Prerequisite

Measure Theory, Probability (Martingales)

Abstract

Laplacian growth is the study of interfaces that move in proportion to harmonic measure. We survey progress over the last decade on discrete models of (internal) Laplacian growth, including the abelian sandpile, internal DLA (first analyzed by Lawler, Bramson and Griffeath in 1992), rotor aggregation, and the scaling limits of these models on the lattice as the mesh size goes to zero. In much of the course, we will develop the tools of classical and discrete potential theory that are needed to analyze these models. Prerequisites will be kept to a minimum; familiarity with the first few chapters of Rudin's "real and complex analysis" and of a graduate probability text such as the one by Durrett, should suffice. One can get an idea of the area from: * https://yuvalperes.com/rotor-router-model/ * http://pi.math.cornell.edu/~levine/gallery/ * http://pi.math.cornell.edu/~levine/what-is-a-sandpile.pdf * Levine, L. and Peres, Y., 2017. Laplacian growth, sandpiles, and scaling limits. Bulletin of the American Mathematical Society, 54, 355-382. https://arxiv.org/abs/1611.00411 (This survey has many references.) Simulations and striking pictures have guided much of the growth of the topic, and many open problems remain.

Lecturer Intro.

Yuval Peres obtained his PhD in 1990 from the Hebrew University, Jerusalem. He was a postdoctoral fellow at Stanford and Yale, and was then a Professor of Mathematics and Statistics in Jerusalem and in Berkeley. Later, he was a Principal researcher at Microsoft. Yuval has published more than 350 papers in most areas of probability theory, including random walks, Brownian motion, percolation, and random graphs. He has co-authored books on Markov chains, probability on graphs, game theory and Brownian motion, which can be found at https://www.yuval-peres-books.com/ . His presentations are available at https://yuval-peres-presentations.com/ Dr. Peres is a recipient of the Rollo Davidson prize and the Loeve prize. He has mentored 21 PhD students including Elchanan Mossel (MIT, AMS fellow), Jian Ding (PKU, ICCM gold medal and Rollo Davidson prize), Balint Virag and Gabor Pete (Rollo Davidson prize). Dr. Peres was an invited speaker at the 2002 International Congress of Mathematicians in Beijing, at the 2008