Geometric approach to random matrices and asymptotic representation theory
A. Okounkov in the celebrated paper [1] has established the common limit of the mixed moments of random Young diagrams with respect to Plancherel measure and of random matrices from Gaussian Unitary Ensemble. He used the fact that moments of random matrices enumerate the maps of topological surfaces while representation theory of symmetric groups that produces Plancherel measure is connected to the description of essentially the same surfaces as the ramified coverings of the sphere.
In this course we will study this construction. Then we will discuss its possible extensions to other classical ensembles of random matrices, such as Laguerre and Jacobi Unitary Ensembles. Their counterparts in representation theory side of the correspondence are connected to Schur-Weyl duality and skew Howe duality.
This course extends the results covered in my minicourse "Random matrices and asymptotic representation theory" from January 2024, and has some overlaps with the lectures by Pavel Nikitin "Asymptotic representation theory and random matrices", but uses geometric approach. The lectures will be self-contained.
This course will be taught from June 20 to July 11, and from Aug. 20 to Sept. 3, 2025.
In this course we will study this construction. Then we will discuss its possible extensions to other classical ensembles of random matrices, such as Laguerre and Jacobi Unitary Ensembles. Their counterparts in representation theory side of the correspondence are connected to Schur-Weyl duality and skew Howe duality.
This course extends the results covered in my minicourse "Random matrices and asymptotic representation theory" from January 2024, and has some overlaps with the lectures by Pavel Nikitin "Asymptotic representation theory and random matrices", but uses geometric approach. The lectures will be self-contained.
This course will be taught from June 20 to July 11, and from Aug. 20 to Sept. 3, 2025.
Lecturer
Date
20th June ~ 3rd September, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday,Friday | 09:50 - 12:15 | A3-3-201 | Zoom 15 | 204 323 0165 | BIMSA |
Prerequisite
Representation theory of symmetric group, Lie groups and Lie algebras, random matrix theory. Acquaintance with the material from the my previous courses "Asymptotic representation theory and random matrices", "From free fermions to limit shapes and beyond", courses of Pavel Nikitin "Asymptotic representation theory and random matrices", "Symmetric functions and representation theory", "Asymptotic representation theory" and course of Fan Yang "Topics in Random matrix theory" would be a plus, but is not required.
Syllabus
1. Short reminder of classical ensembles in random matrix theory
2. GUE random matrices and maps of topological surfaces
3. Reminder on representation theory of symmetric groups, Plancherel measure
4. Ramified coverings of surfaces and Hurwitz numbers
5. Correlation functions in large n limit
6. Laguerre Unitary Ensemble, surfaces and Hurwitz numbers
7. Schur-Weyl duality, limit shapes of P.Biane and Marchenko-Pastur law
8. Jacobi Unitary Ensemble and Hurwitz numbers
9. Skew Howe duality and representations of symmetric groups
10. Limit shape of random square Young tableaux and distribution of JUE eigenvalues.
2. GUE random matrices and maps of topological surfaces
3. Reminder on representation theory of symmetric groups, Plancherel measure
4. Ramified coverings of surfaces and Hurwitz numbers
5. Correlation functions in large n limit
6. Laguerre Unitary Ensemble, surfaces and Hurwitz numbers
7. Schur-Weyl duality, limit shapes of P.Biane and Marchenko-Pastur law
8. Jacobi Unitary Ensemble and Hurwitz numbers
9. Skew Howe duality and representations of symmetric groups
10. Limit shape of random square Young tableaux and distribution of JUE eigenvalues.
Reference
[1] Okounkov, Andrei, Random matrices and random permutations, International Mathematics Research Notices 2000.20 (2000): 1043-1095.
[2] Okounkov, Andrei, and Rahul Pandharipande, Gromov-Witten theory, Hurwitz numbers, and matrix models, I, Proc. Symposia Pure Math. Vol. 80. 2009.
[3] Michel Ledoux, Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. The Continuous Case, Electron. J. Probab. 9 177 - 208, 2004. https://doi.org/10.1214/EJP.v9-191
[4] Gisonni, Massimo, Tamara Grava, and Giulio Ruzza, Laguerre ensemble: correlators, Hurwitz numbers and Hodge integrals, Annales Henri Poincaré. Vol. 21. No. 10. Cham: Springer International Publishing, 2020.
[5] Gisonni, Massimo, Tamara Grava, and Giulio Ruzza, Jacobi ensemble, Hurwitz numbers and Wilson polynomials, Letters in Mathematical Physics 111.3 (2021): 67.
[6] Pittel, Boris, and Dan Romik, Limit shapes for random square Young tableaux, Advances in Applied Mathematics 38.2 (2007): 164-209.
[7] Panova, Greta, and Piotr Śniady, Skew Howe duality and random rectangular Young tableaux, Algebraic Combinatorics 1.1 (2018): 81-94.
[2] Okounkov, Andrei, and Rahul Pandharipande, Gromov-Witten theory, Hurwitz numbers, and matrix models, I, Proc. Symposia Pure Math. Vol. 80. 2009.
[3] Michel Ledoux, Differential Operators and Spectral Distributions of Invariant Ensembles from the Classical Orthogonal Polynomials. The Continuous Case, Electron. J. Probab. 9 177 - 208, 2004. https://doi.org/10.1214/EJP.v9-191
[4] Gisonni, Massimo, Tamara Grava, and Giulio Ruzza, Laguerre ensemble: correlators, Hurwitz numbers and Hodge integrals, Annales Henri Poincaré. Vol. 21. No. 10. Cham: Springer International Publishing, 2020.
[5] Gisonni, Massimo, Tamara Grava, and Giulio Ruzza, Jacobi ensemble, Hurwitz numbers and Wilson polynomials, Letters in Mathematical Physics 111.3 (2021): 67.
[6] Pittel, Boris, and Dan Romik, Limit shapes for random square Young tableaux, Advances in Applied Mathematics 38.2 (2007): 164-209.
[7] Panova, Greta, and Piotr Śniady, Skew Howe duality and random rectangular Young tableaux, Algebraic Combinatorics 1.1 (2018): 81-94.
Audience
Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Anton Nazarov is an associate professor at Saint Petersburg State University, Russia. He completed his PhD at the department of high-energy and elementary particle physics of Saint Petersburg State University in 2012 under the supervision of Vladimir Lyakhovsky. In 2013-2014 he was a postdoc at the University of Chicago. Anton's research interests are representation theory of Lie algebras, conformal field theory, integrable systems, determinantal point processes.