Mathematics

The goal of the course is an excursion on algebraic aspects of integrable systems. The topics are

1. KP equation and Sato-Segal-Wilson Grassmannian.
   Tau function and Hirota bilinear relation.
   Plücker coordinates.
   Schur function expansion.
2. Hermitian random matrix models.
   Orthogonal polynomials, partition function and Schur expansion.
   Combinatorial interpretation of correlator expansion of classical matrix.
   Integrals: ribbon graphs and Hurwitz numbers.

References:

• Tau functions and their applications, Cambridge University press 2021, F. Balogh, J. Harnad.

• Free fermions and tau-functions, Alexander Alexandrov, Anton Zabrodin.

• Combinatorics and random matrix theory, Baik, Jinho; Deift, Percy; Suidan, Toufic, Grad. Stud. Math., 172, American Mathematical Society, Providence, RI, 2016.