Geometry and Topology Seminar 2019-2020

Fall 2010

The seminar will be held in room B901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm

date	speaker	title	host(s)
September 10	Yong-Geun Oh (UW Madison)	Counting embedded curves in Calabi-Yau threefolds and Gopakumar-Vafa invariants	local
September 17	Leva Buhovsky (U of Chicago)	On the uniqueness of Hofer's geometry	Yong-Geun
September 24	Leonid Polterovich (Tel Aviv U and U of Chicago)	Poisson brackets and symplectic invariants	Yong-Geun
October 8	Sean Paul (UW Madison)	Canonical Kahler metrics and the stability of projective varieties	local
October 15	Conan Leung (Chinese U. of Hong Kong)	SYZ mirror symmetry for toric manifolds	Honorary fellow, local
October 22	Markus Banagl (U. Heidelberg)	Intersection Space Methods and Their Application to Equivariant Cohomology, String Theory, and Mirror Symmetry	Maxim
October 29	Ke Zhu (U of Minnesota)	Thick-thin decomposition of Floer trajectories and adiabatic gluing	Yong-Geun
November 5	Sergei Tabachnikov (Penn State)	Algebra, geometry, and dynamics of the pentagram map	Gloria
December 3	Eric Zaslow (Northwestern University)	TBA	Yong-Geun and Conan Leung
December 10	Wenxuan Lu (MIT)	TBA	Young-Geun and Conan Leung
January 21	Mohammed Abouzaid (Clay Institute & MIT)	TBA	Yong-Geun

Abstracts

Yong-Geun Oh (UW Madison)

Counting embedded curves in Calabi-Yau threefolds and Gopakumar-Vafa invariants

Gopakumar-Vafa BPS invariant is some integer counting invariant of the cohomology of D-brane moduli spaces in string theory. In relation to the Gromov-Witten theory, it is expected that the invariant would coincide with the `number' of embedded (pseudo)holomorphic curves (Gopakumar-Vafa conjecture). In this talk, we will explain the speaker's recent result that the latter integer invariants can be defined for a generic choice of compatible almost complex structures. We will also discuss the corresponding wall-crossing phenomena and some open questions towards a complete solution to the Gopakumar-Vafa conjecture.

Leva Buhovsky (U of Chicago)

On the uniqueness of Hofer's geometry

In this talk we address the question whether Hofer's metric is unique among the Finsler-type bi-invariant metrics on the group of Hamiltonian diffeomorphisms. The talk is based on a recent joint work with Yaron Ostrover.

Leonid Polterovich (Tel Aviv U and U of Chicago)

Poisson brackets and symplectic invariants

We discuss new invariants associated to collections of closed subsets of a symplectic manifold. These invariants are defined through an elementary variational problem involving Poisson brackets. The proof of non-triviality of these invariants requires methods of modern symplectic topology (Floer theory). We present applications to approximation theory on symplectic manifolds and to Hamiltonian dynamics. The talk is based on a work in progress with Lev Buhovsky and Michael Entov.

Sean Paul (UW Madison)

Canonical Kahler metrics and the stability of projective varieties"

I will give a survey of my own work on this problem, the basic reference is: http://arxiv.org/pdf/0811.2548v3

Conan Leung (Chinese U. of Hong Kong)

SYZ mirror symmetry for toric manifolds

Markus Banagl (U. Heidelberg)

Intersection Space Methods and Their Application to Equivariant Cohomology, String Theory, and Mirror Symmetry.

Using homotopy theoretic methods, we shall associate to certain classes of singular spaces generalized geometric Poincaré complexes called intersection spaces. Their cohomology is generally not isomorphic to intersection cohomology. In this talk, we shall concentrate on the applications of the new cohomology theory to the equivariant real cohomology of isometric actions of torsionfree discrete groups, to type II string theory and D-branes, and to the relation of the new theory to classical intersection cohomology under mirror symmetry.

Ke Zhu (U of Minnesota)

Thick-thin decomposition of Floer trajectories and adiabatic gluing

Let f be a generic Morse function on a symplectic manifold M. For Floer trajectories of Hamiltonian \e f, as \e goes to 0 Oh proved that they converge to “pearl complex” consisiting of J-holomorphic spheres and joining gradient segments of f. The J-holomorphic spheres come from the “thick” part of Floer trajectories and the gradient segments come from the “thin” part. Similar “thick-thin” compactification result has also been obtained by Mundet-Tian in twisted holomorphic map setting. In this talk, we prove the reverse gluing result in the simplest setting: we glue from disk-flow-dsik configurations to nearby Floer trajectories of Hamitonians K_{\e} for sufficiently small \e and also show the surjectivity. (Most part of the Hamiltonian K_{\e} is \ef). We will discuss the application to PSS isomorphism. This is a joint work with Yong-Geun Oh.

Sergei Tabachnikov (Penn State)

Algebra, geometry, and dynamics of the pentagram map

Introduced by R. Schwartz almost 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. I shall survey recent work on the pentagram map, in particular, I shall demonstrate that the dynamics of the pentagram map is completely integrable. I shall also explain that the pentagram map is a discretization of the Boussinesq equation, a well known completely integrable partial differential equation. A surprising relation between the spaces of polygons and combinatorial objects called the 2-frieze patterns (generalizing the frieze patterns of Coxeter) will be described. Eight new(?) configuration theorems of projective geometry will be demonstrated. The talk is illustrated by computer animation.

Eric Zaslow (Northwestern University)

TBA

Wenxuan Lu (MIT)

TBA

Mohammed Abouzaid (Clay Institute & MIT)

TBA