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

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

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.

page revision: 3, last edited: 02 Jun 2009 22:03