MINISTERO DELL'ISTRUZIONE, DELL'UNIVERSITÀ E DELLA RICERCA

CO.FIN.
GEOMETRIC METHODS IN THE THEORY OF NONLINEAR WAVES
AND THEIR APPLICATIONS.

research program 2006-2008

Boris DUBROVIN obtained his master degree (in 1972) and his Ph. D. (in 1976) from the Department of Mechanics and Mathematics of the Moscow State University. Till 1993 he was a full professor at the same University. Starting from the end of 1993 he is a full professor at SISSA where, since 2001, he is head of the Mathematical Physics Sector.
He is the author of 5 books and of some 70 papers published in renowned mathematical and physical journals.

Recent Visiting Positions

1. MSRI (Berkeley) Visiting Research Professorship, January-May 1999 2. Cambridge University, Rothschild fellowship, September - October 2000
3. ETH (Zurich), October 2000 - February 2001
4. IAS (Princeton), September 2001 - April 2002, member
5. Stanford University, January - February 2003

Honors

Prize of the Moscow Mathematical Society, 1976 (jointly with A.Its and I.Krichever).

Main Invited Talks and Lectures

1. International Congress of Mathematical Physicists, Swansea, 1988 (invited 45 minutes talk)
2. 2nd European Congress of Mathematicians, Budapest, 1996 (plenary talk)
3. International Congress of Mathematicians, Berlin, 1998 (invited 45 minutes talk)
4. Plenary speakers of the International Congress of Mathematical Physics, Rio de Janeiro, 2006.

Participation in Editorial Boards

1. Functional Analysis and its Applications.
2. Differential Geometry and its Applications.
3. Journal of High Energy Physics (electronic)
4. Mathematical Physics, Analysis and Geometry
5. Journal of Geometry and Physics


Other Activities

1. Secretary of Moscow Mathematical Society in 1988 - 92.
2. National coordinator of the Cofin1999 and Cofin2001 projects "Geometry of Integrable Systems" (1999-2003) and of the Cofin2004 project "Geometric methods in the theory of nonlinear waves and their applications" (2004-2006).
3. Co-ordinator of the European Science Foundation project Methods of integrable systems, geometry, applied mathematics (2004-2009).
4. Member of the Steering Committee of the Marie Curie FP6 Research Training Network ENIGMA (2005-2008).
5. Member of the Programme Committee of the 4th European Congress of Mathematicians, Stockholm, June 2004.

PhD students
M. Bertola, G. Carlet, V. Dragovic, T. Grava, D. Guzzetti, P. Lorenzoni, M. Mazzocco, A. Stefanov, M. Ugaglia, V. Vereshchagin, A. Zhivkov.

Recent Conferences and Schools Organized

1. The series of Workshops and Schools Algebraic Geometry and Physics; the first two workshops took place in September 96 and October 2000 at Trieste and September 97 at Medina del Campo, the school took place at Luminy. From 2001 onwards, these workshops/schools take place on a yearly basis, and are hosted by different Institutions in Italy and abroad.
2. CIME Summer School Quantum Cohomology, June - July 97, Cetraro, Italy.
3. Workshop Reflection Groups and Their Applications, Trieste, January, 1998.
4. School/Workshop Differential Geometry, Trieste, April 1999.
5. Workshop Whitham Equations and Their Applications in Mathematics and Physics, Trieste, November-December 2000.
6. Workshop Classification Problems in the Theory of Integrable Systems, Trieste, October 2002.
7. Workshop Riemann - Hilbert problems, integrability and asymptotics, Trieste, September 2005.

