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Research Program Abstracts

Ravi Vakil-

Title: The geometric Littlewood-Richardson rule

Oleg Musin-

Title: The kissing problem in four dimensions

Abstract:

The kissing number k(n) is the maximal number of equal size nonoverlapping spheres in n dimensions that can touch another sphere of the same size.  The number k(3) was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. Newton said that 12 should be the correct answer, while Gregory thought that 13 might be possible. This so-called "thirteen spheres problem" was finally solved by Sch\"utte and van der Waerden in 1953 only.  Philippe Delsarte in 1973 described a "linear programming method" that does allow one to prove good bounds on k(n). Using this method in 1979 Levenshtein and at the same time Odlyzko & Sloane proved that k(8)=240 and k(24)=196560.  The so-called 24-cell, a four-dimensional "platonic solid", yields a configuration of 24 bals that would touch a given one in four dimensional space. It was proved (Arestov & Babenko) that Delsarte's method the best upper bound one could get is 25, i.e. k(4)=24 or 25.  In this talk we present an extension of the Delsarte method and use it to prove that k(4)=24. We also present a new proof that k(3)=12.

Patrick Brosnan-

Title: Matroids, motives and Feynman integration (joint with Prakash Belkale)

Abstract:

I will discuss the connections between matroids and certain varieties arising in the evaluation of Feynman integration.

Jennifer Morse-

Title: Theorems and conjectures about k-Schur functions

Kevin Purbhoo-

Title: Minuscule flag manifolds and the Horn recursion

Neil White-

Title: An introduction to Coxeter matroids

Anders Buch-

 Title: Alternating signs of quiver coefficients

Komei Fukuda-

Title: Old and new ideas of constructing nonrepresentable oriented matroids

Abstract: We look at old and new ideas of constructing nonrepresentable oriented matroids (OMs). We start with the old idea of constructing a nondegenerate cycling in oriented matroid programming (yielding a non-Euclidean OM). Our new ideas make use of the Holt-Klee condition for the LP-orientations in two ways, in the primal way and the dual way. One of the ways turns out to be quite useful and the other is still quite a mystery, i.e. the basic question as to whether it can be used to construct nonrepresentable OMs remains open. (Based on joint work with S. Moriyama and Y. Okamoto)

Nick Proudfoot-

Title: Hypertoric varieties

Abstract: A hypertoric variety is a hyperkahler analogue of a toric variety, defined as the hyperkahler quotient of a quaternionic vector space by a torus. Just as there is a rich interaction between toric geometry and the combinatorics of polytopes, hypertoric geometry is deeply related to hyperplane arrangents and matroids.  I will explain how to construct the hypertoric variety corresponding to a real hyperplane arrangement, emphasizing both the similarities to and the differences from the theory of toric varieties. Iwill then discuss the ordinary and intersection cohomology of hypertoric varieties, with applications to matroid theory.

Laura Matusevich-

Title: Local cohomology of semigroup rings and GKZ systems

Federico Ardila-

Title: Amoebas, matroids, and phylogenetic trees

Abstract: Motivated by a problem about solving systems of polynomial equations, we are led to study the "Bergman complex of a matroid": a certain polyhedral complex related to matroid optimization. We study the combinatorics and topology of this complex.  Somewhat surprisingly, we obtain some new results about the well-known space of phylogenetic trees as a consequence.  (This is joint work with Carly Klivans.)

Eric Sommers-

Title: Exponents for ideals of positive roots

Abstract:  In joint work with Tymoczko, we define a set of exponents for each upper order ideal in the poset of positive roots. This generalizes the usual notion of exponents (which we recover when the ideal is the empty set).  The talk deals with two conjectures for these exponents which generalize classic theorems: one concerns a factorization of an analogue of the Poincare polynomial of the Weyl group, the other concerns a factorization of the characteristic polynomial of a certain hyperplane arrangement. We explain a proof of these conjectures in types A, B, and C.

David Speyer-

Title: Combinatorics of Tropical Linear Spaces

Abstract: Let $K$ denote the field of Laurent Series with exponents and coefficient coefficients and let $v: K^* \to \mathbb{R}$ be the map that takes a power series to the exponent of its lowest degree term. If $X \subset K^n$ is any variety, we define $\mathrm{Trop\ } X$ to be the image of $v : X \cap (K^*)^n \to \mathbb{R}^n$. $\mathrm{Trop\ } X$ is a polyhedral complex, whose geometry is hoped to reflect the geometry of $X \cap (K^*)^n$.

In this talk, we discuss the case where $X$ is a linear subspace of $K^n$, so $X \cap (K^*)^n$ is a hyperplane arrangement. Then the study of $\Trop X$ becomes very combinatoreal and can be described in terms of polyhedral decompositions of the hypersimplex arising from certain collections of matroids. We give a full description of all possible combinatorics when $\dim X=2$ and (relying on work of Ardila and Klivans) of the local geometry of $\Trop X$. We will then describe a precise conjecture and some partial results concerning the maximal $f$-vectors in the general case.

Michael Joswig-

Title: Tropical Halfspaces

Abstract: As a new concept tropical halfspaces are introduced to the (linear algebraic) geometry of the tropical semiring (R,min,+). This yields exterior descriptions of the tropical polytopes that were recently studied by Develin and Sturmfels.  The key tool to the understanding is a newly defined sign of a tropical matrix, which shares remarkably many properties with the sign of the determinant of an ordinary matrix. The methods are used to obtain complete information about the tropical convex hull problem in two dimensions."

Dmitry Kozlov- TBA

Jon McCammond-

Title: Non-crossing partitions for arbitrary Coxeter groups

Abstract: It is well-known that the classical lattice of non-crossing partitions corresponds to a portion of the Cayley graph of the symmetric group and that the Cayley graphs of the other finite Coxeter groups lead to other finite lattices with similar properties. In this talk I will describe how infinite Coxeter groups lead to infinite posets which retain many of the properties enjoyed by their better known cousins (for example they are bounded, graded, finite height, self-dual and locally self-dual and have a particularly nice edge labeling). In addition, they can be used to better understand the fundamental groups of the complexified hyperplane arrangements associated with the infinite Coxeter groups.  These fundamental groups are called Artin groups and the newly defined noncrossing partition lattice for arbitrary Coxeter groups lead to new results about all Artin groups. This is joint work with Noel Brady, John Crisp and Anton Kaul.