##### Department of Mathematics,

University of California San Diego

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### Math 292 - Topology

## Nitu Kitchloo

#### UCSD

## Introduction to the Seminar on the Madsen-Weiss proof of Mumford's conjecture

##### Abstract:

This is the introductory meeting for this quarter's topology learning seminar. The subject of the seminar will be introduced and the talks will be distributed. The goal of the seminar is to study the Madsen-Weiss proof of Mumford's conjecture concerning the cohomology of the stable mapping class group. Knowing the group cohomology of this group means to know the cohomology of the stable moduli space of Riemann surfaces. While Mumford's conjecture deals with the geometry of the moduli space of surfaces, the methods of the proof given by Madsen and Weiss are mostly homotopy theoretic. The proof is a beautiful example for how the rather abstract machinery of homotopy theory can be used to address a concrete geometric question. Also, we think that the techniques used in the proof are interesting by themselves and thus worth studying. The first couple of talks introduce the main players: moduli spaces of Riemann surfaces, mapping class groups, and Mumford's conjecture. Then we turn some fundamental concepts of homotopy theory. This is necessary in order to understand Ulrike Tillmann's work that identifies the computation of the cohomology of the stable mapping class group as a problem of stable homotopy theory. Based on her work, Madsen formulated a conjecture in the world of stable homotopy theory that implies Mumford's conjecture. Finally, the last third of the seminar will be concerned with giving/outlining the proof of Madsen's conjecture by Madsen and Weiss.

Host:

### April 4, 2006

### 10:30 AM

### AP&M 7218

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