Week 08.06.2026 – 14.06.2026

A Ray-Knight theorem is a description of the local time profile of a stochastic process when stopped at some inverse local time. Since a Ray-Knight theorem contains a lot of information about the underlying process, and since a number of results have been obtained for self-interacting random walk models by proving Ray-Knight theorems for the walk, one naturally wonders if a Ray-Knight theorem can be used directly to deduce the scaling limit of the walk. Somewhat surprisingly, a recent result of myself with Kosygina and Mountford shows that this is not the case.
In this talk, I will show that while Ray-Knight theorems are not sufficient for proving scaling limits, one can obtain a functional limit for the walk through what we call joint Ray-Knight theorems. As an application of our main result we prove scaling limits for the “true” self-avoiding walk and the polynomially self-repelling motion. The “true” self-avoiding walk converges to a process called the “true” self-repelling motion, confirming a conjecture of Toth and Werner, while the scaling limit of the polynomially self-repelling random walk appears to be a new stochastic process. This is based on joint work with Elena Kosygina, Laure Mareche, and Tom Mountford.

Title: Geometry, Robots and Society

Abstract: How do we move a robot quickly from one position to another? To answer this question, we need to understand its configuration space, a "map" where we can find every possible position of the robot. Unfortunately, these spaces are very large, they live in very high-dimensions, and they are very difficult to visualize. Fortunately, geometric group theorists and combinatorialists have encountered and studied similar spaces before. Thanks to the tools they've developed, we can build "remote controls" to navigate these complicated spaces; this allows us to move (some) robots optimally. As the imaginary border between "pure" and "applied" mathematics disappears before our eyes, we face an important ethical question that we cannot ignore: What’s the role of mathematicians and scientists in building a more just and equitable society?

This presentation will assume no previous knowledge of these topics, and will be accessible to anyone who is interested. It will include joint work with many people, including Tia Baker, Naya Banerjee, Hanner Bastidas, César Ceballos, John Guo, Megan Owen, Seth Sullivant, Coleson Weir, and Rika Yatchak.