Our research aims are mainly focused on the geometry of nonlinear wave equations, considered as Hamiltonian systems with an infinite number of degrees of freedom. Such systems are described by special - viz. integrable - PDEs of evolutionary type, whose prototype is the celebrated Korteweg - de Vries (KdV) equation, and their perturbations.
The study of such equations had a deep impact both in Physics and in Mathematics. Significant examples can be found in different domains, such as singular limits of weakly dispersive waves, analytic models of water waves, physics of optical fibers, strings and topological field theories. The study of the connections between these physical applications and developments of the mathematical tools opened new horizons also in certain branches of pure mathematics, such as the discovery of mirror symmetry, the theory of Gromov - Witten (GW) invariants and its applications to topology of manifolds, combinatorics of graphs and matrix models, monodromy preserving deformations, quantum groups etc.
Within this research area we focus our project on Classification problems, Solution methods, as well as Applications. A brief description of our aims can be given as follows.
1. Classification problems.
The classification of integrable PDEs in one spatial dimension that admit hydrodynamic limit, Virasoro invariance and a tau function has been achieved in previous works [DZ3]. It has been enlarged in [DLZ] to include all the integrable PDEs admitting a bihamiltonian hydrodynamic limit, but the problem of proving the existence of such bihamiltonian structures for an arbitrary choice of the functional parameters remains open. For instance, the Camassa-Holm equation is included in this second class of hierarchies. We aim to connect the family of PDEs classified in [DZ3] and the family classified in [DLZ] via reciprocal transformations, which act nonlinearly also on the independent variables. At a preliminary level, the construction of new examples of "CH-type" hierarchies, as well as the study of their properties, will be needed.
We also plan to consider formal perturbations of diagonal hyperbolic systems possessing a complete family of commuting flows (the so-called semi-hamiltonian systems). The problem of classification of such systems is still open and we plan to use techniques of the theory of separable systems to study them. One of the aim is to frame Whitham equations within particular deformation schemes of associative algebras.

Due to nonlinearity, the weak dispersion expansions of solutions to these PDEs have only finite range of applicability. More complicated Whitham-type oscillatory asymptotics are to be used after passing through the point of gradient catastrophe of the hydrodynamic approximation.
One has to match solutions with different numbers of oscillatory phases in order to obtain global uniform asymptotics. The matching problem demands a more general approach to the classification.
We plan to use techniques and ideas of the theory of Painleve' equations as well as of the theory of isomonodromic deformations to solve this problem.


2. Solution methods.
From the Hamiltonian point of view, one of the main goals is to achieve a good formulation of the Hamilton-Jacobi (HJ) theory for infinite dimensional systems. Developments in the so-called symplectic field theory [EGH] opened new research perspectives in the HJ theory for nonlinear PDEs, and suggested new links between GW invariants and integrable hydrodynamical equations. These new connections will be used for recursive construction of integrable PDEs.

To develop such a theory, we plan to rely on the connection between bihamiltonian geometry and HJ equations, recently discovered in the context of finite dimensional systems. In this direction, we aim to study bilagrangian distributions defined by Lenard chains, and give their interpretation in terms of solutions to the HJ equations.We expect that the ideas and examples from physical theories of dualities of Seiberg-Witten type, as well as ideas coming from algebraic geometry, will be helpful in this respect.

Our results will be also applied to important finite dimensional systems obtained by reductions of soliton equations, as well as to Heisenberg and Gaudin magnets. In particular, models of Gaudin type will be investigated, dealing with the problems of the construction of integrable time-discretizations, following the approach proposed by Bobenko and Suris and the search for new integrable systems related to universal graded contractions of N-site Gaudin models.

The study of the HJ equations by means of 'classical' tools as well as algebro-geometric methods will be pursued within this research topic.
We will focus both on the so-called "separability structures" associated with the sound differential geometric setting of the HJ theory, and on the role played by special Kaehler structures within the theory of algebraically completely integrable systems.

We plan to devise an umbral calculus suited to study solutions of nonlinear equations through a "nonstandard" perturbation scheme. Rather than perturbing solutions to the linearized equations, we will consider perturbations of a nonlinear system in different variables, namely a discrete nonlinear system which is the umbral correspondent to the differential nonlinear equation.

3. Applications
The first aim concerns the use of tools from the geometric theory of Frobenius manifolds andfrom the analytic theory of the RH problem to study the problem of Universality in the critical behaviour of solutions to systems of dispersive conservative PDEs.
One of our tasks is to prove the universality property conjectured in [D5] about the generic solution to a Hamiltonian perturbation of an hyperbolic PDE near the point of gradient catastrophe of the unperturbed equation. This behavior is described by a particular solution of an ODE of fourth order that belongs to the family of Painleve' equations. Numerical investigations of this conjecture will be performed as well. In this framework we plan to rigorously study the small dispersion limit of the KdV equation at the time of gradient catastrophe. We plan to extend the conjecture formulated for the one component case in [D5], to the multi-component case. Double scaling expansions in matrix models suggest that the conjecture formulated in [D5] remains true for the Toda lattice equations.

