Workshop

Associativity is ubiquitous in mathematics. Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance. But over the past few decades, this concept has blossomed and matured. We start with a brief look at how this has transpired, and then explore the visualization of associativity in the form of the associahedron polytope and its generalizations.

What are the possible face numbers of d-dimensional: polytopes? simplicial polytopes (namely those whose proper faces are all simplices)? flag polytopes (namely simplicial polytopes all whose minimal nonfaces have size 2)? centrally symmetric polytopes (namely those invariant under the map x-->-x)?...

We'll survey results and open problems concerning face numbers (f-vectors) of polytopes and simplicial complexes, including subfamilies of particular interest.

As the area is vast and fast growing, this talk will cover only a portion of it, representing some of the main methods used. These methods range from algebraic to combinatorial, topological and geometric, and will form Part II given by Martina Juhnke-Kubitzke.

This talk builds on Eran Nevo's talk, where he presented results and problems on f-vectors of simplicial complexes and polytopes. We will provide an overview of some of the most important tools used to prove this kind of results. This includes combinatorial methods such as shellings, as well as algebraic and geometric methods such as Stanley-Reisner rings, sigma modules, toric varieties, Lefschetz properties and rigidity of frameworks. We will also hint at specific examples, where those techniques have been applied successfully.

We say a polytope is Ehrhart positive if its Ehrhart polynomial has positive coefficients. There are different examples of polytopes shown to be Ehrhart positive using different techniques. We will survey some of these results. Through work of Danilov/McMullen, there is an interpretation of Ehrhart coefficients relating to the normalized volumes of faces. We try to make this relation more explicit in the particular case of the regular permutohedron. The goal is to prove Ehrhart positivity for generalized permutohedra. If time permits, I will also discuss some related questions. This is joint work with Federico Castillo.

In the field of Ehrhart theory, identification of lattice polytopes with unimodal Ehrhart h*-polynomials is a cornerstone investigation. The study of h*-unimodality is home to numerous long-standing conjectures within the field, and proofs thereof often reveal interesting algebra and combinatorics intrinsic to the associated lattice polytopes. Proof techniques for h*-unimodality are plentiful, and some are apparently more dependent on the lattice geometry of the polytope than others. In recent years, proving a polynomial has only real-roots has gained traction as a technique for verifying unimodality of h-polynomials in general. However, the geometric underpinnings of the real-rooted phenomena for h*-unimodality are not well-understood. As such, more examples of this property are always noteworthy. In this talk, we will discuss a family of lattice n-simplices that associate via their normalized volumes to the n^th-place values of positional numeral systems. The h*-polynomials for simplices associated to special systems such as the factoradics and the binary numerals recover ubiquitous h-polynomials, namely the Eulerian polynomials and binomial coefficients, respectively. Simplices associated to any base-r numeral system are also provably real-rooted. We will put the h*-real-rootedness of the simplices for numeral systems in context with that of their cousins, the s-lecture hall simplices, and discuss their admittance of this phenomena as it relates to other, more intrinsically geometric, reasons for h*-unimodality.

In the first part of this introductory talk, I will give you a hint of how and why the area of linear lifts of polytopes (AKA linear extended formulations) came into being. Starting with Yannakakis' fundamental contribution, which laid dormant for too long, I will take you to the results that revived the interest in lifts of polytopes. In the second part, I will explain some of the many connections that sustain the area, the amazing results that have been obtained recently, the open problems that remain, and the barriers that stand in our way.

Representing polytopes by means of linear matrix inequalities as been a highly successful strategy in combinatorial optimization. Geometrically it corresponds to writing a polytope as the projection of an affine slice of the cone of positive semidefinite (psd) matrices i.e., a spectrahedron. Efforts to understand the theoretical limits of such techniques have connected the existance of such representations to a particular type of matrix factorization, the psd factorization of a nonnegative matrix, and its corresponding notion of psd rank. In this talk we will do a brief survey of the main results in the area, its connections to matrix theory and combinatorics and some of the open problems that remain.

A spectrahedron is an affine slice of the cone of positive semidefinite matrices. Spectrahedra form a rich class of convex bodies that are computationally tractable and appear in many areas of mathematics. Examples include polytopes, ellipsoids, and more exotic convex sets, like the convex hull of some curves. I will introduce the theory of spectrahedra with many examples and discuss some applications in distance geometry and combinatorial optimization.

The Crossing Lemma  of Ajtai, Chvatal, Newborn, Szemeredi (1982) and Leighton (1983)states that if a graph of n vertices and e>4n edges is drawn in the plane, then the number of crossings between its edges must be at least constant times e^3/n^2. This statement, which is asymptotically tight, has found many applications in combinatorial geometry and in additive combinatorics. However, most results obtained using the Crossing Lemma do not appear to be optimal, and there is a quest for improved versions of the lemma for graphs satisfying certain special properties. In this talk, I describe some recent extensions of the lemma to multigraphs (joint work with G. Toth) and to families of continuous arcs in the plane (joint work with N. Rubin and G. Tardos, and with G. Tardos).