Phase transitions and conic geometry

A phase transition is a sharp change in the behavior of a mathematical model as one of its parameters changes. This talk describes a striking phase transition that takes place in conic geometry. First, we will explain how to assign a notion of dimension to a convex cone. Then we will use this notion of dimension to see that two randomly oriented convex cones share a ray with probability close to zero or close to one. This fact has implications for many questions in signal processing. In particular, it yields a complete solution of the compressed sensing problem about when we can recover a sparse signal from random measurements. This talk is designed for a general mathematical audience.

Based on joint works with Dennis Amelunxen, Martin Lotz, Mike McCoy, and Samet Oymak.