Random Matrices and Symmetric Functions
This is a course dedicated to extensive relations of random matrices to the theory of symmetric functions. We will introduce Gaussian random variables, single and multiple, and the basic example of a Gaussian random Hermitian NxN matrix. The relation to symmetric functions will be studied using expectation values in eigenvalue representation. We will compute expectation values of many famous symmetric functions, including the power sum basis, Schur functions, and their generalizations to Jack and Macdonald symmetric functions.
Lecturer
Date
16th September ~ 11th December, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 09:50 - 11:25 | - | - | - |
Prerequisite
It will be helpful to have basic knowledge of linear algebra, polynomials, and probability. Otherwise all relevant notions will be defined in the class.
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
I am a mathematician working in the field of Representation Theory with focus on relation to geometry, particularly Topology (quantum group invariants of knots, links and 3-manifolds, mapping class groups of surfaces, q-skein and double affine Hecke algebras) and Algebraic Geometry (representations of quantum Virasoro and W-algebras associated with cohomology and K-theory of Nakajima quiver varieties). The role and importance of Mathematical Physics (topological quantum field theory, integrable systems, instantons and the String-Math interaction) in these domains has been increasingly recognized; my research is substantially inspired by advances in mathematical physics and benefits from recurring collaboration with physicists. This picture is complemented by my work on the interface with Probability (integrable probabilistic systems, Macdonald processes and random matrices) and a long-standing interest in Classical Invariant Theory (invariants of curves and surfaces, tensors, discriminants and resultants of algebraic equations). Key recent results that illustrate my research program include construction of a genus two analogue of the double affine Hecke algebra [14], formulation of a double elliptic generalization of Macdonald difference equations [8], development of a new method to compute observables of Macdonald processes [22], proof of a determinantal formula for the Cayley invariant of plane curves [24] among others.