Madsen-Weiss MCG Seminar
Spring 2006

Meets Tuesdays 10:30-12:00 in AP&M 7218

 

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. 

 

References:

The key papers:
Weiss - Cohomology of the stable mapping class group
Madsen, Weiss - The stable mapping class group and stable homotopy theory
Madsen, Weiss - The stable moduli space of Riemann surfaces: Mumford's conjecture

Tillmann & hty thy:
Tillmann - On the homotopy of the stable mapping class group
Adams - Stable homotopy and generalised homology
Adams - Infinite loop spaces
Kochman - Bordism, stable homotopy, and Adams spectral sequences

h-principles, etc:
Vassiliev - Complements of discriminants of smooth maps
Vassiliev - Topology of spaces of functions without complex singularities

Schedule:
04/04 - Nitu Kitchloo - Introduction and summary of the proof
04/11 - Nitu Kitchloo - Homotopy theory for Madsen-Weiss
04/18 - Maia Averett - Quillen's +-construction, operads, and more hty thy
04/25 - Mike Gurvich - Tillmann's theorem
05/02 - Sean Raleigh - h-principles
05/09 - Dave Clark - Pontryagin-Thom Construction
05/16 - Ben Cooper -
05/23 - Justin Roberts -
05/30 - ???

Department of Mathematics
University of California, San Diego

UCSD