Pic-graded betti numbers and the Cox rings of Del Pezzo surfaces

The Cox ring of an algebraic variety is a generalization of the homogeneous coordinate rings of toric varieties. An important class of examples of non-toric varieties with finitely generated Cox rings are Del Pezzo surfaces (obtained by blowing up P2 at r general points (3<r<9)). In this talk we will introduce some tools from combinatorial commutative algebra which can be used to obtain presentations for these rings as quotients of polynomial rings proving a conjecture of Batyrev and Popov.

Prime time for primes

As old as Euclid, prime numbers have recently started to yield their secrets. Mathematicians from California to India and elsewhere have shown us that primes regularly fall into strict patterns, they display unusual "clumping," and they are computationally easy to detect. While many mysteries remain, it does seem that this first decade of the new millennium is indeed a prime time for primes.

Covering congruences

A famous old problem of Paul Erdos is whether for each number B the set of integers may be covered with a finite collection of congurence classes with distinct moduli each at least B. In fact, Erdos wrote of this as his "favorite problem". In recent work with Filaseta, Ford, Konyagin, and Yu, we have found some new results in this area; for example, if such finite collections should exist, the largest modulus cannot be O(B). These new results settle some conjectures of Erdos, Graham, and Selfridge.

Computing local cohomology via D-modules

Consider a regular ring R, e.g., R=k[x1, ..., xn] with the ground field k of characteristic 0. Local cohomology modules Hk_I(R) for an ideal I of R can be computed algorithmically viewing them as D-modules. I will present two methods: one for explicit computation of the local cohomology modules, the other for the computation of their characteristic cycles. Understanding local cohomology may be useful, in particular, to answer the question whether a set of generators for an ideal I is minimal. This is joint work with Josep Alvarez.

Koszul Duality and Stability

This is joint work with R. Martinez Villa. The Bernstein-Gelfand-Gelfand correspondence establishes an equivalence between the stable category of finitely generated graded modules over the exterior algebra on n+1 letters and the bounded derived category of coherent sheaves on the n-dimensional projective space. In this non-technical talk we will establish a far-reaching generalization of this result. Our approach is based on Koszul duality and universal constructions. All terms will be defined and explained.

Knot invariants via derived categories of coherent sheaves

I will begin by giving the skein relation definitions of the Jones polynomial and Khovanov homology, two powerful knot invariants which have received a great deal of attention in recent years. Then I will explain how a representation theorist thinks about the Jones polynomial. This will lead us to the problem of categorification of tensor products of representations of sl(2). I will explain a categorification (joint with Sabin Cautis) using the derived categories of coherent sheaves on certain varieties arising in the geometric Langlands program. In particular, this allows us to construct Khovanov homology using algebraic geometry.

Volume and Ehrhart polynomials of polytopes

We discuss the relationship between the volume and the Ehrhart polynomial of an integral polytope. After a brief introduction to background results in the theory of Ehrhart polynomials, we define a family of polytopes which can be considered as a generalization of cyclic polytopes. The coefficients of the Ehrhart polynomial of these polytopes are given by volumes of iterated projections of the polytopes. Next, in joint work with Jesus De Leora and Ruriko Yoshida, we give a formula for the volume of the Birkhoff polytope obtained by a calculation of its Ehrhart polynomial.

Kim and Asia will give previews of their talks at the Graduate Student Combinatorics Conference in Seattle. Asia will talk about Dedekind-Carlitz polynomials and polyhedral geometry, and Kim about growth series of root lattices.

San Francisco State University

An application of exterior face algebras to geometric group theory

My main topic of research is that of tree braid groups: A tree braid group is the fundamental group of the space of all configurations of n distinct points on a given tree. I will state some results about tree braid groups, including a solution to the 'isomorphism problem' for tree braid groups. The proof of 'isomorphism problem' result relies on rigidity results for exterior face algebras and consequently tree braid groups. I will discuss these rigidity results and pose a specific open question about exterior face algebras and their quotients.

2:10 p.m. Milena Hering Institute for Mathematics and its Applications (Minneapolis)		3:10 p.m. Alan von Herrmann Colorado State University
Syzygies of toric varieties Understanding the equations defining algebraic varieties and the relations, or syzygies, between them is a classical problem in algebraic geometry. Green showed that sufficient powers of ample line bundles satisfy property Np, i.e., they induce a projectively normal embedding that is cut out by quadratic equations and whose first p-1 syzygies are linear. I will give an introduction to property Np, and present some sufficient criteria for powers of line bundle on toric varieties to satisfy it.		Constructing the Maximal Subgroups of the Finite Classical Groups In this talk I will discuss background materials that we can use to construct the maximal subgroups of the finite classical groups.

Vertex cover algebras and hypergraphs

In this talk I report on a joint work with Jütrgen Herzog, Takayuki Hibi and Ngo Viet Trung. Herzog, Hibi and Trung introduced vertex cover algebras of weighted simplicial complexes in the paper "Symbolic powers of monomial ideals and vertex cover algebras''. These algebras are special classes of symbolic Rees algebras. They show that symbolic Rees algebras of monomial ideals are finitely generated and such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. They give in general a upper bound for the maximal degree of generators of vertex cover algebras. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. Later, I joined their work and we characterize simplicial complexes which have standard graded vertex cover algebras. It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalizes bipartite graphs and trees. These relationships allow us to obtain very general results on standard graded vertex cover algebras which cover all previous major results on Rees algebras of facet ideals.

3:10 p.m. Cristiano Bocci University of Milano		4:10 p.m. Asia Matthews San Francisco State University
New trends in defectivity Starting from the definitions of defectivity and weak defectivity for a projective algebraic variety, I explain how a deeper study of these concepts permits to discover hidden (and unexpected) properties of the projective varieties. These new properties permit to generalize weak defectivity to sub-defectivity and weakly-3P-defectivity. Finally, I will give a characterization theorem of weakly-3P-defective surfaces in Pr and subdefective surfaces in P6, showing, in this last case, an interesting result which links behaviours of tangent spaces and osculating spaces.		A geometric approach to Carlitz-Dedekind sums A Carlitz polynomial is a polynomial generalization of the Dedekind sum, which in turn is an arithmetic sum playing a central role in various mathematical areas, such as theta functions, group actions on manifolds, and integer-point enumeration in polytopes. The most important property of any Dedekind-like sum is reciprocity which Carlitz proved algebraically for his polynomials. In this paper we give a geometric proof of Carlitz reciprocity and derive Dedekind reciprocity from our result. This approach gives rise to alternate geometric pictures from which we get a new version of Carlitz reciprocity and some new theorems. Finally, using Brion's decomposition theorem for lattice points in polyhedra, we discover two new theorems relating Carlitz sums to the generating function of two and three-dimensional simplices. In three dimensions we rederive the Mordell-Pommersheim theorem, which marks the first appearance of Dedekind sums in Ehrhart polynomials. This talk is Asia's MA thesis defense.