Thursday October 04, 2018 1:00 PM  2:15 PM Math Tower 5127
 Irving Dai, Princeton University
Involutive Floer Homology and Applications to Homology Cobordism groupIn this talk, we discuss some recent applications of involutive Heegaard Floer homology to the homology cobordism group. We establish some nontorsion results and show that homology cobordism group admits an infiniterank summand.

Thursday October 11, 2018 1:00 PM  2:15 PM Math Tower 5127
 Lenny Ng, Duke University
A contact Fukaya categoryI'll describe a way to construct an Ainfinity 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 5127
 Boyu Zhang, Princeton
Compactness for ADHM(1,2) equation on threemanifoldsThe SeibergWitten equation can be generalized to any bundle
with a hyperKahler action of the gauge group. Examples among these are the KapustinWitten equations, VafaWitten equations, and SeibergWitten equations with multiple spinors. It was proved by Taubes and HaydysWalpuski that solutions to these equations satisfy certain compactness properties described by Z/2harmonic spinors. The ADHM equations on threemanifolds were introduced by Doan and Walpuski for the study of G2manifolds, and it was conjectured that they should satisfy the same compactness properties as well. We expand the method of Taubes and HaydysWalpuski 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 SeibergWitten equations in dimension 3. This project is in collaboration with Thomas Walpuski.

Thursday October 25, 2018 1:00 PM  2:15 PM Math Tower 5127
 Oleg Lazarev, Columbia University
Simplifying Weinstein Morse functionsBy 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 handleslides 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, n1, 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 5127
 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 Ainfty algebras associated to L, defined by counting disks in X or counting disks in V (modulo group actions). We construct an Ainfty morphism between them. This morphism is defined by counting the socalled pointlike instantons. This gives an enumerative explanation of the relation between the HoriVafa potential and the Lagrangian Floer potential for a compact toric manifold.

Wednesday November 14, 2018 3:00 PM  4:00 PM Math Tower 5127
 Jingyu Zhao, Harvard
Bulkdeformed potentials for toric Fano surfaces, wallcrossing and periodIn this talk, we provide an inductive algorithm to compute the bulkdeformed potentials for toric Fano surfaces. The computations make use of the wallcrossing formula derived from Fukaya's pseudoisotopies for A∞ algebras. This recovers the work of M. Gross, who verified the closedstring mirror symmetry tropically in the case of P^2. Having computed the bulkdeformed potentials, we also prove a big quantum period theorem which relates descendant GromovWitten invariants with the oscillatory integrals of their LandauGinzburg mirrors. This is a joint work with Hansol Hong and YuShen Lin.

Thursday November 15, 2018 1:00 PM  2:15 PM Math Tower 5127
 Jianfeng Lin, MIT
The geography problem of 4manifolds: 10/8+4A fundamental problem in 4dimensional topology is the
following geography question: "which simply connected topological 4manifolds admit a smooth structure?" After the celebrated work of KirbySiebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4manifold, the ratio of its secondBetti 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 5127
 Yoosik Kim, Boston University
TBATBA

Thursday December 06, 2018 1:00 PM  2:15 PM Math Tower 5127
 Chris Woodward, Rutgers
TBATBA

