Calculating the curvature of a concrete complex
Abstract: Geometric group theorists have been borrowing techniques from differential geometers for more then 15 years. As a result there is now a well developed theory of nonpositively curved metrical simplicial complexes which can be used to study to the "geometry of a group".
Unfortunately for the working group theorist, very few tests exist which determine whether a given finite complex will fit into this framework. More specifically, given an explicit, finite, piecewise Euclidean (PE) complex, there did not (until recently) exist a general algorithm for determining whether this particular complex was nonpositively curved.
After a quick introduction to/review of the theory of metric simplicial complexes, I will present recent results which provide an algorithm to solve this problem in full generality using the theory of Groebner bases from computational algebraic geometry.
(joint work with Murray Elder)