Project Overview


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:

The Research and Training activities of ENIGMA will benefit from the connection with the ESF project MISGAM (Methods of Integrable Systems in Geometry and Applied Mathematics), launched in July 2004 with a duration of five years.

Approch and Methodology

The research problems will be mainly addressed by the approach of pure mathematics, based on the careful analysis of intrinsic connections between the mathematical constructions under consideration, on the precise formulation of the arising conjectures and on proving them as rigorous theorems. The more remote are the mathematical constructions, the deeper and the more unexpected are their interconnections, the higher is the chance of obtaining really new theoretical breakthroughs. The connections between quantum cohomology and integrable PDEs, variations of Hodge structures and Gromov - Witten invariants, Frobenius manifolds and dispersive nonlinear waves are examples of these remote connections to be studied in the framework of the project. Another feature of the research method, intrinsically connected to its interdisciplinary nature, is to use intuition based on the experience of working with various physical models. Mathematical formalization of heuristic ideas based on intuition will be used for unravelling connections between Geometry and Mathematical Physics. This methodological approach and, namely, its interdisciplinarity is the main reason why the connections with fellow researchers, and, especially, the exchange of ideas between geometers and mathematical physicists play a crucial role in the workplan. General tools of the theory of integrable systems such as Riemann - Hilbert problem, bihamiltonian geometry, Lax representations and infinite-dimensional Lie algebras, tau functions and Baker-Akhiezer functions, are common general patterns of the researches on the topics concerning the project. They will be used in quantum cohomology and singularities, in random matrices as well as of the applications to low-dimensional topology, to combinatorics and probability, to statistical models and to nonlinear waves. In the study of certain geometric objects such as moduli spaces of algebraic curves and their mappings, as well as vector and principal bundles, more specific techniques of algebraic geometry and singularity theory will be used. Some analytic tools, like asymptotic expansions, KAM theorem etc. will be used to rigorously justify the applicability of the formulae obtained as a result of geometric researches to real physical models. The comparison of various mathematical models will be, in the appropriate cases, performed by using numerical analysis and symbolic computation tools, such as Mathematica, Maple and Matlab.

Work Plan and Expected Results

The three main Research themes of the research project can be subdivided into more specific Tasks. Namely:
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.