Although the modern theory of integrable systems was initiated by the discovery of integrable PDEs, the problem of their classification remains open. The perturbative approach to the classification problem suggested by B.Dubrovin and Y.Zhang will be applied to completing the classification of the systems of spatially one-dimensional integrable evolutionary PDEs that admit hydrodynamic limit and satisfy certain additional assumptions inspired by the ideas, due to E.Witten and R.Dijkgraaf, of 2D topological field theory. The relationships between the moduli space of such systems and the moduli of Frobenius manifolds will be further studied and exploited.
A transitive action of certain class of bosonic Bogoliubov transformations on the moduli space will be constructed and studied. They generalize the transformations obtained by A.Givental and L.Chekhov in the setting of the Gromov - Witten invariants of projective spaces and in the theory of the Kontsevich - Penner models.
The bihamiltonian structure of the integrable PDEs constructed in this way will be considered in the setting of the theory of deformations of classical W-algebras. Virasoro symmetries of the integrable PDEs will play an important role both in their construction as well as in the classification. One of the goals of this part of the project is to study the full symmetry algebra of these PDEs in order to produce and study deformations of quantum W-algebras.
An alternative construction of the families of integrable PDEs will also be developed in terms of a suitable class of infinite-dimensional Lie algebras. It is understood that an analogue of the Lax representation will be found for the class of integrable PDEs in question.
The results of the symplectic field theory of Ya.Eliashberg, A.Givental and H.Hofer will be applied for further investigating the connections between the theory of Gromov - Witten invariants and classical and quantum integrable structures. A deeper insight into the theory of Gromov - Witten invariants and their generalizations is also envisaged. The more general problem of normal forms of integrable PDEs in a neigborhood of a finite-dimensional invariant manifold fibered into Liouville tori remains open. A natural geometric structure induced on such an invariant manifold by a Hamiltonian structure of the PDE consists of a symplectic foliation equipped with a flat metric on the base of such a foliation. A classification programme of such structures will be developed. The results of this study will be applied to describing normal forms of more general systems of 1+1 evolutionary PDEs not admitting a hydrodynamic limit. The theory of normal forms will also be applied to the geometric analysis of the higher order approximations to the theory of weakly dispersive waves.
An extension of the above perturbative approach to the classification problem onto the class of spatially multidimensional PDEs will require a significant generalization of the technique. The plan is to extend the ideas of the theory of normal forms of integrable PDEs to the theory of nonlocal integrable systems and to apply this new technique to the classification of systems of the multicomponent KP-type equations. The classification of integrable systems arising as symmetry reductions of the ASD Yang-Mills equations or of Anti-Self-Dual conformal structures in four dimensions and their hierarchies will also be addressed. The latter has a direct relationship to dispersionless limits of integrable systems (dKP and infinite Toda are two key examples). This is also related to the problem of classification of those integrable systems that admit a twistor correspondence.
Applications of ideas from the integrability theory and quantum cohomology to singularity theory will be aimed at studying how much of the richer structure of the theory of integrable PDEs and of the theory of Gromov - Witten invariants can be constructed along these lines for the Frobenius manifolds in singularity theory. What are their geometric meanings and what do they say about singularities?
The symplectic loop space formalism and the evolving formulae of Givental will be applied in singularity theory setting as one of the approaches to the above problem.
A specific point of view at the highly nontrivial interactions between integrable systems, Gromov - Witten invariants and the singularity theory suggested by the latter is the relation between Frobenius manifolds and (mixed) Hodge structures. This is related to the "tt*" geometry introduced by S.Cecotti and C.Vafa and developed by B. Dubrovin and C. Hertling. The role of the tt* geometry in the theory of integrable PDEs as well as in the theory of Gromov - Witten invariants will be studied. More generally, the interplay between integrable systems and generalizations of variations of Hodge structures attached to families of non-commutative objects, such as A-infinity algebras or categories continuing the works of S.Barannikov, will be investigated. For the particular case of Calabi - Yau threefold, the plan is to study differential equations related to counting higher genus curves and their connections with mirror symmetry. We also plan to develop a general approach to the problem of relationships between integrable systems with deformation theory of algebraic varieties and singularities. It will be based on the construction, initiated by Barannikov, of integrable systems starting from some algebraic data such as certain extensions of Gerstenhaber or Batalin - Vilkovisky algebras. Integrable systems and deformation quantization of algebraic varieties will also be under investigation. The research activity in quantization of integrable geometry will be focused mainly on two classes of problems. The first one concerns the study of the quantization of many particles and/or spin systems and their relationships to the quantum deformations of Riemann bilinear identities as they appear in the work of F.Smirnov on the Sine-Gordon form factors. The research will concern the study of deformed quantum Calogero-Moser (CM) problems, Lie superalgebras and their relations with WDVV equations.
It will also comprise the study of the topological field theories associated to (quasi) Lie bialgebroids (in particular, twisted Poisson structures) as well as the quantization of Poisson diffeomorphisms and of more general Poisson maps (with a possible application to the quantization of Poisson-Lie groups from the point of view of Kontsevich's formula).
An alternative approach to the classification of 2D integrable systems will be developed as well, based on the ideas of discretization and the consistency approach pushed forward in the work by A.Bobenko and Yu.Suris. The development of (discrete) differential geometry will be intimately related with the development of (discrete) integrable systems. The goal is to exploit the new promising method based on the notion of consistency along the following directions: i)To utilize the consistency approach to three-dimensional discrete equations and to address the question whether there are discrete integrable equations in more than three dimensions. ii)To understand the geometry of the integrability of discrete elliptic (as opposed to hyperbolic) equations. It can be anticipated that additional structures of these geometric systems will be uncovered, such as variational (Lagrangian) formulation and the corresponding symplectic structures. In the study of the geometry of discrete integrable systems and surfaces, other specific problems will be considered, such as the Discrete Lawson Surface Equations. The research will also be focused on complex geometric aspects of integrable systems, particularly those that emerge from the twistor geometry.