##### Department of Mathematics,

University of California San Diego

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### Geometry/Topology Seminar

## Yongbin Ruan

#### University of Michigan

## Witten equation and quantum singularity theory

##### Abstract:

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. At the end of the talk, we will sketch a plan to compute higher genus GW-invariants of Calabi-Yau manifolds.

Host: Mark Gross

### February 25, 2008

### 3:00 PM

### AP&M 7218

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