Probability and Statistical physics
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.
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.
日期
2022年12月10日 至 17日
网站
听众
Undergraduate
, Graduate
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Yuval Peres 1990年于耶路撒冷希伯来大学博士毕业，并先后于斯坦福大学和耶鲁大学任职博士后。此后，他在美国加州大学伯克利分校和耶路撒冷希伯来大学担任数学和统计学教授，并任微软公司首席研究员。Peres在概率论的大部分领域总共发表过超过350篇论文，包括随机游走，布朗运动，渗流和随机图。与他人著有多本专题专著：《概率与分析中的分形》，《布朗运动》，《高斯解析函数的零点与行列式点过程》，《马尔可夫链与混合时间》，《树图与网络中的概率论》，《博弈论》，并被美国数学会和剑桥大学出版社出版。专著涉及马可夫链、概率图、博弈论和布朗运动等方向，具体信息可以在以下网址查找：https://www.yuval-peres-books.com/ . 他的报告可在以下网址查找：https://yuval-peres-presentations.com/。
Peres是Rollo Davidson奖和Loeve奖得主，是2002年北京国际数学家大会、2008年欧洲数学家年会、2017年美国数学家年会邀请报告人，并于2016年当选美国科学院院士。他指导过21名博士，包括Elchanan Mossel (美国麻省理工大学教授, 美国数学家学会会士), 丁剑 (北京大学, 国际华人数学家大会金奖、Rollo Davidson奖得主), 以及Balint Virag和Gabor Pete (Rollo Davidson奖得主).
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.
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.