Thursday September 07, 2017 1:00 PM  2:00 PM Math Tower 5127
 Kei Irie, RIMS, Kyoto University
Chainlevel string topology, pseudoholomorphic disks, and Kuranishi structures I will talk about an application of chainlevel string topology to pseudoholomorphic curve theory in symplectic topology.
Specifically, for any closed, oriented and spin Lagrangian submanifold $L$ in a symplectic vector space,
one can define a MaurerCartan element of the loop bracket defined at chain level,
using virtual fundamental chain of the moduli space of pseudoholomorphic disks with boundaries on $L$.
This idea is due to Fukaya, who also pointed out its important consequences in symplectic topology.
I will explain how to rigorously carry out this idea, using a novel chain model of the free loop space and the theory of Kuranishi structures.

Thursday September 21, 2017 1:00 PM  2:00 PM 5127
 Artem Kotelskiy, Princeton
Bordered theory for pillowcase homology.Pillowcase homology is a geometric construction, which was developed in order to better understand and compute a knot invariant called singular instanton knot homology. We will describe an algebraic extension of pillowcase homology. We will compute partially wrapped Fukaya category of the pillowcase enlarged by immersed Lagrangians, and thus translate geometric pieces of pillowcase homology construction into algebraic world. The main ingredient will be a certain type DD structure, which allows one to recover Lagrangian Floer homology from modules in a computable and geometrically clear way.

Thursday October 05, 2017 1:00 PM  2:00 PM 5127
 Umut Varolgunes, MIT
MayerVietoris sequence for relative symplectic cohomologyI will first recall the definition of an invariant that assigns to any compact subset K of a closed symplectic manifold M a module SH_M(K) over the Novikov ring. I will go over the case of M=two sphere to illustrate various points about the invariant. Finally I will state the MayerVietoris property and explain under what conditions it holds.

Thursday October 12, 2017 1:00 PM  2:00 PM MAT 5127
 Mohammad Tehrani, Stony Brook University
Compactification of moduli spaces of Jholomorphic maps relative to snc divisorsIn this talk, I will describe an efficient way of compactifying moduli space of Jholomorphic maps relative to simple normal crossings (snc) symplectic divisors, including the holomorphic case. The primary goal of this construction is to define GromovWitten invariants relative to snc divisors, and to establish a GWdegeneration formula for any semistable degeneration with an snc central fiber.

Thursday October 19, 2017 1:00 PM  2:00 PM MAT 5127
 Chris Scaduto, Stony Brook University
An odd Khovanov homotopy typeAssociated to a link, Lipshitz and Sarkar constructed a refinement of Khovanov homology that takes the form of a stable homotopy type. I'll explain how to construct similar refinements for other versions of Khovanov homology, including the "odd" theory, and take some time to explain some motivations from mathematical gauge theory. This is joint work with S. Sarkar and M. Stoffregen.

Thursday November 09, 2017 1:00 PM  2:00 PM MAT 5127
 Siqi He, caltech
The Extended Bogomolny Equations, Generalized Nahm Pole and SL(2,R) Higgs Bundle.We will discuss Witten's gauge theory approaches to Jones polynomial and Khovanov homology by counting solutions to some gauge theory equations with singular boundary conditions. When we reduce these equations to 3dimensional, we call them the extended Bogomolny equations.
We develop a DonaldsonUhlenbeckYau type correspondence for the moduli space of the extended Bogomolny equations on Riemann surface Σ times R^+ with Nahm pole singularity at Σ × {0} and the Teichmuller component of the stable SL(2, R) Higgs bundle, this verifies a conjecture of Gaiotto and Witten. The proof is based on an observation that the extended Bogomolny equations can be reduced to a KazdanWarner type equation. We will also discuss a partial correspondence for solutions with knot singularities in this program, corresponding to the nonTeichmuller components in the moduli space of stable SL(2, R) Higgs bundles. This is joint work with Rafe Mazzeo.

Thursday November 16, 2017 1:00 PM  2:00 PM MATH 5127
 Francesco Lin, Princeton
Connected sums in Pin(2)monopole Floer homology.While not much is known about the structure of the homology cobordism group, Pin(2)symmetry in SeibergWitten theory seems to provide a promising direction to unveil many of its properties. In this talk, we discuss the behavior under connected sums of the Floer theoretic invariants arising from it  and discuss possible applications.

Thursday November 30, 2017 1:00 PM  2:00 PM MAT 5127
 Dingyu Yang, IAS
Integral virtual fundamental chainsTo define invariants using moduli spaces of holomorphic curves in general symplectic manifolds, a virtual technique is typically required, such as Kuranishi theory or polyfolds. All the methods in full generality use perturbation or duality, involve locally breaking the symmetry then taking the weighted averages, and thus yield virtual fundamental chains over rationals. We carry out a program of FukayaOno outlined in their 2001 paper. The key notions are a groupnormal structure that one can always construct for a good coordinate system, and a groupnormal complex structure that is always present on the moduli space of holomorphic curves, and their combined groupnormal complex good coordinate system. Using this, one can perform a singlevalued groupnormally polynomial perturbation to yield integral virtual fundamental chains/pseudocycles for Floer/GW moduli spaces on general symplectic manifolds. This method is expected to be applicable to all moduli spaces based on holomorphic curves. This is a joint work with Guangbo Xu.

Thursday December 07, 2017 1:00 PM  2:00 PM SCGP 103
 Erkao Bao, Stony Brook University
Semiglobal Kuranishi charts and contact homologyContact homology was proposed and studied by Eliashberg, Givental and Hofer 16 years ago. It is a very powerful tool to distinguish different contact structures. However, the rigorous definition did not come out until 2015. In this talk, we will first see that the naive definition does not work because the moduli spaces of Jholomorphic curves that we count to define the differential of contact homology are not transversely cut out. In order to achieve transversality, we will use a simplified version of the FOOO's Kuranishi perturbation theory, consisting of "semiglobal Kuranishi charts". This is a joint work with Ko Honda.

