Talk titles and Abstracts:

Mihai Ciucu, Cruciform regions and a conjecture of Di Francesco
The problem of finding formulas for the number of tilings of lattice regions goes back to the early 1900's, when MacMahon proved (in an equivalent form) that the number of lozenge tilings of a hexagon is given by an elegant product formula. In 1992, Elkies, Kuperberg, Larsen and Propp proved that the Aztec diamond (a certain natural region on the square lattice) of order n has 2^n(n+1)/2 domino tilings. A large related body of work developed motivated by a multitude of factors, including symmetries, refinements and connections with other combinatorial objects and statistical physics. It was a model from statistical physics that motivated the conjecture which inspired the regions we will discuss in this talk.

In recent work on the twenty vertex model, Di Francesco was led to a conjecture which states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted T_n, are obtained by starting with a square of side-length 2n, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order n-1. Inspired by the regions T_n, we construct a family of cruciform regions C_m,n^a,b,c,d generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of the region T_n is a divisor of the number of tilings of the cruciform region C_2n-1,2n-1^n-1,n,n,n-2, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco's conjecture.

Richard Ehrenborg, Sharing pizza in n dimensions
We introduce and prove the n-dimensional Pizza Theorem. Let H be a real n-dimensional hyperplane arrangement. If K is a convex set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement H. We prove that if H is a Coxeter arrangement different from A₁ⁿ such that the group of isometries W generated by the reflections in the hyperplanes of H contains the negative of the identity map, and if K is a translate of a convex set that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of H that we call the even restricted arrangement. We get stronger results in the case of balls. We prove that the pizza quantity of a ball containing the origin vanishes for a Coxeter arrangement H with |H|-n an even positive integer. This is joint work with Sophie Morel and Margaret Readdy.

Eric Katz, Models of matroids
Matroids are known for having many equivalent cryptomorphic definitions, each offering a different perspective. A similar phenomenon holds in the algebraic geometric approach to them. We will discuss different ways of viewing matroids, motivated by group actions, intersection theory, and K-theory, each of which has been generalized from the realizable case to a purely combinatorial approach.

Lei Xue, A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds
In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s ≤ 2d vertices has at least φ_k(d+s, d) = {d+1 choose k+1} + {d choose k+1} - {d+1-k \choose k+1} k-faces. In the talk, we will prove this conjecture and discuss equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on d-dimensional polytopes with 2d+1 or 2d+2 vertices.