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.
Lecturer
Date
24th February ~ 16th May, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Friday | 08:50 - 10:35 | A3-1-101 | ZOOM 05 | 293 812 9202 | BIMSA |
Prerequisite
Measure Theory, Probability (Martingales)
Syllabus
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
Reference
* 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.)
Audience
Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
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. In 2023, he joined Beijing Institute of Mathematical Sciences and Applications. He 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. He 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). He 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.