The discovery of the integrable behavior of certain classical and quantum systems with infinite degrees of freedom is one of the most exciting developments of mathematics during the second half of the 20th century. Geometry plays a central role in the mathematical foundations of the theory of integrability, and a number of hard geometrical problems have been resolved using methods of integrable systems. The connections between this theory and differential geometry, especially Poisson geometry, and algebraic geometry, enriched these branches of classical mathematics with new techniques and formulations of new problems; they were also instrumental in developing and refining tools applicable to the theory of integrable systems.

Poisson structures, in finite and infinite dimensions, generalize both symplectic structures and the duals of Lie algebras. They constitute a very general framework for the study of Hamiltonian systems. In particular the geometry and algebra of bihamiltonian structures, and their relation to Lax pairs and classical r-matrices can be applied to the study of completely integrable systems. Recent advances concern Lie algebroid theory and quantum bihamiltonian systems (bihamiltonian structures in non-commutative geometry).

The importance of the connections between algebraic geometry and the theory of integrable systems became clear at a very early stage of the creation of that theory. Indeed, the discovery of the finite-gap integration method transformed the classical geometry of Riemann surfaces and theta-functions into a truly applicable tool for the theory of nonlinear waves. Applications of these techniques to the asymptotic analysis of rapidly oscillating solutions of Hamiltonian PDEs will require further development of the theory of integrable hierarchies, which first appeared in the theory of Gromov-Witten invariants, associated with semisimple Frobenius manifolds. The connections between the theory of integrable systems and the algebraic geometry of multidimensional complex varieties have so far been studied almost exclusively by pure mathematicians, and we believe that this direction of research deserves to be promoted by being included in the programme of this Conference and thus brought to the attention of a wider circle of researchers.

Various methods of differential geometry have been used in the theory of integrable systems since the very early stages of its development. To give just few a examples we can mention Darboux-Backlund transformations, vector bundles on infinite dimensional Grassmannians, relations between the theory of infinite-dimensional Hamiltonian systems of hydrodynamic type and the finite-dimensional Riemannian geometry of the underlying manifolds, geometry of isomonodromic deformations, etc. Recent discoveries of discrete analogues of these differential geometric methods seem very promising not only from the point of view of geometry but also for numerous applications.

Moreover, the search for integrable systems has a long history. Kovalevskaya's method was used to search for the cases where the solutions of the equation of motion, analytically continued to the complex plane, have no singularity other than poles. Painleve did the same for second-order nonlinear ordinary differential equations, slightly relaxing the conditions on the possible singularities of solutions. The celebrated six Painleve equations were derived. These equations are having applications in numerous domain and possess many interesting properties. Analogues of these equations in the discrete and ultra-discrete setup have now been discovered, and are closely related to many branches of mathematics and physics. Recent research shows that Nevanlinna theory in complex analysis, Diophantine approximation in number theory, growth properties in the algebraic geometry of rational surfaces, the bilinear method, various Lie-algebraic methods, Darboux transformations and birational maps can all be used effectively to study discrete equations. The underlying symmetries also reveal significant results.

This conference will focus on a number of geometric and applied aspects of the theory of integrable systems which are likely to become of major importance for subsequent developments. In addition to the lectures by invited speakers (see the list of acceptances), four introductory mini-courses are planned on the following topics: Discrete and continuous integrable systems, Integrable systems and Poisson geometry, Multi-dimensional algebraic geometry of integrable systems, Numerical analysis for integrable systems.

Scientific Organizers: Boris Dubrovin (SISSA, Italy), Basil Grammaticos (Paris VII - XI, France), Yvette Kosmann-Schwarzbach (Ecole Polytechnique, France), M. Lakshmanan (Bharathidasan University, India), J. Satsuma (Aoyama Gakuin University, Japan) and K.M. Tamizhmani (Pondicherry University, India).

Local Organizing Committee: T.Duraivel (Pondicherry University), V. Indumathi (Pondicherry University), K.Joseph Kennedy (Pondicherry University), M.Lakshmanan (Bharathidasan University), H.P.Patil (Pondicherry University), K.Porsezian (Pondicherry University), A.M.S.Ramasamy(Pondicherry University), M.Sitaramayya (Hyderabad University), K.M.Tamizhmani (Pondicherry University) and Thamizharasi Tamizhmani (K.M.Centre for P.G. Studies, Puducherry).

Organized by: Department of Mathematics, Pondicherry University

Local Secretary: K.M.Tamizhmani (Pondicherry University).