Topology and Symplectic Geometry / Math of Gauge Fields seminar

from Monday
January 01, 2018 to Thursday
May 31, 2018
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Thursday
January 25, 2018

1:00 PM - 2:00 PM
MAT 5-127
Kenji Fukaya, Stony Brook University
Gluing analysis in relative Gromov Witten theory

In this talk, I will explain certain points about the construction of Kuranishi structure (or anything similar to obtain virtual fundamental chains) on the compactification of moduli spaces appearing in relative Gromov-Witten theory and their variants in Floer homology. Various points of the construction are similar to the case of stable map compactification for ordinary Gromov Witten theory. However, there are certain points which are not discussed in the literature and are not so much trivial.


Thursday
February 01, 2018

1:00 PM - 2:00 PM
MAT 5-127
Linh Truong, Columbia University
Truncated Heegaard Floer homology and concordance invariants

Heegaard Floer homology has proven to be a useful tool in the study of knot concordance. Ozsvath and Szabo first constructed the tau invariant using the hat version of Heegaard Floer homology and showed tau provides a lower bound on the slice genus. Later, Hom and Wu constructed a concordance invariant using the plus version of Heegaard Floer homology; this provides an even better lower-bound on the slice genus. In this talk, I discuss a sequence of concordance invariants that are derived from the truncated version of Heegaard Floer homology. These truncated Floer concordance invariants generalize the Ozsvath-Szabo and Hom-Wu invariants.


Thursday
April 26, 2018

1:00 PM - 2:00 PM
MAT 5-127
Sara Venkatesh, Columbia University
Quantitative symplectic cohomology Quantitative symplectic cohomology

Mirror symmetry predicts the existence of Floer invariants that yield “local” information about the Fukaya category. Guided by this, we construct a quantitative symplectic cohomology theory that detects Floer-essential Lagrangians within subdomains. This theory conjecturally specializes to a flavor of Rabinowitz Floer homology. We illustrate the quantitative behavior of this theory by examining negative line bundles over toric symplectic manifolds.


Thursday
May 03, 2018

1:00 PM - 2:00 PM
MAT 5-127
Michael Miller, UCLA
Equivariant instanton homology and group cohomology

Floer's celebrated instanton homology groups are defined for integer homology spheres, but analagous groups in Heegaard Floer and Monopole Floer homology theories are defined for all 3-manifolds; these latter groups furthermore come in four flavors, and carry extra algebraic structure. Any attempt to extend instanton homology to a larger class of 3-manifolds must be somehow equivariant - respecting a certain SO(3)-action. We explain how ideas from group cohomology and algebraic topology allow us to define four flavors of instanton homology for rational homology spheres, and how these invariants relate to existing instanton homology theories.


Thursday
May 10, 2018

1:00 PM - 2:00 PM
SCGP 313
Guangbo Xu, Princeton
Bershadsky--Cecotti--Ooguri--Vafa torsion in Landau--Ginzburg models

In the celebrated work of Bershadsky--Cecotti--Ooguri--Vafa the genus one string amplitude in the B-model is identified with certain analytic torsion of the Hodge Laplacian on a Kähler manifold. In a joint work with Shu Shen (IMJ-PRG) and Jianqing Yu (USTC) we study the analogous torsion in Landau--Ginzburg models. I will explain the corresponding index theorem based on the asymptotic expansion of the heat kernel of the Schrödinger operator. I will also explain the rigorous definition of the BCOV torsion for homogeneous polynomials on ${\mathbb C}^N$. Lastly I will explain the conjecture stating that in the Calabi--Yau case the BCOV torsion solves the holomorphic anomaly equation for marginal deformations.


Thursday
May 24, 2018

1:00 PM - 2:00 PM
MAT 5-127
Andrew Hanlon, UC Berkeley
Categorical monodromy for mirrors of toric varieties

We will discuss a monodromy action on the Fukaya-Seidel category of a Laurent polynomial from varying the arguments of the polynomial's coefficients. We will see how this monodromy corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated. This will require a new interpretation of the Fukaya-Seidel category in this setting.


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