Hom-polytopes Quotient polytopes Hom-varieties for polytopal algebras Counting lattice polytopes by lattice points Infinitesimal nonlinear extension of GL_n Toric maximum likelihood estimation Detailed description is here and here.

Almost-transitive actions on sphere products

Abstract: We determine the structure of transitive actions of compact Lie groups on products of two spheres where one of the two spheres is one-dimensional, or more generally on spaces which have the same rational homotopy as such a sphere product. Furthermore, we show that if a non-compact simply connected Lie group acts transitively on such a space, then the orbits of the maximal compact subgroups are simply connected rational cohomology spheres.

Cohen-Macaulayness of initial ideals Degrees of Graver basis elements Ehrhart theory The volume of cyclic polytopes Roots of polynomials Magic Detailed description is here and here.

Hyperplane arrangements, partial cubes, well graded families of sets, and preference structures

Abstract: There will be a rather informal presentation of some relations existing between arrangements, preference modeling, and other related structures.

Graded Retractions of a Stanley-Reisner Ring

Abstract: Linear algebra can be viewed as a special case of a more general theory based on simplicial complexes, linear algebra itself corresponding to a single simplex. We will explore the possibility of extending to this more general setting the basic linear algebra fact that an idempotent matrix is conjugate to a subunit matrix. We will also talk on motivation, previous work, and the methods used to attack this problem, which include the theory of Stanley- Reisner rings and rudiments of K-theory.

1 p.m. Charles Cochet Universite Paris 7 - Denis Diderot		2 p.m. Seth Sullivant University of California, Berkeley
The number of integral points in convex polytopes and applications to Lie theory and network flows Abstract: The enumeration of integral points appears in numerous parts of mathematics, such as Lie theory, network flows, magic squares, partitions of integers, representation theory, crystal bases of quantum groups, and statistics. We will review several classical algorithms, and then describe a new one for polytopes arising from Lie theory.		Compressed polytopes and statistical disclosure limitation Abstract: An integral polytope is called compressed if all of its pulling triangulations are unimodular. Somewhat surprisingly, this strong condition on the structure of the triangulations of a polytope is equivalent to a strong condition about the facets of that polytope. This equivalence has been rediscovered a number of times, and has a number of applications. I will explain how compressed polytopes arise when statistical agencies try to protect confidential survey data.

Reconstructing generalized quadrangles

Abstract: Generalized polygons were introduced by Jacques Tits around 1960 in order to give geometric interpretations for all simple Lie groups and simple algebraic groups including the exceptional types. Among these polygons are generalized 3-gons, also called projective planes, which had been studied for a long time, but most fascinating are generalized 4-gons, also called generalized quadrangles, because they form the richest class. It is a classical and easy to prove result that it is equivalent to study projective planes (generalized 3-gons) and affine planes, which are defined using a variation of Euclid's famous parallel axiom. Affine planes are important in the theory of projective planes, since, for example, many of the known examples are given by constructing an affine plane first and then completing it to a projective plane in a general process. In this talk a possible notion of affine quadrangles, analog to that of affine planes, is presented, and it is shown that some of the known generalized quadrangles have an easy description in terms of affine quadrangles, which makes them easy to visualize. In the second part of the talk the notion of affine quadrangles is applied to the theory of topological generalized quadrangles.

Algebraic Statistics for Computational Biology

Abstract: We discuss recent interactions between algebra and statistics and their emerging applications to computational biology. Statistical models of independence and alignments for DNA sequences will be illustrated by means of a fictional character, DiaNA, who rolls tetrahedral dice with face labels A, C, G, T. For a picture of DiaNA and an on-line version of our book on this subject see here. (This lecture is aimed at a general interdisciplinary audience.)

Gorenstein Liaison in P4

Abstract: The theory of liaison (or linkage) of curves in projective 3-space has been a very good tool in the study of curves, their Hilbert schemes, special curves and so on. The idea is to study properties of curves by looking at "simpler" curve that is in some way similar to the original one. This theory does not generalize well to curves in higher dimensional spaces unless we introduce notion of Gorenstein Liaison. In this talk I will give introduction to the Gorenstein Liaison theory in projective 4-space and discuss some open problems. I'll focus on the results holding on some 3-folds in a projective 4-space.

1 p.m. Fu Liu Massachusetts Institute of Technology		2 p.m. Kevin Woods University of California, Berkeley
Identities of Ehrhart polynomials arising from algebraic geometry Abstract: Mochizuki's work on torally indigenous bundles yields combinatorial identities by generating to different curves of the same genus. We rephrase these identities in combinatorial languages and strengthen them. The main result of our work is that given a certain way to construct a polytope from a connected trivalent graph, the odd values of the Ehrhart quasi-polynomials of this polytope are given by a single polynomial which depends only on the number of vertices and edges of the original trivalent graph. This is joint work with Brian Osserman.		Counting with generating functions Abstract: Here is an example: given vectors a1, ..., an, let f(s) be the number of ways to write the vector s as a nonnegative integer combination of the ai. This function f(s) is called the vector partition function, and it can be encoded nicely as a generating function. But suppose we would like to find an explicit representation of the function f(s), using, say, the greatest integer function. I will show how to find this representation quickly (in polynomial time). Indeed, given a rational generating function encoding of any sort of counting function c(s), we can find an explicit representation for c(s) quickly. Another interesting example of such a function is the Ehrhart quasi-polynomial, which I will also discuss. This is joint work with Sven Verdoolaege.

A completely accessible and historically motivated introduction to the theory of partitions and its connections to number theory, combinatorics, representation theory, complex analysis, group theory, modular forms, continued fractions, patience, chess, particle physics, and ping-pong

Abstract: In this talk, we take a whirlwind tour of the theory of partitions. Beautiful results from this area's rich history will presented, and the connections between partition theory and many other fields will be discussed. The talk will be aimed at the partition-theoretically uninitiated, and should be accessible to everyone.

Codimension-2 Hilbert Bases

Abstract: Hilbert Bases were born in 1979 in the context of discrete optimization, and it has become clear they are involved in many basic combinatorial problems. Aside from this they are interesting objects in their own right, with rich discrete-geometric and algebraic flavors. I'll discuss Hilbert Bases with few generators with respect to dimension. This brings us into the realm of Gale Diagrams and initial ideals of toric ideals. Gale Diagrams are a tool for visualizing high dimensional polytopes. Both of these topics may be of independent interest.