Research

I am interested in discrete geometryWikipedia, computational geometryWikipedia, and graph theoryWikipedia.

For my thesis, I worked on the following question: Given a configuration of the vertices of a graph in N-dimensional Euclidean space, when can you find a configuration of the graph in 3-dimensions with the same edge lengths? I classified all such graphs with the help of my advisor Robert Connelly.

My current research is focused on questions related to the Kneser-Poulsen conjecture: Consider a collection of possibly overlapping balls in Euclidean space. Suppose that the balls are rearranged so that the distances between the centers of the balls has not decreased. Kneser and Poulsen independently conjectured in the 1950's that the volume of the union of the balls has not decreased. This conjecture has been proven in dimension 2 by Bezdek and Connelly, but it is still open in higher dimensions.

Papers

Published papers:

• Realizability of graphs (with Robert Connelly). Discrete and Computational Geometry. 37 (2007), no. 2, 125-137.

  This is a paper with my Ph.D. advisor Robert Connelly. Given a configuration of the vertices of a graph in N-dimensional Euclidean space, when can you find a configuration of the graph in 3-dimensions with the same edge lengths? We answer this question, assuming the 3-realizability of two specific graphs. The link goes to the official journal version.

• Realizability of graphs in three dimensions. Discrete and Computational Geometry. 37 (2007), no. 2, 139-162.

  This is a proof that the two specific graphs are 3-realizable.

Submitted papers:

• Making contractions continuous: a problem related to the Kneser-Poulsen conjecture (with Robert Connelly). Submitted to Contributions to Discrete Mathematics in July 2007.

  In this paper, we give an example of a contraction in dimension n such that there is no continuous contraction in dimension 2n-1. It is known that there is always a continuous contraction in dimension 2n.

• Monotonicity of the volume of four balls in hyperbolic space. Submitted to Contributions to Discrete Mathematics in December 2007.

  This paper proves that 4 balls in hyperbolic space satisify the Kneser-Poulsen conjecture.

Ph.D. Thesis

My Ph.D. thesis, Applications of stress theory: realizing graphs and Kneser-Poulsen, includes material from my two published papers and the second of the submitted papers.