Posts Tagged ‘김상현’

Sang-hyun Kim, Acute Triangulations of the Sphere

Monday, March 18th, 2013
Acute Triangulations of the Sphere
2013/04/19 Friday 4PM-5PM – ROOM 3433
We prove that a combinatorial triangulation L of a sphere admits an acute geodesic triangulation if and only if L does not have a separating three- or four-cycle. The backward direction is an easy consequence of the Andreev–Thurston theorem on orthogonal circle packings. For the forward direction, we consider the Davis manifold M from L. The acuteness of L will provide M with a CAT(-1) (hence, hyperbolic) metric. As a non-trivial example, we show the non-existence of an acute realization for an abstract triangulation suggested by Oum; the degrees of the vertices in that triangulation are all larger than four. This approach generalizes to triangulations coming from more general Coxeter groups, and also to planar triangulations. (Joint work with Genevieve Walsh)

Sang-hyun Kim (김상현), On Gromov Conjecture and Topological Jigsaw Puzzle

Wednesday, December 23rd, 2009

(Joint Topology & Discrete Math Seminar)

On Gromov Conjecture and Topological Jigsaw Puzzle
Sang-hyun Kim
Dept. of Mathematics, University of Texas at Austin, Austin, Texas
2009/12/28 Mon, 4PM-5PM
Inspired by the famous virtual Haken conjecture in 3–manifold theory, Gromov asked whether every one-ended word-hyperbolic group contains a surface group. One simple, but still captivating case, is when the word-hyperbolic group is given as the double of a free group with a cyclic edge group. In the first part of the talk, I will describe the polygonality of a word in a free group, and a relation between polygonality and Gromov’s question. Polygonality is a combinatorial property, which is very much like solving a “topological jigsaw puzzle”. In the second part, I will describe a reduction to a purely (finite) graph theoretic conjecture using the Whitehead graph. Part I is a joint word with Henry Wilton (Caltech).