipyramid operators. When the Major MacMahon map is applied to the ab-index of a simplicial poset, it yields the q-analogue of n! times the h-polynomial of the poset.
Applying the map to the Boolean algebra gives the distribution of the major index on the symmetric group, a seminal result due to MacMahon.
Similarly, when applied to the cross-polytope we obtain the distribution of one of the major indexes on the signed permutations, due to Reiner.
This is joint work with Margaret Readdy
Title: The lattice of bipartitions
Abstract: Bipartitional relations were introduced by Foata and Zeilberger, who showed these are precisely the relations which give rise to equidistribution of the associated inversion statistic and major index. In this talk we consider the natural partial order on bipartitional relations given by inclusion, and prove that the Möbius function of each of is intervals is 0, 1, or -1. To achieve this goal we will explore the topology of the order complex. We will see that bipartitional relations on a set of size n form a graded lattice of rank 3n-2. The order complex of this lattice is homotopy equivalent to a sphere of dimension n-2. Each proper interval in this lattice has either a contractible order complex, or it is isomorphic to the direct product of Boolean lattices and smaller lattices of bipartitional relations. The main tool in the proofs of these results is discrete Morse theory as developed by Forman, and an application of this theory to order complexes of graded posets, designed by Babson and Hersh.
This is joint work with Christian Krattenthaler.
Title: Winding Numbers and the Generalized Lower-Bound Conjecture
Abstract: Consider a set V of n distinct points in affinely general position in R^e. For 0 ≤ k < n−e , a k-splitter is the convex hull of a set of k points whose affine span separates the remaining points into two sets, one of which has size k. Let p be an additional point in affinely general position with respect to V . In this talk, we will discuss w_k(p), the kth winding number, which counts how many times k-splitters wrap around p in the counter clockwise direction. It is known that w_k(p) ≥ 0, as a consequence of the g-theorem. There are elementary proofs (Lee, Welzl) for some special cases (e.g. e = 2). We will discuss the ideas behind these proofs. Then we will discuss the relationship of this work to the g-theorem. We will then pose possible directions for further research
Title: Examining the Ehrhart Series of Reflexive Polytopes
Abstract: The Ehrhart series of a lattice polytope encodes combinatorial data about its integer scalings. From this series, we can determine properties of the polytope that may have been otherwise obscured, such as when a corresponding semigroup algebra is Gorenstein. In this talk, we will discuss an open question about the form of the Ehrhart series for integrally closed, reflexive polytopes and describe progress in the case of simplices.
Title: Rademacher--Carlitz Polynomials
Abstract: We introduce and study the Rademacher--Carlitz polynomial These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view the Rademacher—Carlitz polynomial as a polynomial analogue (in the sense of Carlitz) of the Dedekind--Rademacher sum, which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher--Carlitz polynomials, we show how they are the only nontrivial ingredients of integer-point transforms of any rational polyhedron P, and (if time allows) we derive a novel reciprocity theorem for Dedekind--Rademacher sums, which follows naturally from our setup.
This is joint work with Matthias Beck.
