Lecture 10:  Thursday, May 18, 2023

We are pleased to have Prof. Russell Lyons, Department of Mathematics, Indiana University, as our honored speaker.

THURSDAY, MAY 18, 2023, at 4:30 PM
Math/Stat 112, 5727 South University Avenue

Reception following the lecture at 5:30 pm, in the Reading Room, Math/Stat 101, 5727 South University Avenue


"Voronoi Tessellations without Nuclei"

ABSTRACT:  Given a discrete set of points in a metric space, called nuclei, one associates to each such nucleus its Voronoi cell, which consists of all points closer to it than to other nuclei. This construction is widely used in mathematics, science, and engineering; it is even used in baking. In Euclidean space, one commonly uses a homogeneous Poisson point process to assign the locations of the nuclei. As the intensity of the point process tends to 0, the nuclei spread out and disappear in the limit, with each pair of points eventually belonging to the same cell. Surprisingly, this does not happen in other settings such as hyperbolic space; instead, one obtains a Voronoi tessellation without nuclei! We describe properties of such a limiting tessellation, as well as analogous behavior on Cayley graphs of finitely generated groups. We will illustrate results with many pictures and several animations. The talk is based on work of Sandeep Bhupatiraju and joint work in progress with Matteo d'Achille, Nicolas Curien, Nathanael Enriquez, and Meltem Unel. We will not assume knowledge of Poisson point processes or of hyperbolic space.

Prof. Lyons will also give a talk at the following:

Probability and Statistical Physics Seminar
FRIDAY MAY 19, 2023, 2:30 PM, in Eckhart 202, 5734 S. University Avenue
"Monotonicity for Continuous-Time Random Walks"

ABSTRACT:  Variable-speed, continuous-time random walk on a graph is given by an assignment of nonnegative rates to its edges. There are independent Poisson processes associated to the edges with the given rates. When a walker is at a vertex, it jumps to a neighbor at the time of the next event that occurs for the corresponding incident edges. In the case of a Cayley graph of a finitely generated group, we are particularly interested in the setting where the edge rates depend only on the corresponding generators. Our lecture is concerned with monotonicity in the rates for various fundamental properties of random walks. We will survey results, counterexamples, and open questions. We will give general ideas of proofs, but avoid technicalities. Most of the talk will be devoted to two questions on Cayley graphs: On infinite graphs, we ask about the limiting linear rate of escape, i.e., the limit of the distance divided by the time. Does this increase when the rates are increased? On finite graphs, we ask about the convergence to the stationary (uniform) distribution. Does this happen faster when the rates are increased? It turns out that both questions have surprising answers. This is joint work with Graham White.