Research Interests

My most recent research interests involve the study of nonlinear partial differential equations and their applications to the differential topology and geometry of spaces of dimension three and four. The basic issue here is to provide an exhaustive list of sorts for four-dimensional spaces. (With time included, the world is observedly four-dimensional, but the large scale structure of our particular four-dimensional space is not known. This issue is cosmological. The related mathematical issue is to list all possible four-dimensional structures.) The time-tested strategy for telling spaces apart is to creatively count the number of solutions of a natural differential equation on the space. When two spaces yield different counts, they must differ. My research has centered mostly on the development of workable strategies for making such counts. (On the other hand, there are no known techniques that unfailingly decide when two spaces are the same.) More generally, I am interested in applications of geometry, topology, and differential equations to problems in physics and in other natural sciences.

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Section 11: Mathematics