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.
