A Common Topological Approach to Randomness in the Structure of Brains and the Cosmos

Human brains and the cosmos differ in size by many orders of magnitude, serve completely different purposes, and are rarely considered together. Nevertheless, when both are reduced to the size of a single computer screen, it turns out that they share certain topological properties that lead to the development of common models for the scientific testing of hypotheses related to randomness in their behaviour.

These models are best built from what are known as Gaussian and related random functions, and the hypothesis testing is based on integral and differential geometric concepts applied to certain sets that the functions generate. It turns out that this combination of probability and geometry generates new theory of intrinsic mathematical interest with recent new, and deep, results.

The aim of this talk will be to describe applications, and then the theory via these applications, with hints as to how to take the theory to higher levels and how to broaden the classes of applications that it can cover.