- Differential Geometry
- Riemannian Geometry
- Metric Geometry/Alexandrov Spaces
- Geometry of Fractals
- Discrete Homotopy Theory
- Topological Groups
- Generalized Covering Spaces
Ph.D., The University of Maryland
I am Head of the Mathematics Department at the University of Tennessee. My current research is connected to discrete homotopy theory, which replaces curves and homotopies in standard algebraic topology with discrete chains and homotopies. One obtains covering spaces and fundamental groups "at a scale". Applications include explicit fundamental group finiteness theorems and generalizations of the Covering Spectrum, which is a significant subset of the Length Spectrum. My most recent work extends some of these ideas to resistance metrics on fractals, for which there is a well-defined notion of Laplace operator but generally no non-constant rectifiable curves. The relationship between the Laplace spectrum and Length Spectrum has been a subject of research for over fifty years, and there remain some interesting open questions. The generalized spectra in my recent work provide proxies for the Length Spectrum in such spaces when there is no actual length to work with.
In my career I have taught more than 25 different courses, including calculus at all levels, linear algebra, geometry, topology, algebra, analysis, and one of my favorites, Honors Introduction to Abstract Math. One disadvantage of being Department Head is that the very heavy administrative load greatly reduces the amount of teaching and research I am able to do.