Parameterized Morse theory in low-dimensional and symplectic topology

The first part of the talk will be a quick survey on the homotopy type of the group of Diffeomorphisms of a 3-manifold. Pioneering work was done by Hatcher and Ivanov in the late 70s and early 80s, computing this for Haken 3-manifolds and famously the Smale conjecture. Following Thurston, the natural question is whether Diff is homotopy equivalent to the group of isometries, for geometric 3-manifolds. Gabai showed this is true in the hyperbolic case. McCullough and others studied Seifert fibred spaces and many spherical classes of examples. In the second part, I will talk about Heegaard splittings, a natural view of Morse theory for 3-manifolds. Casson-Gordon in the mid 1980s introduced a key idea of strong irreducibility and this was extended by Scharlemann-Thompson to telescoping. There have subsequently been many developments, including classification of splittings for all 7 non hyperbolic geometries of Thurston. Comparing splittings was introduced by Scharlemann and I, and distance of splittings by Hempel. If time permits the relationship with hyperbolic geometry will be sketched. Speaker: Josh Sabloff Title: Families of Legendrian Submanifolds via Generating Families Abstract: I will introduce a framework to investigate families of Legendrian submanifolds using generating family homology through an application of the families theory to the analysis of a loop of Legendrian nspheres in the standard contact space that is contractible in the smooth, but not Legendrian, category; this is joint work with Mike Sullivan. The computation of generating family homology necessary for the application comes from joint work with on Lagrangian cobordisms withFrederic Bourgeois and Lisa Traynor. Speaker: Martin Scharlemann Title: The Schönflies Conjecture and its spin-offs Abstract: We briefly review the resolution of the Schnflies Conjecture in all dimensions other than four, discuss why the remaining conjecture is important, and the classic approach to its resolution. This approach has spawned much beautifully pictorial mathematics, without actually succeeding. An underlying theme is that, although the conjecture has not yet been settled, it interlocks with and has inspired much interesting topology in dimensions three and four. Speaker: Chris Schommer-Pries Title: From the cobordism hypothesis to higher Morse theory Abstract: This talk will survey some recent developments in our understanding of extended topological field theories and their classification. This includes the cobordism hypothesis and related results. In the course of this talk we hope to make clear the role of higher Morse theory in this story. Speaker: Lisa Traynor Title: An Introduction to Symplectic and Contact Topology and the Technique of Generating Families Abstract: I will give a brief introduction to some of the major objects in symplectic and contact topology: symplect and contact manifolds, Lagrangian and Legendrian submanifolds, and symplectic and contact diffeomorphisms. Then I will describe the technique of generating families: this is a way to encode a Lagrangian or Legendrian submanifold by a parameterized family of functions. Morse-theoretic constructions then lead to generating family (co)homology groups for a Legendrian submanifold and wrapped generating family (co)homology groups for a Lagrangian cobordism. I will also describe how from a Lagrangian cobordism with a generating family, one obtains a cobordism map that satisfies some of the typical properties of a TQFT. Speaker: Jamie Vicary Title: Computations with topological defects Abstract: I will show how some fundamental computational processes, including encrypted communication and quantum teleportation, can be defined in terms of the higher representation theory of defects between 2d topological cobordisms, giving insight into fundamental questions in classical and quantum computation. No knowledge of computer science will be required to understand this talk. Speaker: Katrin Wehrheim Title: How to extend 2+1 (symplectic but not quite) field theories to 2+1+1