Abstracts - Fall 2012

Speaker:  Sabin Cautis

Title:  The geometry of infinite spherical twists

Abstract: The contangent bundle to the projective line T^*P^1 has an interesting derived autoequivalence given by a spherical twist. Applying this twist many times turns out to have a well defined limit. We will describe this limit and relate it to the geometry of the quadric cone. This construction generalizes to other Nakajima quiver varieties and can subsequently be used to define homological knot invariants.

Speaker:  Emanuele Macri

Title:  MMP for moduli spaces of sheaves on K3 surfaces and Cone Conjectures

Abstract:  We report on joint work in progress with A. Bayer on how one can use wall-crossing techniques to study the birational geometry of a moduli space M of Gieseker-stable sheaves on a K3 surface X. In particular:

(--) We will give a "modular interpretation" for all minimal models of M.

(--) We will describe the nef cone, the movable cone, and the effective cone of M in terms of the algebraic Mukai lattice of X.

(--) We will establish the so called Tyurin/Bogomolov/Hassett-Tschinkel/Huybrechts/Sawon Conjecture on the existence of Lagrangian fibrations on M.

Speaker:  Andrei Negut

Title:  Moduli of Sheaves and Shuffle Algebras

Abstract:  Due to a classical result of Nakajima and Grojnowski (rank 1 case) and Baranovsky (general case), the infinite-dimensional Heisenberg Lie algebra acts in a geometric way on the cohomology groups of the moduli spaces of torsion free sheaves on a surface. We define a geometric action of gl_1^ on the K-theory groups of the same spaces, all the while uncovering and making explicit the action of a richer structure: the shuffle algebra.

Speaker:  Ben Williams

Title:  Obstruction theory and the Brauer group 

Abstract:  The generalization to a scheme, X, of central-simple algebras over a field are Azumaya algebras; that is to say locally-free O_X-algebras A that are \'etale-locally isomorphic to a matrix algebra. An analogous concept, that of a topological Azumaya algebra, may be defined for a topological space. When X is a complex variety, complex realization allows us to view an Azumaya algebra as a topological Azumaya algebra. We may use obstruction theory to prove non-existence results for Azumaya algebras; for instance we can give lower bounds on the ranks of Azumaya algebras in terms of the singular cohomology of a complex variety X. When X is of dimension 3, these bounds suggest the period-index conjecture may be false.