## Frobenius manifolds: analytic theory

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.

## Excursion in Integrability

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

- KP equation and Sato-Segal-Wilson Grassmannian.

Tau function and Hirota bilinear relation.

Plücker coordinates.

Schur function expansion. - 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

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)

## Past Activities of the Integrable Systems, Frobenius Manifolds and Nonlineare Waves group

(partially funded by MISGAM and ENIGMA)

- International Conference Geometrical Methods in Mathematical Physics , 12-17 December 2011, Moscow, Russia
- Integrable systems in pure and Applied Mathematics, a conference in honour of Boris Dubrovin's 60 birthday, 8-12 June 2010, Sardinia, Italy
- Conference on Integrable Systems, Geometry, Matrix Models and Applications, Trieste, 14-18 October 2008
- Frobenius Structures and Singularity Theory, Poznan, 25-29 August 2008
- Conference on "Random and Integrable models in Mathematics and Physics", Solvay Institute, Brussels, 11-15 September 2007
- Conference on "Algebraic aspects of integrable systems", Islay, Scotland, July 1-7, 2007
- "ENIGMA Conference on Mathematical Physics", KTH, Stockholm, June 25-28, 2007
- Summer School on "Aspects of Membrane Dynamics", KTH, Stockholm, June 11-23, 2007
- Conference on "Tropical Geometry and Applications", Loughborough University, Leicestershire (UK), April 20-23, 2007
- Workshop on "Higher Structures in Geometry and Physics", IHP, Paris, January 15-19, 2007
- Conference "Random Matrices", Centre International de Rencontres Mathématiques (CIRM), Luminy, France, October 30 - November 3 2006
- "Integrable Systems in Applied Mathematics", Colmenarejo (Madrid) 4-8 September 2006. A satellite Conference of the International Congress of Mathematicians (ICM 2006, Madrid 22-30 August 2006)
- "Affine Hecke algebras, the Langlands Program, Conformal Field Theory and Matrix Models", Centre International de Rencontres Mathématiques (CIRM), Luminy, France, June 19 - July 14 2006
- Conference on "Riemann-Hilbert Problems Integrability and Asymptotics", SISSA, Trieste, 20-25 September 2005
- Workshop on "Random Matrices and Other Random Objects", FIM, ETH-Zurich, 17-21 May 2005
- Workshop "Geometry and Integrable Systems", Berlin, 3-7 June 2005