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 Kneser-Poulsen

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 3-realizable graphs, and we show some results related to the Kneser-Poulsen conjecture in hyperbolic space.

A graph is d-realizable 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 2-realizable if and only if it does not have K_4 as a minor.  We show that a graph is 3-realizable 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 non-continuous expansion in hyperbolic space.  We show that the conjecture holds for 4 balls in 3-dimensional hyperbolic space.