Orthogonal Polynomials, Double Affine Hecke Algebras, and Quantum Character Varieties
This is an introductory course on the Macdonald theory of orthogonal polynomials, Double Affine Hecke Algebras (DAHA), and their role in modern mathematics and physics. The main goal of the course is to provide foundational knowledge of orthogonal polynomials, preparing students to engage with modern research literature in representation theory, algebraic combinatorics, and related fields.
We will begin with the basics of Macdonald theory, introducing the ring of symmetric functions as well as Schur and Macdonald polynomials. A key highlight will be the connection between Macdonald polynomials and Dunkl operators, which arise naturally in the study of quantum many-body systems. Building on this foundation, we will explore Cherednik's nonsymmetric generalizations of orthogonal polynomials and introduce the Double Affine Hecke Algebra (DAHA).
In the final part of the course, we will uncover a surprising connection between DAHA and the geometry of quantum character varieties, which explains the action of the modular group $SL(2,Z)$ by automorphisms of DAHA. This interplay between algebra, geometry, and physics will provide a glimpse into the deep and beautiful structures underlying modern mathematical research.
We will begin with the basics of Macdonald theory, introducing the ring of symmetric functions as well as Schur and Macdonald polynomials. A key highlight will be the connection between Macdonald polynomials and Dunkl operators, which arise naturally in the study of quantum many-body systems. Building on this foundation, we will explore Cherednik's nonsymmetric generalizations of orthogonal polynomials and introduce the Double Affine Hecke Algebra (DAHA).
In the final part of the course, we will uncover a surprising connection between DAHA and the geometry of quantum character varieties, which explains the action of the modular group $SL(2,Z)$ by automorphisms of DAHA. This interplay between algebra, geometry, and physics will provide a glimpse into the deep and beautiful structures underlying modern mathematical research.
Lecturer
Date
25th February ~ 20th May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 10:40 - 12:15 | A14-202 | ZOOM 06 | 537 192 5549 | BIMSA |
Prerequisite
Undergraduate Abstract Algebra (required); Lie Algebras (recommended).
Syllabus
Tentative plan:
1. Symmetric functions
1.1 The ring of symmetric functions
1.2 Elementary symmetric functions, Complete Symmetric functions, and Power sums
1.4 Schur functions
1.3 Scalar product and orthogonal polynomials
2. Macdonald polynomials
2.1 Root systems and Weyl Characters
2.2 Scalar product
2.3 Macdonald polynomials
2.4 Dunkl operators
3. Nosymmetric Macdnomald polynomials
3.1 Cherednik's scalar product
3.2 Nonsymmetric Macdonald polynomials
3.3 Cherednik-Dunkl operators
4. Double Affine Hecke Algebras
4.1 Affine root systems
4.2 Duality and Fourier transforms
4.3 Applications to combinatorics and Macdonald constant term conjecture
4.4 Double Affine Hecke Algebras and Quantum Character varieties
1. Symmetric functions
1.1 The ring of symmetric functions
1.2 Elementary symmetric functions, Complete Symmetric functions, and Power sums
1.4 Schur functions
1.3 Scalar product and orthogonal polynomials
2. Macdonald polynomials
2.1 Root systems and Weyl Characters
2.2 Scalar product
2.3 Macdonald polynomials
2.4 Dunkl operators
3. Nosymmetric Macdnomald polynomials
3.1 Cherednik's scalar product
3.2 Nonsymmetric Macdonald polynomials
3.3 Cherednik-Dunkl operators
4. Double Affine Hecke Algebras
4.1 Affine root systems
4.2 Duality and Fourier transforms
4.3 Applications to combinatorics and Macdonald constant term conjecture
4.4 Double Affine Hecke Algebras and Quantum Character varieties
Reference
1) I.G.Macdonald Symmetric Functions and Orthogonal Polynomials. Vol. 12. American Mathematical Soc., 1998
2) I.Cherednik Double Affine Hecke Algebras Vol. 319. Cambridge University Press, 2005.
2) I.Cherednik Double Affine Hecke Algebras Vol. 319. Cambridge University Press, 2005.
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
I studied Applied Mathematics and Physics at the Moscow Institute of Physics and Technology, where I earned both my B.Sc. and M.Sc. degrees. In 2013, I joined the graduate program in Mathematics at Rutgers, The State University of New Jersey, and completed my Ph.D. in 2018 under the guidance of Prof. V. Retakh. After earning my doctorate, I held postdoctoral positions at the University of California Berkeley, the Centre de Recherches Mathématiques in Montreal, and the University of Toronto. In July 2024, I became an Associate Professor at the Beijing Institute of Mathematical Sciences and Applications (BIMSA)