Southern California Algebraic Geometry Seminar Fall 2011 Abstracts

Speaker: Ionut Ciocan-Fontanine

Title: Stable quasimaps to GIT quotients

Abstract:  I will discuss certain new compactifications with good properties of moduli spaces of maps from nonsingular marked curves to a class of GIT quotients, generalizing from a unified perspective many particular examples considered earlier in the literature. If time permits, I will also present some applications to Gromov-Witten theory. The talk is based on joint works with Bumsig Kim and Davesh Maulik. 

Speaker: Alexander Duncan

Title: Versal varieties and twisting

Abstract:  Let G be a linear algebraic group. Informally, a versal G-torsor is a G-torsor to which any other G-torsor maps (all torsors are over a field). Versal torsors behave similarly to universal objects, but fail to satisfy certain uniqueness propreties. We have the amusing equation: "uni(que)+versal=universal." In this talk, I introduce several variations of the notion of versality for a G-variety X. These notions have equivalent formulations as properties of twisted forms on X: existence of k-points, density of k-points, k-unirationality and so on. I will discuss applications of this equivalence to the versality of homogeneous spaces, toric varieties, quadratic forms and cubic forms. This is joint work with Zinovy Reichstein. 

Speaker: Aaron Lauda

Title: Odd categorification and cohomology

Abstract:  The nilHecke algebra plays a fundamental role in the cohomology of partial flag varieties and the theory of symmetric functions. More recently this algebra appears as one of the basic building blocks for categorifying quantum sl(2). We will discuss an `odd' analog of the nilHecke algebra and explain how it gives rise to a new non-commutative theory of symmetric functions, cohomology of flag varieties, and categorification of quantum sl(2).  This is a joint work with Alexander Ellis and Mikhail Khovanov.

Speaker: Yu-jong Tzeng

Title: Counting curves with arbitrary singularities on surfaces

Abstract:  A famous problem in classical algebraic geometry is how many r-nodal curves there are in a linear system |L| on an algebraic surface. If the line bundle L is sufficiently ample, Gottsche conjectured that the number of r-nodal curves is a universal polynomial in the Chern numbers of L and S.  This conjecture was proven independently by Tzeng and Kool-Shende-Thomas. In this talk we will generalize Gottsche's conjecture and show the number of curves with any number of arbitrary isolated singularities are also given by universal polynomials.