Mathematics

The aim of the course is to introduce the audience to the analytic theory of Dubrovin-Frobenius manifolds.

The theory of Frobenius manifolds was constructed by B. Dubrovin to formulate in geometrical terms the WDVV equations of associativity of 2D topological field theories.

It has links to many branches of mathematics, like singularity theory and reflection groups, algebraic and enumerative geometry, quantum cohomology, theory of isomonodromic deformations, boundary value problems and Painlevé equations, integrable systems and non-linear waves.

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:

Integrable systems are special  systems which can be solved exactly in some sense. They arise in a variety of settings, ranging from Hamiltonian systems, nonlinear wave equations   and probability. This  course covers the origins of the subject as well as modern topics in  nonlinear waves  and  integrable probability.

1.  The Korteweg de Vries equation (KdV)