Maria Belk
Maria Belk

Ph.D. (2005) Cornell University

First Position
Postdoctoral position, Texas A&M University
Dissertation
Applications of Stress Theory: Realizing graphs and KneserPoulsen
Advisor:
Research Area:
Discrete Geometry
Abstract:We use the ideas of stress theory and tensegrities to answer some questions in discrete geometry. In particular, we classify 3realizable graphs, and we show some results related to the KneserPoulsen conjecture in hyperbolic space.
A graph is drealizable if, for every configuration of its vertices in E_N, there exists another corresponding configuration in E_d with the same edge lengths. A graph is 2realizable if and only if it does not have K_4 as a minor. We show that a graph is 3realizable if and only if it does not have K_5 or K_{2,2,2} as a minor.
Kneser and Poulsen independently conjectured that the volume of the union of balls in Euclidean space will not decrease under an expansion. Csikós proved that this is true in Euclidean space under a continuous expansion, and Connelly and Bezdek proved it for the Euclidean plane (not necessarily a continuous expansion). It is still an open question for higher dimensions. The conjecture can also be stated for balls in hyperbolic space. Csikós proved that it is true under a continuous expansion in hyperbolic space, but it is still an open question for a noncontinuous expansion in hyperbolic space. We show that the conjecture holds for 4 balls in 3dimensional hyperbolic space.