Abstracts - Spring 2018

Speaker:  Ben Antieau

Title: The topological period-index problem

Abstract: I will survey recent breakthroughs on the topological period-index problem for Brauer groups and their connection to and implications for the algebraic period-index problem.

Speaker: Barbara Fantechi

Title:  Deformations of varieties with transversal ADE singularities and their crepant resolutions

Abstract: This is joint work in progress with Alex Massarenti. ADE singularities are canonical surface singularities; they are normal, quotient singularities, have no moduli, and their minimal resolution is crepant. A complex variety X has transversal ADE singularities along a smooth subvariety Z if \'etale locally the pair (X,Z) is isomorphic to the product of of an isolated surface singularity (S,p) with affine space. We denote by Y the crepant resolution of X. We study the relationship between the functor Def_X of infinitesimal deformations of X and Def_Y with LTDef_X, the functor of locally trivial deformations of X. While the problem can be stated completely in the language of schemes, the solution involves the Deligne-Mumford stack associated to X. 

Speaker: Tommaso de Fernex

Title:   The complex Plateau problem

Abstract:  The complex Plateau problem has been studied in the context of CR manifolds in works of Rossi (1965) and Harvey and Lawson (1975) where it is proved that every embedded strongly pseudoconvex compact CR manifold of dimension at least 3 is the boundary of a Stein space with isolated singularities. What remains to be determined are intrinsic conditions on the CR manifold that guarantee that the Stein filling is smooth. In the talk I will survey this question and discuss some old and recent developments on the problem.

Speaker: Hiraku Nakajima

Title: Moore-Tachikawa 2d TQFTs whose values are holomorphic symplectic varieties

Abstract: An Atiyah-Segal 2d TQFT associate a vector space V to (oriented) S^1, a tensor product of copies of V and its dual to disjoint union 

of (oriented) S^1's, and a linear map for a cobordism.  Given a complex simple algebraic group G, Moore-Tachikawa conjectured an existence 

of 2d TQFT which takes values in holomorphic symplectic manifolds with hamiltonian G-actions (arXiv:1106.5698). Here the composition of 

cobordisms is given by the hamiltonian reduction by the diagonal subgroup G in GxG corresponding to the glued boundary. This conjecture was 

solved by Ginzburg-Kazhdan (unpublished). I will explain Moore-Tachikawa's conjecture and an equivalent construction in terms of ring objects 

in the equivariant derived Satake category. This is a joint work with Braverman and Finkelberg.