Mini-School program

A preparatory mini-school on Open-Closed Field Theories will be held on March 19-20 in 107 Harris Hall, with lecture courses to be delivered by David Ayala (Montana State University) and Nate Bottman (MPIM). There will also be single lectures by Ezra Getzler and Boris Tsygan.

You can find a copy of the schedule for the school here or below:

David Ayala: Factorization homology in low-dimensions: 1-tangles in 3-spaces

Notes: Day 1, Day 2
Abstract: Initial developments of factorization homology abstract the global observables of perturbative TQFTs, in which global observables are determined by point-observables/operators.  In this series, we will discuss upgrades of factorization homology tailored for a larger class of TQFTs, in which global observables are determined by point- and line-observables/operators — the discussion will be focused on low-dimensions.  In-route, we will outline a proof of the tangle-hypothesis in the case of 1-dimensional tangles in ambient 3-manifolds.  This is related to Skein-modules.  A key technical step will be a (fun!) construction of a the orthogonal group O(3) on a braided-monoidal (\infty,1)-category with duals.

 

Nate Bottman : Functoriality in categorical symplectic geometry

Abstract:  

Talk 1: An introduction to categorical symplectic geometry. (Notes) In this talk, I will introduce the central objects of categorical symplectic geometry: Floer cohomology HF^*(L,K), which is an intersection theory of Lagrangian submanifolds that is enriched by counts of pseudoholomorphic bigons and triangles; and the Fukaya category Fuk(M), which is a categorification of HF^*(L,K) that plays a starring role in Kontsevich’s Homological Mirror Symmetry conjecture. I will illustrate some of the basic properties and behavior in low-dimensional examples. Time permitting, I will survey the settings in which the Fukaya category can be computed with existing techniques.

Talk 2: The operadic principle. (Notes) The operadic principle in symplectic geometry says that if we define an invariant by counting some sort of pseudoholomorphic map, then its algebraic nature is inherited from the operadic nature of the underlying configuration spaces of domains. I will illustrate this in the case of the Fukaya category and in the case of functors associated to Lagrangian correspondences. Time permitting, I will introduce the open-closed and closed-open string maps, and use them to sketch a proof of Abouzaid’s generation criterion for Fukaya categories.

Talk 3: (Notes) Functors from Lagrangian correspondences, 2-associahedra, and the symplectic (A-infinity,2)-category (Symp). The functoriality properties of the Fukaya category are notoriously elusive. Over the last 15 years, work of Wehrheim—Woodward, Ma’u, Fukaya, and myself has led up to a well-behaved functoriality package for the Fukaya category. This package is Symp, the symplectic (A-infinity,2)-category, and in this talk I will survey this object. In particular, I will explain the central role of the 2-associahedra, which are compactified configuration spaces of marked vertical lines in R^2.

Talk 4: (Notes) Applications of functoriality. There is a wide variety of work that has applied the functoriality of the Fukaya category, either directly or indirectly. I will discuss some of these applications, depending on the interests of the audience. These applications may include Abouzaid—Seidel—Smith’s work on symplectic Khovanov homology, Lekili—Perutz’s work on Heegaard—Floer homology, Pascaleff and Subotic’s work on monoidal structures on Fukaya categories, Evans—Lekili’s work on Fukaya categories of G-manifolds, and Seidel’s formal group-valued invariant of closed monotone symplectic manifolds.

 

Ezra Getzler: Moduli spaces of real Riemann surfaces and open-closed topological field theory

Abstract: Topological field theory in two dimensions in the sense of Dijkgraaf and Witten and of Atiyah amounts to the study for a Riemann surface S of a graph whose vertices are pants decompositions of S, and whose edges correspond to Dehn-Lickorish moves. We may study this graph using Teichmüller theory, that is, by placing a hyperbolic metric on the surface. Bill Harvey explained how to view Teichmüller space as the interior of a manifold with corners, whose lowest dimensional strata correspond to pants decompositions of the surface. (He considers the closed case; the extension to surfaces with punctures was outlined by Kimura, Stasheff and Voronov.) Adjoining to these the next lowest dimensional strata gives a stratified space corresponding to the graph of pants decompositions.

In the first part of the talk, we show that there is a filtration on this manifold with corners such that the inclusion of the k-th filtrand is a k-connected map. For k=0, this filtrand is just the union of the lowest dimensional corners, and this is the statement that every hyperbolic surface has a pants decomposition; for k=1, this filtrand is precisely the result of adjoining the next lowest dimensional corners, and this becomes the statement that any pair of pants decompositions are connected by a sequence of Dehn-Lickorish moves; for k=2, we recover the Moore-Seiberg theorem, which gives a 2-dimensional simplicial complex whose fundamental group is the mapping class group of the surface.
In the remainder of the talk, we extend this story to open-closed topological field theory. We do this by replacing the surface with boundary by its double, which is a Klein surface, or real Riemann surface (a Riemann surface with orientation reversing automorphism). Every real Riemann surface has a real pants decomposition. There are five moves between them, including the famous Cardy condition, compared to two Dehn-Lickorish moves. We will show how to construct a filtration on the real analogue of Harvey’s moduli space.

 

Boris Tsygan: Topological quantum mechanics and index

Abstract: I will explain how the simplest example of TQFT, topological quantum mechanics, is related to homological algebra and index theorems.

 

 

We hope to be able to provide funding to support the attendance of junior participants at the school.