Title: Two results in Discrete Differential Geometry
Speaker: Aaron Trout, Chatham University
When: April 3, 12:30-13:30
Where: Porter Hall 125B
Abstract:
We will discuss two results in discrete differential geometry: 1) A combinatorial 3-manifold with edges of degree at most five has edge-diameter at most five. 2) Suppose such a combinatorial 3-manifold $M$ has vertices $v$ and $w$ at the maximum edge-distance. Then, $M$ is a 3-sphere whose triangulation is completely determined by the star of $v$. These theorems are analogous to two important results in the differential geometry of positively curved spaces. The first is analogous to the Bonnet-Myers theorem, which bounds the diameter of Riemannian manifolds whose Ricci curvature is everywhere greater than a fixed positive constant. The second result is a discrete version of the Toponogov-Cheng rigid-sphere theorem, which shows that the maximum diameter allowed by the Bonnet-Myers theorem is achieved only for the standard sphere. The proofs we present are entirely combinatorial in nature, use only elementary arguments, and follow closely the proofs of the corresponding classical results. We will also talk about some enumeration algorithms which can be used to investigate the geometry of combinatorial manifolds. This talk is based on my Ph.D. thesis, which was done at Rice University under the direction of Robin Forman.
