# Integrable systems, Frobenius manifolds and nonlinear waves

## Research Topics

**Geometry of Integrable Systems**- Frobenius manifolds,Topological Field Theory and integrable hierarchies.
- Frobenius manifolds classification problems and genus expansion. Applications to Gromov - Witten invariants of higher genera.
- Reflection groups and integrable hierarchies.
- Hamiltonian structures of evolutionary PDEs.
- Geometry of bihamiltonian manifolds.
- Algebraic geometry and integrable systems.

**Integrability phenomena in nonlinear differential equations**- Theory and applications of Kadomtsev - Petviashvili (KP) equation. Theta-functions and water waves.
- Whitham equations: variational principles, algebraic geometry, analytic theory.
- Oscillations in weakly dispersive systems.
- Geometric and Analytic aspects of equations of the Painleve' type.

## Overview

Integrable evolutionary differential equations were discovered at
the end of the 60's in the study of dispersive waves. It has been realized afterwards
that integrable systems of analytical mechanics, and also classical Painleve'
equations can be embedded into the general theory of integrable
systems as particular reductions of integrable PDEs, and their
study received a brand new impulse by the use of methods introduced
in the analysis of PDEs(e.g., the Lax method).

Mathematical
progresses of the theory of integrable systems was related to
Riemann - Hilbert boundary value problems, to algebraic and
differential geometry, to infinite dimensional Lie algebras, and to
quantum groups. As an example, we recall the striking solution of
the classical Riemann - Schottky problem via the proof (due to
Dubrovin, Arbarello - De Concini and Shiota) of Novikov's conjecture about
the solutions of soliton equations.

An important general approach to integrable systems was introduced
by F. Magri. It is based on the geometry of bihamiltonian
systems. After the discovery in the early 90s of the fundamental
role of the Korteweg-de Vries equation (KdV) in the matrix and
topological models of two-dimensional gravity, and, in particular,
after the solution, by E. Witten and M. Kontsevich, of the old
problem of the topology of the moduli spaces of algebraic curves in
terms of KdV hierarchy, the theory of integrable systems has become
one of the unifying principles in the mathematics of the beginning
of the 21st century. Further developments of two-dimensional
topological field theory supplied the theory of integrable systems
with a new tool based on the geometric and analytic theory of
Frobenius manifolds. They were introduced by B. Dubrovin as the
geometrical setting for the so-called WDVV equations of
associativity, but, subsequently, have become a common paradigma
for rather distant branches of mathematics, such as Gromov - Witten
invariants of symplectic manifolds, singularity theory,
isomonodromy deformations, and the theory of reflection groups and
their extensions.

Actually, various fundamental problems of
these branches of mathematics, such as the problem of Gromov -
Witten invariants at higher genera and the problem of integrability
in four dimensional supersymmetric field theories, require a new
approach to the problem of classification of integrable
systems. Our approach to this problem is naturally based on the
theory of Frobenius manifolds. This theory suggests to classify
integrable bihamiltonian hierarchies of evolutionary PDEs which
possess a tau-function and Virasoro invariance. The KdV hierarchy
and the Drinfeld-Sokolov hierarchies corresponding to simply laced
simple Lie algebras are examples of these hierarchies, but this
list can be strongly extended. A closely related topic is the
study, by means of differential geometrical and algebro geometrical
techniques, of finite-dimensional integrable Hamiltonian systems
(especially those arising as suitable reductions of
infinite-dimensional ones). Among the many open problems of this
theory we can mention the properties of the action-angle variables
in the framework of special Kaehler geometry and the theory of
Frobenius manifolds. The solution of these problems is also
important for applications of integrable systems to the Quantum
Theory.

## COFIN and PRIN

- Geometric methods in the theory of nonlinear waves and their applications - 2006
- Geometric methods in the theory of nonlinear waves and their applications - 2004

## European Networks

- Marie Curie RTN "European Network in Geometry, Mathematical Physics and
Applications"
**(ENIGMA 2004-2008)**(Project Coordinator G. Falqui) - ESF Scientific Programme "Methods of Integrable Systems, Geometry, Applied Mathematics"
**(MISGAM 2004-2009)**(Proponents B. Dubrovin and P. Van Moerbeke) - ERC Advanced Grant FroM-PDE "Frobenius Manifolds and Hamiltonian Partial Differential Equations"
**(Advanced Investigator Grant Scheme 2009-2013)**(Principal investigator B. Dubrovin)

## Workshops

(partially funded by MISGAM and ENIGMA)

- Workshop "Geometry and Integrable Systems", Berlin, 3-7 June 2005
- Workshop on "Random Matrices and Other Random Objects", FIM, ETH-Zurich, 17-21 May 2005
- Conference on "Riemann-Hilbert Problems Integrability and Asymptotics", SISSA, Trieste, 20-25 September 2005
- "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
- "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)
- Conference "Random Matrices", Centre International de Rencontres Mathématiques (CIRM), Luminy, France, October 30 - November 3 2006
- Workshop on "Higher Structures in Geometry and Physics", IHP, Paris, January 15-19, 2007
- Conference on "Tropical Geometry and Applications", Loughborough University, Leicestershire (UK), April 20-23, 2007
- Summer School on "Aspects of Membrane Dynamics", KTH, Stockholm, June 11-23, 2007
- "ENIGMA Conference on Mathematical Physics", KTH, Stockholm, June 25-28, 2007
- Conference on "Algebraic aspects of integrable systems", Islay, Scotland, July 1-7, 2007
- Conference on "Random and Integrable models in Mathematics and Physics", Solvay Institute, Brussels, 11-15 September 2007
- Frobenius Structures and Singularity Theory, Poznan, 25-29 August 2008
- Conference on Integrable Systems, Geometry, Matrix Models and Applications, Trieste, 14-18 October 2008