Mathematical Interests

My research focuses on a number of problems in the geometric topology of polyhedra. My initial area of interest was the study of simplexwise linear maps of disks in 2 and 3 dimensions. Simplexwise linear maps of disks first arose in the approach to smoothing theory of S. S. Cairns from the 1940's, and more recently in the work of N. Levitt; they are also relevant to smooth maps of disks via work of Kuiper. In contrast to piecewise linear maps, spaces of simplexwise linear maps are not groups, and so most of the standard techniques of topology do not apply to their study; hence very little is known about these spaces. The techniques involved tend to be quite geometric and ad hoc. I began my study of simplexwise linear maps in 2 dimensions in my dissertation, and I have since obtained some further 2 dimensional results. There is more to be proved in 2 dimensions, and the field is wide open in 3 dimensions and above.

Ever since my dissertation, each time I finished working on one problem I found myself natually led to another. For example, in the course of working on a proof about simplexwise linear maps, I was led to the question of the curvature of polyhedra, and that topic has been my main focus in recent years. There is a classical method of computing the curvature of polyhedra in 3 dimensions, known as the angle defect, that goes back at least as far as Descartes. The angle defect has been generalized to higher dimensions, and has been studied from both a combinatorial approach, for example in the work of B. Grünbaum and G. C. Shephard, and from a differential geometric point of view, for example in the work of Banchoff. My approach has focussed primarily on a new gneralization of the angle defect that I have defined and studied. My work on the angle defect has in turn led to a number of interesting questions that I would like to explore in the near future: give an axiomatic characterization of my angle defect; determine the properties of the stratified version of the Euler characteristic that arises in my approach; relate that stratified Euler characteristic to the well known Poincare-Hopf Theorem; generalize the various definitions of curvature used for polyhedra to partially ordered sets.

In addition to thinking about polyhedra, my other professional interest has been writing textbooks for college mathematics courses. So far I have written three books: "A First Course in Geometric Topology and Differential Geometry" (Birkhäuser 1996), "Proofs and Fundamentals: A First Course in Abstract Mathematics," (Birkhäuser 2000, second edition Springer-Verlag 2010), and "The Real Numbers and Real Analysis" (Springer-Verlag 2011). I have also written lecture notes for my math-for-liberal-arts course "Topics in Geometrical Mathematics," and perhaps one day these notes will be turned into a book as well.