Moduli Spaces and Algebraic Structures in Homotopy Theory

Abstract

Award: DMS-0705428

Principal Investigator: Craig C. Westerland

This research agenda has three parts. In the first, the

principal investigator proposes to study algebraic structures

inherent in the geometry of moduli spaces, particularly those

that have not been studied from an operadic point of view and

including several families of moduli spaces from differential

geometry, algebraic geometry, and physics. The second part of

the proposal concerns applications of this study of moduli spaces

to objects in homotopy theory, including equivariant homology for

the free loop space on a manifold, string topology operations for

a topological version of cyclic homology, and string topology of

classifying spaces of Lie groups. The third major project will

apply techniques from stable homotopy theory to the study of

Hurwitz spaces, in a collaboration with number theorists.

Moduli spaces are geometric objects that describe the variability

of other geometric objects. For example, any element of the

collection of all spheres centered at the origin in Euclidean

three-dimensional space is completely determined by the radius of

the circle, a positive number, so the collection of all these

spheres (each of which is a two-dimensional object) is described

by the positive half of the real number line (a one-dimensional

object). A moduli space of particular interest in this and other

ongoing mathematical research describes the variable geometry of

a surface such as the surface of a two-holed doughnut with

several points labeled or marked on it, a construction which

provides access to important questions of quantum field theory,

algebra, and geometry.