# Probability and Statistical physics

**Lecturer**

Yuval Peres (Chair Professor) Senya Shlosman (Visiting Research Fellow)

**Date & Time**

14:00 - 15:30, each Saturday from 2022-12-10 to 2022-12-17

**Location**

**Room:** TSIMF A103 **Zoom:** 518 868 7656 **Password:** BIMSA

**Record**

Yes

**Introduction**

The first two lectures are described below. We can continue with further lectures and also working on research questions with small groups, according to the interest of the students.

Asking questions during lectures is strongly encouraged!!!

**Lecture 1: **The hunter, the rabbit and Kakeya sets

**Lecturer: **Yuval Peres

**Abstract: **A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in measure theory and harmonic analysis since Besicovich (1919); we find a new connection to combinatorics and game theory. A hunter and a rabbit move on an n-vertex cycle without seeing each other until they meet. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such Kakeya sets. I’ll conclude with an open problem: is the capture time of a weak rabbit (that can only jump to distance 1) on a general n-vertex graph, linear in n ?

**Lecture 2: **Percolation and phase transitions

**Lecturer: **Senya Shlosman

**Abstract: **We are all familiar with some phase transitions in the real world, such as the boiling of water, when increasing the temperature continuously yields a qualitative change. The first mathematical models of phase transitions emerged in the 20th century, with some recent spectacular progress that has led to three Fields medals in this century. We will present the fundamentals of the topic, the remarkable new insights obtained in two dimensions, leading up to the central open question of continuity of the percolation probability in three dimensions.

**Level**

Graduate & Undergraduate

**Language**

English

**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 European congress of Math, and at the 2017 Math Congress of the Americas. In 2016, he was elected to the US National Academy of Science.

Senya Shlosman obtained his PhD in 1978, from the St.-Petersburg branch of Steklov institute. His adviser was Roland Dobrushin. He obtained his second PhD (habilitation) in 1989, from the Ukrainian Institute of Mathematics. From 1991 he was Professor of the Dept. of Math., UC Irvine. He moved to France in 1996, getting the position of Directeur de Recherche in CPT, CNRS, Luminy, Marseille. Currently he is a leading scientific researcher in the Institute for Information Transmission Problems of the Academy of Science, Moscow, and Professor of the Center for Advance Studies in Skolkovo Inst. of Technology, Moscow.

**TA**

Dr. Qi Zhang (qzhang@bimsa.cn)

**Video**

**Notes**