COLLEGE OF ARTS AND SCIENCES Department of Mathematics and Statistics
PhD, Mathematics, Washington State University - in progress
Bachelor of Science, Mathematics, St. Mary's College, Maryland, 2013
Symmetric Univalent Harmonic Maps. BYU REU, Summer 2011. Advisor: Dr. Michael Dorff. Proving that a complex function is one to one has traditionally been a more involved process than proving a real function is one to one. For analytic functions, many different approaches have been developed in order to show a function is one to one. However, for harmonic functions, there are relatively few tools available. Existing methods rely on convexity in a direction, or on the harmonic argument principle. This summer, we looked at symmetric harmonic maps, and in particular, maps with symmetry. In particular we developed a transformation that would increase the symmetry of a map. In order to prove univalence of these functions, we adapted existing tools. We are currently working on proving univalence from this transformation in general.
An Undergraduate Approach to Curve Shortening Flow, St. Mary's College, Fall 2011-Spring 2012. Advisor: Dr. Casey Douglas. In 1986, Abresch and Langer published, among other things, a classification of self similar solutions to curve shortening flow. However, their solution makes use of quite deep tools from Differential Geometry. Using tools appropriate for undergraduates, we found alternative proofs of these classification results. We also investigate embedded solutions to a related problem.
A Different Sort of Polygon St. Mary's College, Fall 2012-Spring 2013. Advisor: Dr. Alex Meadows. We investigated tube formulae, in particular one by Steiner. One proof of the previous formula relies on approximation by convex polygon. In an effort to generalize the above, we define $\alpha$-polygons, which generalize convex polygons. These generalized polygons also obey a tube formula which generalizes Steiner's formula. They also obey many of the properties that convex polygons obey. However, unlike traditional polygons, $\alpha$-polygons have an intrinsic scale, which changes the way they behave geometrically.
I'm interested in the geometry of metric spaces. In particular, I've been reading A course in Metric Geometry by Barago, Barago, and Ivanov. At the same time, I'm beginning an investigation into the analysis of metric spaces. I get a lot of information from my office mates, but I've started looking into Geometric Measure Theory by Federer.
There are a lot of different ways of developing tangent spaces for arbitrary metric spaces. Its extremely interesting to read about the different types. Some people define them as equivalence classes of paths; others define them using Gromov-Hausdorff limits. Still others define them as equivalence classes of sequences converging to a point. I'm excited to figure out why these different ways of looking at tangent spaces all exist, and the merits of each.