Event Type:

Geometry-Topology Seminar

Date/Time:

Monday, April 11, 2016 - 12:00 to 13:00

Location:

Gilkey 113

Guest Speaker:

Peter Veerman

Institution:

Portland State University

Abstract:

For distinct points p and q in a two-dimensional connected Riemannian manifold M, we define their mediatrix Lpq as the set of points equidistant to p and q. It is known that mediatrices have a cell decomposition consisting of a finite number of branch points connected by Lipschitz curve. We show additional geometric regularity properties of mediatrices: at each point they have the radial linearizability property, which means that they are tangent to a finite collection of lines meeting in the origin. In the case of mediatrices on the sphere, where mediatrices are simple closed Lipschitz curves, we show these curves have at most countable singularities, and the total angular deficiency has a finite upper bound related to the total curvature of the metric on the sphere. On the other hand mediatrices have the topological property that they separate the manifold M into two parts and that non proper subset of them does. This allows for their topological classification. This classification is in some sense a generalization of the Jordan Brouwer Theorem. We will briefly discuss the classification of minimal separating sets in the orientable surfaces of genus 0, 1, and 2. In 2014, mediatrices found an application in a territorial conflict between Peru and Chile. We’ll show the opinion of the International Court of Justice in The Hague.