This is an introductory course on the theory of Macdonald orthogonal polynomials, double affine Hecke algebras (DAHA) and their role in modern mathematics and physics. We will start with the basics of Macdonald theory, introducing the ring of symmetric functions, as well as Schur functions and Macdonald polynomials. The key point will be the connection between the Macdonald polynomials and the Cherednik-Dunkl operators, which naturally arise in the study of quantum integrable many-body systems, namely, the trigonometric Ruijsenaars system. We consider nonsymmetric version of Macdonald polynomials and introduce the double affine Hecke algebra (DAHA). We will also study the rational version, the so-called rational Cherednik algebra, and discuss applications and connections with orthogonal polynomials, integrable systems, and combinatorics. Similar constructions exist at the elliptic level; these will be discussed towards the end of the course.

Lecturer
Matushko Mariya Georgievna

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, agreement no. 075-15-2025-303).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center