# Fall-2010-Geometry-Topology

## Contents

- 1 Fall 2010
- 2 Abstracts
- 2.1 Yong-Geun Oh (UW Madison)
- 2.2 Leva Buhovsky (U of Chicago)
- 2.3 Leonid Polterovich (Tel Aviv U and U of Chicago)
- 2.4 Sean Paul (UW Madison)
- 2.5 Conan Leung (Chinese U. of Hong Kong)
- 2.6 Markus Banagl (U. Heidelberg)
- 2.7 Ke Zhu (U of Minnesota)
- 2.8 Sergei Tabachnikov (Penn State)
- 2.9 Ma Chit (Chinese U. of Hong Kong)
- 2.10 Eric Zaslow (Northwestern University)
- 2.11 Wenxuan Lu (MIT)

## Fall 2010

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

## 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.

### Ma Chit (Chinese U. of Hong Kong)

*A growth estimate of lattice points in Gorenstein cones using toric Einstein metrics*

Using the existence of Einstein metrics on toric Kahler and Sasaki manifolds, a lower bound estimate on the growth of lattice points is obtained for Gorenstein cones. This talk is based on a joint work with Conan Leung.

### Eric Zaslow (Northwestern University)

*Ribbon Graphs and Mirror Symmetry*

I will define, for each ribbon graph, a dg category, and explain the conjectural relation to mirror symmetry. I will being by reviewing how T-duality relates coherent sheaves on toric varieties to constructible sheaves on a vector space, then use this relation to glue toric varieties together. In one-dimension, the category of sheaves on such gluings has a description in terms of ribbon graphs. These categories are conjecturally related to the Fukaya category of a noncompact hypersurface mirror to the variety with toric components.

I will use very basic examples. This work is joint with Nicolo' Sibilla and David Treumann.

### Wenxuan Lu (MIT)

*Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli*
Spaces

We study two instanton correction problems of Hitchin's moduli spaces along with their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space can be put into an instanton-corrected form according to physicists Gaiotto, Moore and Neitzke. The problem boils down to the construction of a set of special coordinates which can be constructed as Fock-Goncharov coordinates associated with foliations of quadratic differentials on a Riemann surface. A wall crossing formula of Kontsevich and Soibelman arises both as a crucial consistency condition and an effective computational tool. On the other hand Gross and Siebert have succeeded in determining instanton corrections of complex structures of Calabi-Yau varieties in the context of mirror symmetry from a singular affine structure with additional data. We will show that the two instanton correction problems are equivalent in an appropriate sense. This is a nontrivial statement of mirror symmetry of Hitchin's moduli spaces which till now has been mostly studied in the framework of geometric Langlands duality. This result provides examples of Calabi-Yau varieties where the instanton correction (in the sense of mirror symmetry) of metrics and complex structures can be determined.

[[1] Spring 2011 Geometry-Topology]