Joint IAS-PU Symplectic Geometry Seminar

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

Sobhan Seyfaddini
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.

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.

Lagrangian Whitney sphere links

Ivan Smith
University of Cambridge
November 1, 2016
Let $n > 1$. Given two maps of an $n$-dimensional sphere into Euclidean $2n$-space with disjoint images, there is a $\mathbb Z/2$ valued linking number given by the homotopy class of the corresponding Gauss map. We prove, under some restrictions on $n$, that this vanishes when the components are immersed Lagrangian spheres each with exactly one double point of high Maslov index. This is joint work with Tobias Ekholm.

Towards a theory of singular symplectic varieties

Aleksey Zinger
Stony Brook University
October 25, 2016
Singular algebraic (sub)varieties are fundamental to the theory of smooth projective manifolds. In parallel with his introduction of pseudo-holomorphic curve techniques into symplectic topology 30 years ago, Gromov asked about the feasibility of introducing notions of singular (sub)varieties suitable for this field. I will describe a new perspective on this question and motivate its appropriateness in the case of normal crossings singularities.

Length and width of Lagrangian cobordisms

Joshua Sabloff
Haverford College; Member, School of Mathematics
October 11, 2016
In this talk, I will discuss two measurements of Lagrangian cobordisms between Legendrian submanifolds in symplectizations: their length and their relative Gromov width. The Gromov width, in particular, is a fundamental global invariant of symplectic manifolds, and a relative version of that width helps understand the geometry of Lagrangian submanifolds of a symplectic manifold.

Monotone Lagrangians in cotangent bundles

Luis Diogo
Columbia University
October 11, 2016
We show that there is a 1-parameter family of monotone Lagrangian tori in the cotangent bundle of the 3-sphere with the following property: every compact orientable monotone Lagrangian with non-trivial Floer cohomology is not Hamiltonian-displaceable from either the zero-section or one of the tori in the family. The proof involves studying a version of the wrapped Fukaya category of the cotangent bundle which includes monotone Lagrangians. Time permitting, we may also discuss an extension to other cotangent bundles. This is joint work with Mohammed Abouzaid.

Packaging the construction of Kuranishi structure on the moduli space of pseudo-holomorphic curve

Kenji Fukaya
Stonybrook University
October 4, 2016
This is a part of my joint work with Oh-Ohta-Ono and is a part of project to rewrite the whole story of virtual fundamental chain in a way easier to use. In general we can construct virtual fundamental chain on (basically all) the moduli space of pseudo-holomorphic curve. It depends on the choices. In this talk I want to provide a statement to clarify which is the data we need to start with and in which sense the resulting structure is well defined. A purpose of writing such statement is then it can be a black box and can be used without looking the proof.