# Seminars & Events for Joint PU/IAS Symplectic Geometry Seminar

##### Floer-like complexes for surfaces, maximally unlinked braids, and finite energy foliations

In this talk I will present an approach to constructing finite energy foliations by pseudo-holomorphic curves with prescribed asymptotic orbits in the symplectization of a mapping torus. The idea is that so called maximally unlinked braids of periodic orbits support a Floer-like chain complex. The concept of unlinkedness comes from LeCalvez work on surface homeomorphisms, as I will explain. The upshot is that it allows us to essentially characterize finite energy foliations for mapping tori: also, these chain complexes should be of independent interest. I will draw a lot of pictures.

##### On symplectic homology of the complement of a normal crossing divisor

In this talk, we discuss our work in progress about how degeneration of the divisor at infinity into a normal crossing divisor affects the symplectic homology of an affine variety. From an anti-surgery picture, by developing an anti-surgery formula for symplectic homology similar to work by Bourgeois-Ekholm-Eliashberg, we show that essentially, the change in symplectic homology is reflected by the Hochschild invariants of the Fukaya category of a collection of Lagrangian spheres on the smooth divisor.

##### Equivalent notions of high-dimensional overtwistedness

In recent joint work with Borman and Eliashberg, a new definition of overtwisted contact structures was given for high-dimensional contact manifolds, which were then classified up to isotopy. However, the definition is fairly cumbersome, so much so that it is essentially impossible to check in any explicit example. This talk will focus on various criteria which are shown to be equivalent to overtwistedness, giving numerous explicit examples of overtwisted contact manifolds, and relating overtwisteness to a number of older works. In particular we show that negatively stabilized open books, looseness of the Legendrian unknot, and existence of a "nice" plastikstufe are all equivalent to overtwistedness. This project is joint work with R. Casals and F. Presas.

##### Unlinked fixed points of Hamiltonian diffeomorphisms and a dynamical construction of spectral invariants

Hamiltonian spectral invariants have had many interesting and important applications in symplectic geometry. Inspired by Le Calvez’ theory of transverse foliations for dynamical systems of surfaces, we introduce a new dynamical invariant, denoted by N, for Hamiltonians on surfaces (except the sphere). We prove that, on the set of autonomous Hamiltonians, this invariant coincides with the classical spectral invariant. This is joint work with Vincent Humilière and Frédéric Le Roux.

##### Periodic Symplectic Cohomologies

Periodic cyclic homology group associated to a mixed complex was introduced by Goodwillie. In this talk, I will explain how to apply this construction to the symplectic cochain complex of a Liouville domain and obtain two periodic symplectic cohomology theories, which are called periodic symplectic cohomology and finitely supported periodic symplectic cohomology, respectively. The main result is that there is a localization theorem for the finitely supported periodic symplectic cohomology.