Laplacian Growth
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.
Please find our Piazza course page at https://piazza.com/bimsa/spring2023/sp23
You can sign-up for the course with the access code: Laplacian.
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.
Please find our Piazza course page at https://piazza.com/bimsa/spring2023/sp23
You can sign-up for the course with the access code: Laplacian.
讲师
日期
2023年02月24日 至 05月16日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二,周五 | 08:50 - 10:35 | A3-1-101 | ZOOM 05 | 293 812 9202 | BIMSA |
修课要求
Measure Theory, Probability (Martingales)
课程大纲
1. A Guided Tour Of Laplacian Growth
1.1 Growth with multiple sources
1.2 Abelian property
1.3 Rotor-routing: derandomized random walk
1.4 The discrete Laplacian
1.5 Odometer function
1.6 Tiptoeing around the Laplacian
1.7 The obstacle problem
1.8 Green function
1.9 Smash sum in the continuum
1.10 The abelian sandpile model
2. Potential Theory
2.1 Harmonic Functions
2.2 Superharmonic Functions
2.3 Poisson Kernel: From Small Balls to All Balls
2.4 Harnack Inequality
2.5 Superharmonic Potentials
2.6 Dirichlet Problem: Perron’s Method
2.7 Dirichlet Problem: Wiener’s Method
3. Potential Theory in the lattice
3.1 Discrete Harmonic Functions
3.2 Simple Random Walk
3.3 Martingales; optional stopping
3.4 Green Function
3.5 Discrete Potential Theory
3.6 Dirichlet Problem: Discrete To Continuous
3.7 Discrete Harmonic Polynomials
4. External DLA
4.1 Harmonic measure from infinity
4.2 Hitting a half-line
4.3 Beurling estimate
4.4 Kesten’s Diameter Bound
5. The Obstacle Problem
5.1 Least Superharmonic Majorant
5.2 Boundary Regularity for the Obstacle Problem
5.3 Convergence of Obstacles, Majorants and Domains
6. Divisible Sandpile
6.1 Least Action Principle
6.2 Point Source
6.3 General Toppling Procedures
6.4 Multiple Sources
7. Internal diffusion-limited aggregation
7.1 Point source IDLA, inner estimate: The two-phase procedure
7.2 No Thin Tentacles
7.3 Concentration Inequalities: Bernstein and Freedman
8. Logarithmic Fluctuations
8.1 Sharp upper bound on the probability of thin tentacles
8.2 Early and late points 181
8.3 Multiple Sources 192
1.1 Growth with multiple sources
1.2 Abelian property
1.3 Rotor-routing: derandomized random walk
1.4 The discrete Laplacian
1.5 Odometer function
1.6 Tiptoeing around the Laplacian
1.7 The obstacle problem
1.8 Green function
1.9 Smash sum in the continuum
1.10 The abelian sandpile model
2. Potential Theory
2.1 Harmonic Functions
2.2 Superharmonic Functions
2.3 Poisson Kernel: From Small Balls to All Balls
2.4 Harnack Inequality
2.5 Superharmonic Potentials
2.6 Dirichlet Problem: Perron’s Method
2.7 Dirichlet Problem: Wiener’s Method
3. Potential Theory in the lattice
3.1 Discrete Harmonic Functions
3.2 Simple Random Walk
3.3 Martingales; optional stopping
3.4 Green Function
3.5 Discrete Potential Theory
3.6 Dirichlet Problem: Discrete To Continuous
3.7 Discrete Harmonic Polynomials
4. External DLA
4.1 Harmonic measure from infinity
4.2 Hitting a half-line
4.3 Beurling estimate
4.4 Kesten’s Diameter Bound
5. The Obstacle Problem
5.1 Least Superharmonic Majorant
5.2 Boundary Regularity for the Obstacle Problem
5.3 Convergence of Obstacles, Majorants and Domains
6. Divisible Sandpile
6.1 Least Action Principle
6.2 Point Source
6.3 General Toppling Procedures
6.4 Multiple Sources
7. Internal diffusion-limited aggregation
7.1 Point source IDLA, inner estimate: The two-phase procedure
7.2 No Thin Tentacles
7.3 Concentration Inequalities: Bernstein and Freedman
8. Logarithmic Fluctuations
8.1 Sharp upper bound on the probability of thin tentacles
8.2 Early and late points 181
8.3 Multiple Sources 192
参考资料
* 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.)
* 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.)
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).