Joint IAS-PU Symplectic Geometry Seminar

Birational Calabi-Yau manifolds have the same small quantum products.

Mark McLean
Stony Brook University
April 30, 2018

We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.

Mirror spaces from formal deformation of Lagrangians and their gluing.

Hansol Hong
Harvard University
April 23, 2018

For given a Lagrangian in a symplectic manifold, one can consider deformation of A-infinity algebra structures on its Floer complex by degree 1 elements satisfying the Maurer-Cartan equation. The space of such degree 1 elements can be thought of as giving a local chart of the mirror. In this talk, I will explain how to glue local charts from different Lagrangians using isomorphisms between Lagrangians in the Fukaya category.
As an application, we will discuss the mirror construction for Gr(2,4) that recovers its Lie-theoretical mirror.

Fukaya categories of Calabi-Yau hypersurfaces

Paul Seidel
Massachusetts Institute of Technology; Member, School of Mathematics
April 9, 2018

Consider a Calabi-Yau manifold which arises as a member of a Lefschetz pencil of anticanonical hypersurfaces in a Fano variety. The Fukaya categories of such manifolds have particularly nice properties. I will review this (partly still conjectural) picture, and how it constrains the field of definition of the Fukaya category.

Higher ribbon graphs

David Nadler
University of California, Berkeley
March 12, 2018

Ribbon graphs capture the topology of open Riemann surfaces in an elementary combinatorial form. One can hope this is the first step toward a general theory for open symplectic manifolds such as Stein manifolds. We will discuss progress toward such a higher dimensional theory (joint work with Alvarez-Gavela, Eliashberg, and Starkston), and in particular, what kind of topological spaces might generalize graphs. We will also discuss applications to the calculation of symplectic invariants.

Recent developments in knot contact homology

Lenny Ng
Duke University
December 11, 2017
Knot contact homology is a knot invariant derived from counting holomorphic curves with boundary on the Legendrian conormal to a knot. I will discuss some new developments around the subject, including an enhancement that completely determines the knot (joint work with Tobias Ekholm and Vivek Shende) and recent progress in the circle of ideas connecting knot contact homology, recurrence relations for colored HOMFLY polynomials, and topological strings (joint work in progress with Tobias Ekholm).

Open Gromov-Witten theory of $(\mathbb{CP}^1,\mathbb{RP}^1)$ in all genera and Gromov-Witten Hurwitz correspondence

Amitai Zernik
Member, School of Mathematics
December 4, 2017

In joint work with Buryak, Pandharipande and Tessler (in preparation), we define equivariant stationary descendent integrals on the moduli of stable maps from surfaces with boundary to $(\mathbb{CP}^1,\mathbb{RP}^1)$. For stable maps of the disk, the definition is geometric and we prove a fixed-point formula involving contributions from all the corner strata. We use this fixed-point formula to give a closed formula for the integrals in this case.

Open Gopakumar-Vafa conjecture for rational elliptic surfaces

Yu-Shen Lin
Harvard University
November 27, 2017
We will explain a definition of open Gromov-Witten invariants on the rational elliptic surfaces and explain the connection of the invariants with tropical geometry. For certain rational elliptic surfaces coming from meromorphic Hitchin system, we will show that the open Gromov-Witten invariants with boundary conditions near infinity (up to some transformation) coincide with the closed geodesic counting invariants defined by Gaiotto-Moore-Neitzke, which are integer-valued.

Morse-Bott cohomology from homological perturbation

Zhengyi Zhou
University of California, Berkeley
November 6, 2017
Abstract: In this talk, I will give a new construction of the Morse-Bott cochain complex, where the underlying vector space is generated by the cohomology of the critical manifolds. This new construction has two nice features: (1) It requires the minimum amount of transversality. (2) The choices made in the construction do not depend on the moduli spaces. I will explain its relation to three other constructions in literature, namely Austin-Braam's push-pull construction, Fukaya's push-pull construction and the cascades construction.

Wrapped Fukaya categories and functors

Yuan Gao
Stonybrook University
October 23, 2017
Inspired by homological mirror symmetry for non-compact manifolds, one wonders what functorial properties wrapped Fukaya categories have as mirror to those for the derived categories of the mirror varieties, and also whether homological mirror symmetry is functorial. Comparing to the theory of Lagrangian correspondences for compact manifolds, some subtleties are seen in view of the fact that modules over non-proper categories are complicated.