# Joint IAS-PU Symplectic Geometry Seminar

## Symplectic homology for cobordisms

Alexandru Oancea
Université Pierre et Marie Curie; Member, School of Mathematics
February 23, 2017
Symplectic homology for a Liouville cobordism (possibly filled at the negative end) generalizes simultaneously the symplectic homology of Liouville domains and the Rabinowitz-Floer homology of their boundaries. I intend to explain a conceptual framework within which one can understand it, and give a sample application which shows how it can be used in order to obstruct cobordisms between contact manifolds. Based on joint work with Kai Cieliebak and Peter Albers.

## $C^\infty$ closing lemma for three-dimensional Reeb flows via embedded contact homology

Kei Irie
Kyoto University
February 16, 2017
$C^r$ closing lemma is an important statement in the theory of dynamical systems, which implies that for a $C^r$ generic system the union of periodic orbits is dense in the nonwondering domain. $C^1$ closing lemma is proved in many classes of dynamical systems, however $C^r$ closing lemma with $r > 1$ is proved only for few cases. In this talk, I'll prove $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds. The proof uses recent developments in quantitative aspects of embedded contact homology (ECH).

## Relative quantum product and open WDVV equations

Sara Tukachinsky
University of Montreal
February 2, 2017
The standard WDVV equations are PDEs in the potential function that generates Gromov-Witten invariants. These equations imply relations on the invariants, and sometimes allow computations thereof, as demonstrated by Kontsevich-Manin (1994). We prove analogous equations for open Gromov-Witten invariants that we defined in a previous work. For ($\mathbb CP^n$ ,$\mathbb RP^n$), the resulting relations allow the computation of all invariants. The formulation of the open WDVV requires a lift of the big quantum product to relative cohomology.

## $C^0$ Hamiltonian dynamics and a counterexample to the Arnold conjecture

Member, School of Mathematics
November 29, 2016
After introducing Hamiltonian homeomorphisms and recalling some of their properties, I will focus on fixed point theory for this class of homeomorphisms. The main goal of this talk is to present the outlines of a $C^0$ counterexample to the Arnold conjecture in dimensions four and higher. This is joint work with Lev Buhovsky and Vincent Humiliere.

## Rectification and the Floer complex: quantizing Lagrangians in $T^*N$

Claude Viterbo
Ecole Normale Supérieure
November 29, 2016
We shall give a construction of the quantized sheaf of a Lagrangian submanifold in $T^*N$ and explain a number of features and applications.

## The gauged symplectic sigma-model

Constantin Teleman
University of California, Berkeley and Oxford University
November 15, 2016
I will recall the construction of the space of states in a gauged topological A-model. Conjecturally, this gives the quantum cohomology of Fano symplectic quotients: in the toric case, this is Batyrev’s presentation of quantum cohomology of toric varieties. Time permitting, I will discuss the role of “Coulomb branches” in gauge theory in relation to equivariant quantum and symplectic cohomology.