Research Statement
I. Combinatorics and Discrete Geometry
My primary interest within Discrete Geometry is polytopes. In the plane, polytopes are simply polygons. In three space, they include cubes, “snub cubes”, pyramids, octahedra and many other solids. (These three dimensional versions are sometimes called “polyhedra”.) While polytopes are relatively easy to define, these objects quickly increase in complexity as the dimension increases. Four dimensional polytopes possess many surprising properties and even three dimensional polytopes provide many interesting, open research questions. I am particularly fascinated how the finite structures that encode the combinatorics of polytopes are able to capture so many aspects of higher dimensional Euclidean geometry.
Currently, my research in this area is focused on producing a new type of computer search program. By utilizing the Gale transform, I was able to derive a novel method, called “clique replacement”, that can produce a new polytope from an existing one. The technique is robust in the sense that any two polytopes of the same dimension with the same number of facets must be connected to each other by a finite sequence of clique replacements.
Over the past five months, I have developed a computer search program based on clique replacement that produces all polytopes within a specified number of clique replacement steps of a given starting polytope. This represents the first stage of a two stage computer investigation that will form the basis for an Research Experience for Undergraduates this summer at Lafayette College. My hope is that the software ultimately will act as a kind of research laboratory for investigating polytopes.
Here you will find more details.
Some other interests within Combinatorics are polytopal digraphs, combinatorial games and magic as well as antimagic graph labelings. My most recent paper is a collaboration with two professors at the University of Montana on the topic of antimagic graph labelings.
II. Data Visualization and Temple MVV
TempleMVV is a method for graphing multivariate data that was developed at Temple University over fifteen years ago. The technique is natural and relatively simple yet the graphs it produces are able to retain much of the complex structure of multidimensional data. Over the past year, I have reacquainted myself with the technique and am optimistic about its potential for a variety of applications in mathematics and education. In addition to talks at the Graduate/Faculty Conference at the University of Montana and in the department’s Geometry Seminar, I have begun some preliminary collaboration with professors from the UM Math department to analyze pharmaceutical as well as genetic data. Here you will find an introduction to the technique as well as a few mathematical applications.
I believe the technique has tremendous potential for interdisciplinary collaboration. Moreover, if properly introduced, the technique also has promise for general education, especially for multivariate statistics.