Topology and Symplectic Geometry / Math of Gauge Fields seminar

from Thursday
June 01, 2017 to Sunday
December 31, 2017
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Thursday
September 07, 2017

1:00 PM - 2:00 PM
Math Tower 5-127
Kei Irie, RIMS, Kyoto University
Chain-level string topology, pseudo-holomorphic disks, and Kuranishi structures

I will talk about an application of chain-level string topology to pseudo-holomorphic curve theory in symplectic topology.
Specifically, for any closed, oriented and spin Lagrangian submanifold $L$ in a symplectic vector space,
one can define a Maurer-Cartan element of the loop bracket defined at chain level,
using virtual fundamental chain of the moduli space of pseudo-holomorphic 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
5-127
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
5-127
Umut Varolgunes, MIT
Mayer-Vietoris sequence for relative symplectic cohomology

I 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 Mayer-Vietoris property and explain under what conditions it holds.


Thursday
October 12, 2017

1:00 PM - 2:00 PM
MAT 5-127
Mohammad Tehrani, Stony Brook University
Compactification of moduli spaces of J-holomorphic maps relative to snc divisors

In this talk, I will describe an efficient way of compactifying moduli space of J-holomorphic maps relative to simple normal crossings (snc) symplectic divisors, including the holomorphic case. The primary goal of this construction is to define Gromov-Witten invariants relative to snc divisors, and to establish a GW-degeneration formula for any semistable degeneration with an snc central fiber.


Thursday
October 19, 2017

1:00 PM - 2:00 PM
MAT 5-127
Chris Scaduto, Stony Brook University
An odd Khovanov homotopy type

Associated 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 5-127
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 3-dimensional, we call them the extended Bogomolny equations.

We develop a Donaldson-Uhlenbeck-Yau 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 Kazdan-Warner type equation. We will also discuss a partial correspondence for solutions with knot singularities in this program, corresponding to the non-Teichmuller 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 5-127
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 Seiberg-Witten 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 5-127
Dingyu Yang, IAS
Integral virtual fundamental chains

To 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 Fukaya-Ono outlined in their 2001 paper. The key notions are a group-normal structure that one can always construct for a good coordinate system, and a group-normal complex structure that is always present on the moduli space of holomorphic curves, and their combined group-normal complex good coordinate system. Using this, one can perform a single-valued group-normally 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
Semi-global Kuranishi charts and contact homology

Contact 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 J-holomorphic 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 "semi-global Kuranishi charts". This is a joint work with Ko Honda.


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