Abstracts - Fall 2015

Speaker:  Jim Bryan

Title: Curve counting on Abelian surfaces and threefolds, and Jacobi forms

Abstract: We explain how generating functions for curve counting problems on Abelian surfaces and threefolds are given by a certain nice Jacobi form.  A new computational technique mixes motivic and toric methods and makes a connection between the topological vertex and Jacobi forms.

Speaker: Dragos Oprea

Title: On the tautological ring of the moduli space of K3 surfaces

Abstract: I will discuss joint work with Alina Marian and Rahul Pandharipande aimed at studying the tautological ring of the moduli space of K3 surfaces. The method involves equivariant localization of virtual fundamental classes of certain relative Quot schemes. We produce in this fashion relations intertwining the kappa classes and the Noether-Lefschetz loci. Along the way, we also obtain closed formulas for the top Segre classes of tautological bundles over the Hilbert schemes of points of a fixed K3 surface, confirming a prediction of Lehn.

Speaker: Giulia Sacca

Title:  Geometry of O'Grady's six dimensional example

Abstract:  There are not many known examples of compact irreducible hyperk\"ahlermanifolds. Two series of examples appear in dimension 2n, for every n \ge 2, and are related to the Hilbert scheme of points on a K3 or an abelian surface; and in dimension 6 and 10 there is one extra, or exceptional, deformation class, each of which was found by O'Grady.  While considerable work has been devoted to studying hyperkahler manifolds belonging to the first two deformation classes, not much is known for the exceptional deformation classes. In this talk I will present joint work with G. Mongardi and A. Rapagnetta, regarding the geometry of O'Grady's six dimensional example. By realizing these examples as ``quotients'' of another hyperkahler manifold by a birational involution, we are able to compute all the Hodge numbers and study properties of their moduli spaces.

Speaker:  Xinwen Zhu

Title: Principle B for de Rham representations

Abstract: I will discuss the following rigidity theorem for Q_p etale local systems on connected algebraic varieties over p-adic fields: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. This is joint work with Ruochuan Liu.