The following talks will be held at the Simons Center in Room 102. Live streaming is available for virtual participants here: http://scgp.stonybrook.edu/live
Virtual Lecture 1: Wednesday 6/23
1) Introductory ECH and cobordism map difficulties
Abstract: In these two lectures I will introduce Embedded Contact Homology, a useful Floer theory for contact 3-manifolds, obtained by counting pseudoholomorphic curves and Reeb orbits. ECH is known to be isomorphic to versions of monopole Floer homology, and can be used to probe both the topology of 3-manifolds and dynamics of contact forms. After further explaining its properties, I will discuss the issues with building “ECH cobordism maps” induced from symplectic 4-manifolds with contact boundary, and potential/known resolutions in special cases. Some foundations of ECH rely on its isomorphism with monopole Floer homology, but it would be useful to have a direct construction inherent to pseudoholomorphic geometry.
Virtual Lecture 2: Wednesday 7/7 SCGP Room 102
2) Building invariants of 4-spheres and Lagrangian tori using ECH = SWF
Abstract: In these two lectures I will use ECH (embedded contact homology) and its relation to SWF (monopole Floer homology) to probe homotopy 4-spheres and also Lagrangian tori in symplectic 4-manifolds. The invariants built for Lagrangian tori will count pseudoholomorphic curves in the complement of the tori, and they recover old invariants: perhaps the rephrasal will yield new applications? The “invariants” built for homotopy 4-spheres will count pseudoholomorphic curves in the complement of circles, but at the moment they are not sensitive enough to distinguish/detect 4-spheres: I will discuss ideas to possibly refine these “invariants”.
Virtual Lecture 3: Wednesday 8/25
3) 3-dimensional Weinstein conjecture and 2-or-infinity conjecture
Abstract: In these two lectures I will introduce the famous (3-dimensional) Weinstein conjecture, asserting the existence of closed Reeb orbits in contact 3-manifolds, as well as refinements to the conjecture. The proof is due to Taubes and heavily uses Seiberg-Witten theory. I will discuss SW theory in the presence of contact forms and give a rough sketch of Taubes’ proof, in order to set up and explain on-going work to establish refinements to the conjecture. One big refinement is the “2-or-infinity” conjecture, asserting that there are either two or infinitely many Reeb orbits in any contact 3-manifold (and this is known in a lot of cases).