Donu Arapura The Beilinson-Hodge conjecture
The conjecture is that weight 2i cycles on the ith
cohomology of smooth complex variety come from motivic cohomology. I'll spend about half of the talk explaining what all of this means, and the other half explaining why it's true for semiabelian varieties. This is joint work with Manish Kumar. Joseph Lipman
A bivariant theory of Hochschild homology
A bivariant theory based on sheafified Hochschild homology
and Grothendieck duality will be described, on the category of schemes that are flat, separated and essentially of finite type over a fixed noetherian scheme S. The cohomology and homology groups thereby associated to such an x:X --> S, with Hochschild complex H_X, are Ext^i(H_X, H_X) and Ext^{-i}(H_X, x^!O_S). The hardest result is the existence of orientations for the class of flat S-morphisms f:X --> Y, by canonical relative fundamental class maps H_X --> f^!H_Y. Together with the canonical map from top-degree relative differentials to Hochschild homology, the fundamental class map gives, for Y=S, a fundamental class Omega^{top}_{X/S} --> x^!O_S, known previously only in characteristic 0 or for smooth x. Nero Budur
Singularities of hyperplane arrangements
We review different techniques for studying singularities of varieties. We give an overview of the status of computing some singularity invariants. We present some new results on singularity invariants of hyperplane arrangements and some concrete open problems. These problems are related to: combinatorics, to an interpolation problem for points, and to the number of solutions modulo prime powers of an equation. Tom Nevins Hilbert schemes and cotangent bundles The Hilbert scheme of n points on the affine plane---or, more generally, the cotangent bundle of a smooth curve---has a concrete construction via Hamiltonian reduction. One sees by this construction that the Hilbert scheme is itself "almost a cotangent bundle." I'll describe some work in progress aimed at better understanding the discrepancy between the Hilbert scheme and a cotangent bundle, and what happens when one quantizes the construction. Bill Fulton Character formulas
In this expository talk, we show how a Riemann-Roch formalism leads to a simple proof of a general formula for restrictions of equivariant line bundles to fixed points. On homogeneous varieties it gives Weyl's character formula, and on toric varieties it gives Brion's formula for lattice points in polytopes. This is based on ideas of George Quart in the 1970's and recent conversations with Bill Graham. Lev Borisov In search of families of dg-algebras related to resolutions of Gorenstein toric singularities. A Gorenstein toric singularity can be described by simple combinatorial data, namely a convex polytope $P$ in ${\bf Z}^n$ with integer vertices. Different triangulations of $P$ with vertices given by integer points of $P$ give rise to different resolutions of the singularity. It has been shown that bounded derived categories of coherent sheaves on these resolutions are equivalent. It is reasonable to expect that there is in fact a continuous family of triangulated categories that includes these categories as its limit points. This is very much work in progress, and the main questions are still wide open. It is my hope that by bringing this problem to your attention I can inspire someone to find such construction. |