Seminars Winter 2008

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Fall 2009

Thursday, January 10, 2008

Matthew Ballard, Washington

Thursday, January 17, 2008

Ezra Getzler,

TFT in 2 dimensions and Deligne-Mumford moduli spaces

Thursday, January 31, 2008

Gabriel Kerr, Northwestern University

Relative Fukaya Categories

I will describe recent progress in defining Fukaya categories of superpotentials with run-off behavior and show how these may be thought of as relative Fukaya Categories. After reviewing the classical case (i.e. the case that has been known now for ten years) I will illustrate how many of the same theorems work in the relative case (in particular the existence of exceptional collections and their mutations). As this is a work in progress, I will allow myself to dream a bit in the last part of the talk. Some dreams will include: reconstructing Fukaya categories of smooth varieties from degenerations (Gross/Seibert Theory + Superpotential) and Mirror Symmetry for non-toric Fano manifolds.

Thursday, February 7, 2008

Dmitry Tamarkin, Northwestern University

Microlocal criterion for non-displaceability of Lagrangians

Let $L$ be a compact Lagrangian submanifold in $T^*X$. We then produce a full subcategory $C_L$ in the category of sheaves on $X\times \mathbb{R}$ ($\mathbb{R}$ is the real axis). Given two Lagrangians $L_1,L_2$, take arbitrary objects $F_1$ from $C_{L_1}$ and $F_2$ from $C_{L_2}$. One can define $\text{Rhom}(F_1,F_2)$ as an object of the derived category of sheaves on $\mathbb{R}$ (because $\mathbb{R}$ acts on $X\times \mathbb{R}$ by shifts). Furthermore, one can define a map from $\text{Rhom}(F_1,F_2)$ to its shift along $\mathbb{R}$ by any positive number. We prove that as long as this map is non-zero for all shifts, any hamiltonian shift of $L_2$ intersects $L_1$ (i.e. $L_1$ and $L_2$ are non-displaceable).

As an application, we prove that the Clifford torus in $\mathbb{C}P^n$ is non-displaceable as well as the Clifford torus and $\mathbb{R}P^n$.

Thursday, February 14, 2008

Ronnie Hadani, University of Chicago

Quantization of symplectic vector spaces: An algebra-geometric approach

Quantization is a fundamental procedure in mathematics and in physics. From the physical side, quantization is the procedure by which one associates to a classical mechanical system its quantum counterpart. From the mathematical side, it seems that quantization is a way to construct interesting Hilbert spaces out of symplectic manifolds, suggesting a method for constructing representations of the corresponding groups of symplectomorphisms.

In my lecture, I will consider the problem of quantization in the simplified setting of symplectic vector spaces over the finite field $\mathbb{F}_{p}$. Specifically, I will construct a quantization functor, $\mathcal{H}$, associating a Hilbert space $\mathcal{H}_{V}$, to a finite dimensional symplectic vector space $\left( V,\omega \right)$ over $\mathbb{F}_{p}$. As a result, we will obtain a canonical model for the Weil representation of the symplectic group $Sp\left( V\right)$.

The main technical result, is a proof of a strong form of the Stone-von Neumann theorem for the Heisenberg group over$\mathbb{F}_{p}$. This result, roughly, concerns the existence of a canonical flat connection on a certain vector bundle $\mathcal{H}$, defined on $Lag\left( V\right)$. In this terminology, the space $\mathcal{H}_{V}$ is obtained as the space of horizontal sections in $\mathcal{H}$.

The connection is constructed as follows: It is given explicitly as asystem of isomorphisms between pairs of fibers, $F_{M,L}:\mathcal{H}_{|L}\overset{\simeq}{\rightarrow }\mathcal{H}_{|M}$, for $M,L\in Lag\left( V\right)$ which are in transversal position, i.e., $M\cap L=0$. The construction of $F_{M,L}$, for general $M,L\in Lag\left( V\right)$, is obtained from the transversal formulas using the algebra-geometric operation of (perverse) extension.

Thursday, February 21, 2008

Andrew Neitzke, IAS

Comments on Holomorphic Anomalies

Thursday, February 28, 2008

Yongbin Ruan, University of Michigan

Witten equation and quantum singularity theory

A long standing problem in Gromov-Witten theory is to compute higher genus Gromov-Witten invariants of compact Calabi-Yau manifold such as quintic 3-folds. The defining equation of these Calabi-Yau manifold has a natural interpretation in Landau-Ginzburg/singularity theory. More than 15 years ago, Witten proposed a PDE as a replacement of familiar Cauchy-Riemann equation in the Laudau-Ginzburg/singularity setting. Furthermore, he proposed two remarkable conjectures for his conjectural theory for ADE-singularity. In the talk, we will present a moduli theory of the solution spaces of Witten equation. As a consequence, we solve Witten's conjectures for quantum theory of ADE-singularities. In the end of talk, we will sketch a plan to compute higher genus GW-invariants of Calabi-Yau manifolds.

Thursday, March 6, 2008

Dmitry Tamarkin, Northwestern University

Microlocal criterion for non-displaceability of Lagrangians - Part 2

Thursday, March 13, 2008

Alexander Ritter, MIT

Novikov-symplectic cohomology and exact Lagrangian embeddings

We are interested in finding topological obstructions to the existence of exact Lagrangian submanifolds $L$ inside a cotangent bundle $T^*N$. We assume $N$ is simply connected, but we make no assumptions on the Maslov class of $L$. We prove that $H^2(N)$ injects into $H^2(L)$ and that the image of $\pi_2(L)$ in $\pi_2(N)$ has finite index. A more complicated statement holds in the non-simply connected case. Our approach builds on Viterbo's work: by using symplectic cohomology we construct a transfer map on the Novikov homologies of the free loop spaces of $N$ and $L$. The above application is then a consequence of the vanishing of the Novikov homology of the free loopspace with respect to non-zero 1-forms.

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