Seminar in Topology and Symplectic Geometry

from Friday
June 01, 2018 to Monday
December 31, 2018
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Thursday
October 04, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Irving Dai, Princeton University
Involutive Floer Homology and Applications to Homology Cobordism group

In this talk, we discuss some recent applications of involutive Heegaard Floer homology to the homology cobordism group. We establish some non-torsion results and show that homology cobordism group admits an infinite-rank summand.


Thursday
October 11, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Lenny Ng, Duke University
A contact Fukaya category

I'll describe a way to construct an A-infinity category associated to some contact manifolds, analogous to a Fukaya category for a symplectic manifold. The objects of this category are Legendrian submanifolds equipped with augmentations. Currently we're focusing on standard contact R^3 but we're hopeful that we can extend this to other contact manifolds. As time permits, I'll discuss some properties of this contact Fukaya category, including generation by unknots and a potential application to proving that "augmentations = sheaves". This is joint work in progress with Tobias Ekholm and Vivek Shende.


Thursday
October 18, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Boyu Zhang, Princeton
Compactness for ADHM(1,2) equation on three-manifolds

The Seiberg-Witten equation can be generalized to any bundle
with a hyperKahler action of the gauge group. Examples among these are the Kapustin-Witten equations, Vafa-Witten equations, and Seiberg-Witten equations with multiple spinors. It was proved by Taubes and Haydys-Walpuski that solutions to these equations satisfy certain compactness properties described by Z/2-harmonic spinors. The ADHM equations on three-manifolds were introduced by Doan and Walpuski for the study of G2-manifolds, and it was conjectured that they should satisfy the same compactness properties as well. We expand the method of Taubes and Haydys-Walpuski to prove that the compactness property holds for ADHM(1,2) equations. Our method offers a unified proof for all the known compactness results of generalized Seiberg-Witten equations in dimension 3. This project is in collaboration with Thomas Walpuski.


Thursday
October 25, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Oleg Lazarev, Columbia University
Simplifying Weinstein Morse functions

By work of Cieliebak and Eliashberg, any Weinstein structure on Euclidean space that is not symplectomorphic to the standard symplectic structure necessarily has at least three critical points; an infinite collection of such exotic examples were constructed by McLean. I will explain how to use handle-slides and loose Legendrians to show that this lower bound on the number of critical points is exact; that is, any Weinstein structure on Euclidean space R^{2n} has a compatible Weinstein Morse function with at most three critical points (of index 0, n-1, n). This implies topological bounds on the rank of the Grothendieck group of the wrapped Fukaya category of a Weinstein R^{2n} and therefore restrictions on which categories can arise as the wrapped Fukaya category of an exotic Weinstein R^{2n}. Similar results hold for general Weinstein domains of dimension at least six.


Thursday
November 08, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Guangbo Xu, SCGP
Open Quantum Kirwan Map

(Joint with Woodward) Let L be a Lagrangian submanifold of a GIT quotient X of a linear space V. There are two versions of Fukaya A-infty algebras associated to L, defined by counting disks in X or counting disks in V (modulo group actions). We construct an A-infty morphism between them. This morphism is defined by counting the so-called point-like instantons. This gives an enumerative explanation of the relation between the Hori-Vafa potential and the Lagrangian Floer potential for a compact toric manifold.


Wednesday
November 14, 2018

3:00 PM - 4:00 PM
Math Tower 5-127
Jingyu Zhao, Harvard
Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period

In this talk, we provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces. The computations make use of the wall-crossing formula derived from Fukaya's pseudo-isotopies for A∞ algebras. This recovers the work of M. Gross, who verified the closed-string mirror symmetry tropically in the case of P^2. Having computed the bulk-deformed potentials, we also prove a big quantum period theorem which relates descendant Gromov-Witten invariants with the oscillatory integrals of their Landau-Ginzburg mirrors. This is a joint work with Hansol Hong and Yu-Shen Lin.


Thursday
November 15, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Jianfeng Lin, MIT
The geography problem of 4-manifolds: 10/8+4

A fundamental problem in 4-dimensional topology is the
following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8. Furuta proved the ''10/8+2''-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to Furuta's problem by analyzing the Pin(2)-equivariant Mahowald invariants. In particular, we improve Furuta's result into a ''10/8+4''-Theorem. Furthermore, we show that within the current existing framework, this is the limit. This is joint work with Mike Hopkins, XiaoLin Danny Shi and Zhouli Xu.


Thursday
November 29, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Yoosik Kim, Boston University
TBA

TBA


Thursday
December 06, 2018

1:00 PM - 2:15 PM
Math Tower 5-127
Chris Woodward, Rutgers
TBA

TBA


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Instructions for subscribing to Stony Brook Math Department Calendars