Explore cutting-edge research problems in random matrix theory through this comprehensive collection of lectures from the 27th Annual PCMI Summer Session. Delve into the interdisciplinary field that connects mathematics, physics, computer science, and statistics through presentations by leading researchers. Master advanced topics including algebraic structures for multilevel eigenvalue densities, random walks in random environments and the Kardar-Parisi-Zhang class, Gaussian random matrices and stationary random functions, and limit shapes beyond dimers. Investigate sophisticated mathematical techniques such as transfer matrix approaches to 1D random band matrices, dynamical methods in random matrices, local limits of random sorting networks, and stochastic Loewner evolution with branching. Examine theoretical foundations covering distributions of random matrices and their limits, methods for lower bounds on fluctuations of random variables, and applications ranging from number theory and integral geometry to the Sherrington-Kirkpatrick model. Learn from expert mathematicians including Yi Sun, Timo Seppäläinen, Gérard Ben Arous, Richard Kenyon, Horng-Tzer Yau, Vadim Gorin, Roland Speicher, Sourav Chatterjee, Jinho Baik, and Persi Diaconis as they present their latest research findings and open problems. Gain insights into connections between random matrix theory and Toeplitz operators, superprocesses, and various physical models through this intensive mathematical experience designed for researchers, graduate students, and mathematics educators at the post-secondary level.

Syllabus

Yi Sun (2) Algebraic structures for multilevel eigenvalue densities
Yi Sun (1) Algebraic structures for multilevel eigenvalue densities
Timo Seppäläinen (2) Random walk in random environment and the Kardar-Parisi-Zhang class, part 2
Timo Seppäläinen (1) Random walk in random environment and the Kardar-Parisi-Zhang class, part 1
Gérard Ben Arous (1.2) Gaussian Random Matrices and stationary random functions, part 1.2
Gérard Ben Arous (1.1) Gaussian Random Matrices and stationary random functions, part 1.1
Richard Kenyon (1.12 Limit shapes beyond dimers
Richard Kenyon (1.1) Limit shapes beyond dimers
M. Shcherbina (1.2) Transfer matrix approach to 1d random band matrices.
M. Shcherbina (1.1) Transfer matrix approach to 1d random band matrices.
Horng-Tzer Yau (1.2) Dynamical Methods In Random Matrices
Horng-Tzer Yau (1.1) Dynamical Methods In Random Matrices
Karl Liechty, (1.2) Research Problem
Karl Liechty, (1.1) Research Problem
Vadim Gorin (1.2) Local limits of Random Sorting Networks
Vadim Gorin (1.1) Local limits of Random Sorting Networks
Govind Menon (1.2) Stochastic Loewner evolution with branching and the Dyson superprocess
Govind Menon (1.1) Stochastic Loewner evolution with branching and the Dyson superprocess
Roland Speicher (1.2) Distributions of Random Matrices and Their Limits, part 1.2
Roland Speicher (1.1) Distributions of Random Matrices and Their Limits, part 1.1
Sourav Chatterjee (1.2) Method for lower bounds on fluctuations of random variables, part 1.2
Sourav Chatterjee (1.1) Method for lower bounds on fluctuations of random variables, part 1.1
Peter Forrester - Decomposition of measure in RMT applied to number theory and integral geometry
Jhino Baik (1.2) Fluctuations of the free energy of the Sherrington-Kirpatrick model, part 1.2
Jhino Baik (1.1) Fluctuations of the free energy of the Sherrington-Kirpatrick model, part 1.1
Persi Diaconis-FROM RANDOM MATRIX THEORY TO TOEPLITZ OPERATORS