Homotopy theoretic methods in the study of moduli spaces

Stanford University, Stanford CA

Investigators

Abstract

The proposed activity will continue the development of a relatively new area of mathematics in which algebraic topology is used to study moduli spaces of Riemann surfaces and related objects. The main result so far in this area is the solution, by Madsen and Weiss, of a generalization of a conjecture of Mumford. This subject lies in the overlap between algebraic topology and other branches of mathematics, with expected applications in algebraic geometry, symplectic geometry, and possibly theoretical physics. The proposed activity will further investigate and develop the result of Madsen and Weiss. Specifically, the proposed activity will define a d-dimensional cobordism category C^d and determine the homotopy type of its classifying space. This will imply the result of Madsen and Weiss, but is a much more general result, opening for many new possible applications. At the same time, it will be conceptually simpler and allows for a much simpler proof. This is one project of the proposal. The remaining four proposed projects are related. The theorem of Madsen and Weiss is a very important theorem, that has already recieved much interest. It sheds a completely new light on the moduli spaces M_g of Riemann surfaces of genus g and on the 2-dimensional cobordism category. These objects has been studied and used for long time, and in many different areas of mathematics. On the other hand it is clear that the result is not definitive, for example it is restricted to dimension d=2. The development of the proper generalization should be of great interest and importance for algebraic topology as well as for related fields.