Home > Research

Maria Belk

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.

Talks

Here are powerpoint slides for four talks that I have given:

• Realizability of Graphs: This talk is based on my thesis research, which is joint work with my advisor Robert Connelly. These slides are a slightly modifed version of the 20 minute talk that I gave at the Horizon of Combinatorics conference at Lake Balaton, Hungary, in July 2006.

  Abstract: A graph is d-realizable if, for every realization of the graph in some Euclidean space, there exists a realization in d-dimensional Euclidean space with the same edge lengths. For example, any tree is 1-realizable, but a triangle is not. In this talk, we will classify all 3-realizable graphs.

• Realizability of Graphs: This is a version of the above talk that was aimed at undergraduates attending the Carleton Summer Math Program during the summer of 2007.
• Making your house safe from zombie attacks: I gave this talk in Fall 2006 at the Texas A&M Postdoc Colloquium. This talk is based on ongoing research with my husband Jim Belk.

  Abstract: We play a game on a graph where zombies wander around the graph trying to catch and eat a person. The zombie number of the graph is the number of zombies needed to catch the person, assuming the person can move much faster than the zombies. I will show how zombie number is related to treewidth and discuss some of the forbidden minors.

• Problems related to the Kneser-Poulsen conjecture: I gave this talk in Fall 2007 in the Algebra and Combinatorics seminar at Texas A&M.

  Abstract: Consider a collection of overlapping balls in Euclidean space. If we change the positions of the balls, then the volume of the union may change. In the 1950's, Kneser and Poulsen conjectured that if the distances between the centers do not decrease, then the volume of the union must increase or remain the same. In 2002, Bezdek and Connelly proved this conjecture for discs on the Euclidean plane. The conjecture remains open for higher dimensional Euclidean spaces, as well as for spherical and hyperbolic spaces. In this talk, we will examine the difficulties involved in extending the proof to these other settings.

Home > Research