Moduli Spaces and Algebraic Structures in Homotopy Theory

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AbstractAward: DMS-0705428Principal Investigator: Craig C. WesterlandThis research agenda has three parts. In the first, theprincipal investigator proposes to study algebraic structuresinherent in the geometry of moduli spaces, particularly thosethat have not been studied from an operadic point of view andincluding several families of moduli spaces from differentialgeometry, algebraic geometry, and physics. The second part ofthe proposal concerns applications of this study of moduli spacesto objects in homotopy theory, including equivariant homology forthe free loop space on a manifold, string topology operations fora topological version of cyclic homology, and string topology ofclassifying spaces of Lie groups. The third major project willapply techniques from stable homotopy theory to the study ofHurwitz spaces, in a collaboration with number theorists.Moduli spaces are geometric objects that describe the variabilityof other geometric objects. For example, any element of thecollection of all spheres centered at the origin in Euclideanthree-dimensional space is completely determined by the radius ofthe circle, a positive number, so the collection of all thesespheres (each of which is a two-dimensional object) is describedby the positive half of the real number line (a one-dimensionalobject). A moduli space of particular interest in this and otherongoing mathematical research describes the variable geometry ofa surface such as the surface of a two-holed doughnut withseveral points labeled or marked on it, a construction whichprovides access to important questions of quantum field theory,algebra, and geometry.