Geometry/Physics Seminar

Title: Moduli Spaces of Microlocal Sheaves and Cluster Combinatorics
Speaker: Harold Williams
Speaker Info: UT Austin
Brief Description:
Special Note:

We explore a relationship between combinatorics and certain moduli spaces appearing in symplectic geometry. The combinatorics comes from the theory of cluster algebras, a kind of unified theory of canonical bases in representation theory and algebraic geometry. Some basic features of cluster algebras are that they are defined from purely combinatorial data (for example, a quiver) and they are coordinate rings of varieties covered by algebraic tori with transition functions of a special, universal form. Despite the originally representation-theoretic motivation for the subject, connections between cluster theory and symplectic geometry emerged later through the appearance of similar formulae in wall-crossing and mirror symmetry. We will discuss recent work expanding on this connection, in particular providing a universal framework for interpreting cluster varieties as moduli spaces of objects in Fukaya categories of Weinstein 4-manifolds. In simple examples these moduli spaces reduce to well-known ones, such as character varieties of surfaces and positroid cells in the Grassmannian. An accompanying theme, which plays a key role both technically and in relating the symplectic perspective to more standard representation-theoretic ones, is the role of categories of microlocal sheaves as topological counterparts of Fukaya categories. This is joint work with Vivek Shende, David Treumann, and Eric Zaslow.
Date: Thursday, October 1, 2015
Time: 4:00pm
Where: Lunt 107
Contact Person: Theo Johnson-Freyd
Contact email: theojf@math.northwestern.edu
Contact Phone:
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