The principal scientific aim of the Network is to create a fertile research and training environement for scientists working in the fields of Geometry, Mathematical Physics, and Applications to Physics. It is expected that the broad interdisciplinary basis and intertwining of methods of Geometry and Mathematical Physics lead to solutions of hard geometrical problems and to formulations of new mathematical models of physical phenomena. The Consortium will attract young geometers to the exciting realm of mathematical physics and applied mathematics and will equip young mathematical physicists with the powerful tools based on the techniques of modern geometry and of the theory of random matrices.

The research programme is structured along three main Themes.

A. | Geometry of Integrable Systems |

B. | Random Matrices |

C. | Applications: Integrability and Random matrices in mathematical models of physical phenomena |

The primary research objectives are the following:

- Further understanding of the Mathematics of Integrability by completing the classification of the systems of spatially one-dimensional integrable evolutionary Partial Differential Equations (PDEs) that admit hydrodynamic limit.
- Establishing of the relationships between the moduli space of such systems of PDEs, the moduli of Frobenius manifolds, and the theory of Gromow-Witten invariants.
- Achievement of significant progress in the theory of discrete differential geometry and discrete integrable systems.
- Solution of the classification problem of systems arising as symmetry reductions of the ASD Yang-Mills equations, in relation with twistor geometry.
- Rigorous analysis of N body systems and of the algebraic structure of topological models of quantum gravity and quantum field theory
- Analytical and geometrical study of matrix models beyond the realm of formal power series.
- Understanding of the general universality classes appearing in critical regimes of Matrix models as well as the computation of significant properties.
- Understanding of the analytical aspects of asymptotic integrability of systems depending on a parameter, especially in the realm of the theory of water waves equations.
- Application of methods of Algebraic geometry to study structural and modulation properties of families of physically relevant PDEs

The Research and Training activities of ENIGMA will benefit from the connection with the ESF project

- A.1 - Integrable Evolutionary PDEs and Geometry of Hamiltonian Systems.
- A.2- Geometry (Algebraic, Complex and Differential) and Integrability.
- A.3- Quantization of integrable geometry
- B.1- Random Matrices, orthogonal polynomials and Integrable Systems
- B.2- Random Matrices, topology and combinatorics
- C.1- Weakly dispersive waves and applications
- C.2- Applied Algebraic Geometry

The expected scientific results in the first phase of the project's activities are:

In task A.1, the bihamiltonian and the r-matrix structures of the 2D Toda hierarchy will be determined. The description of hyperelliptic reductions of Benney equations for arbitrary genus will be given. The classification of multidimensional integrable systems of hydrodynamic type will be refined.

In task A.2, rigorous results in the classification of integrable discrete 3D systems are expected. The classification of connections on the complex plane with vanishing holonomy along lines will be achieved. The characterization of the links between twistors and Frobenius manifolds will be given. Furthermore, general results on the geometry of Hodge structures and their orbits will be obtained.

In task A.3, the properties deformation quantization of classical algebras of functions (possibly, as A algebras) will be determined. Quantum canonical transformations between hierarchies of integrable PDEs and `direct sums' of KdV hierarchies will be constructed. New rigorous solutions to the quantum Separation of Variable problems and new algorithms to solve quantum systems of generalized Calogero type will be given.

In task B.1, extensions of the Sato - Miwa -Jimbo isomonodromic tau function to the differential systems satisfied by biorthogonal polynomials will be studied. The relationship of universality classes in random matrix theory with integrable PDEs and Virasoro constraints will be established and analytic solutions of dispersionless 2D Toda hierarchies will be given as integrals over normal matrices.

In task B.2, Fay identities satisfied by matrix models with generic Coulomb interaction will be determined, and the new integrable lattices associated with such identities will be studied. In task C.1, the theory of dispersionless PDEs will be applied to the problem of interface dynamics. In task C.2 the link between singularity and integrability will be clarified. In the context of the integrability of the Camassa-Holm equation, a detailed analysis of the motion in Jacobi varieties will be performed.

By the second phase it is foreseeable that new directions of research in this interdisciplinary field might emerge in such an elapse of time. With this proviso in mind, the expected results of Phase II are the following:

Task A.1: Extension of the classification theorems of bihamiltonian hierarchies of hydrodynamic type to more general classes of integrable bihamiltonian hierarchies of PDEs; Determination of the relations between bi-spectrality, tau-functions, and W-algebras

Task A.2: Application of properties of invariant submanifolds of integrable PDEs to the moduli spaces of vector bundles over Riemann Surfaces. Descripton of the ordinary matrix model, the double scaling limit of matrix models, and Kontsevich-like matrix model in the framework of the geometry of extended object (branes) in specific local Calabi-Yau three-folds; Determination of the geometrical properties of discrete elliptic equations and of the additional structures of these systems .

Task A.3: Computation of the cohomology of quantum affine Jacobi varieties appearing in the Gelfand - Dickey hierarchies. Study of the field theories associated with twisted Poisson structures. Study of the deformation of quantum N-body problems in terms of the algebraic structures.

Task B.1: Study of the large N expansion of matrix model partition functions, in particular the two-matrix model and the chain of matrices, in terms of algebraic geometry.

Task B.2 Characterization of the universality classes of the large N expansion of general matrix models, as well as of specific classes of solvable matrix model in connection with knot theory.

Task C.1: Study of measure-valued solutions for specific classes of Camassa-Holm like evolutionary PDEs. Application of Riemann-Hilbert problem methods to the asymptotics of solutions of integrable evolution equations as well as statistical quantities arising in the theory of random matrices. Analysis of statistical properties of one dimensional growth processes in the framework of random matrix theory. Task C.2: Construction by algebro geometric methods of the reductions of dispersionless integrable hierarchies.

ENIGMA deals with questions in mathematics and applications to theoretical physics; no ethical issues are involved in this project.