Time: August 31, 4:30 – 5:30 pm EDT
Title: Dynamical First-Passage Percolation
Abstract: In first-passage percolation (FPP), we place i.i.d. nonnegative weights on the edges of the cubic lattice Z^d and study the induced weighted graph metric T = T(x,y). Letting F be the common distribution function of the weights, it is known that if F(0) is less than the threshold p_c for Bernoulli percolation, then T(x,y) grows like a linear function of the distance |x-y|. In 2015, Ahlberg introduced a dynamical model of first-passage percolation, in which the weights are resampled according to Poisson clocks, and considered the growth of T(x,y) as time varies. He showed that when F(0) < p_c, the model has no “exceptional times” at which the order of the growth is anomalously large or small. I will discuss recent work with J. Hanson, D. Harper, and W.-K. Lam, in which we study this question in two dimensions in the critical regime, where F(0) = p_c, and T(x,y) typically grows sublinearly. We find that the existence of exceptional times depends on the behavior of F(x) for small positive x, and we characterize the dimension of the exceptional sets for all but a small class of such F.
Time: September 14, 4:30 – 5:30 pm EDT
Title: Pandemic REUs
Abstract: The pandemic changed many things, REU Programs included. I will discuss challenges and advantages of mentoring undergraduates in math research from afar. Some results about interacting particle systems—namely, the frog model and ballistic annihilation—from this summer will also be presented.