After the point of gradient catastrophe the solution of the PDE becomes oscillatory. To determine the formal asympototic solution in the neighborhood of a finite-dimensional invariant manifold fibered into Liouville tori it is necessary to develop a nonlinear analogue of the WKB method. A Hamiltonian structure of the PDE induces a symplectic foliation on its invariant submanifolds, equipped with a flat metric on the base of such a foliation. We plan to develop a classification programme of such structures and to apply it to describing normal forms of more general systems of 1+1 evolutionary PDEs not admitting a hydrodynamic limit.
Next we plan to study the asymptotic solution near singular points (the focal points in the linear WKB theory) and the corresponding matching of the asymptotic solutions in the oscillatory and non-oscillatory regimes, and the role played by Painleve' transcendents at the point of gradient catastrophe and beyond.

Finally, a specific subject concerns the relations between the Weil-Petersson measures on the moduli spaces of Riemann surfaces with marked points, Frobenius structures, and the triangulated surfaces used in the simplicial approach to quantum gravity and topological field theory. We will study cellular decompositions of M(g,N) defined by ribbon graphs and their links with dispersionless integrable hierarchies. The connections of PDEs associated with 2D Ricci flow with integrable hierarchies describing the topology of M(g,N) will be clarified

The mathematical modeling of important phenomena arising in physics and biology often leads to nonlinear wave equations. It is quite remarkable that, within a certain range of initial data, many of these universal equations exhibit a regular behavior, typical of integrable systems. Actually, integrable behaviors in evolutionary PDEs were discovered at the end of 1960s as the result of the analysis of the numerical experiments (dating to the beginning of 1950s) by Fermi, Pasta, and Ulam.
The Korteweg - de Vries (KdV) equation, well known in the theory of dispersive waves since the end of XIX century, was the first instance in which the mathematical theory of integrability of PDEs appeared, starting from the celebrated paper by Gardner, Green, Kruskal, and Miura [GGKM].
Numerous integrable systems important in physics and mathematics were discovered later (see, e.g., [CD] and references therein). An approach to the classification of integrable PDEs based on the symmetry analysis has been initiated in the end of 70s by A.Shabat, A.Mikhailov, V.Sokolov [MSS]. This approach proved to be a powerful tool in the classification of low order systems of integrable PDEs. However, significant technical difficulties make it impossible to obtain the classification results for integrable PDEs of higher orders.

The Hamiltonian nature of the KdV equation has been realized in 1971 by Gardner and by Zakharov and Faddeev. A remarkable bihamiltonian property of KdV has been discovered by Magri in 1978 [Mag] on the basis of the Lenard recursion scheme. The existence of a bihamiltonian structure for a PDE gives the possibility to produce an infinite hierarchy of symmetries and conservation laws. A relationship of the bihamiltonian recursion procedure with the geometry of the Nijenhuis torsion was clarified in [GD], [FF]. As it has been realized later, the Virasoro symmetry of integrable PDEs can also be naturally explained on the basis of the bihamiltonian structure [ZM,AvM].

The progress in the theory of algebro-geometric solutions of integrable PDEs and also in the understanding of their relationships to integrable models of statistical mechanics and quantum field theory at the beginning of 80s gave rise to the discovery, due to Hirota and Sato and his group (see e.g. [JM]), of the notion of tau-function. Such a notion proved to be important also in various other mathematical applications of the theory of integrable systems. In particular, the partition function of the Hermitian random matrix model studied in the theories of 2D gravity coincides with a particular tau-function of the Toda lattice hierarchy. The critical behavior of the partition function, suggested in the beginning of 90s [BK, DS, GM] can be described, in the setting of the so-called double scaling limit, in terms of certain particular solutions to the Painleve' equations and their generalizations. Comparison of these results with the Witten's approach to topological gravity led to the remarkable discovery [Ko, Wit] of the role of the tau-function of the Virasoro invariant solution to the KdV hierarchy in the intersection theory of certain cycles on the moduli spaces of punctured Riemann surfaces. Tau-functions of other particular solutions to the KdV more recently proved to be important in the theory of the Weil - Petersson measures on the moduli spaces of Riemann surfaces [MZ].

Further progress in the application of the ideas and results from the theory of matrix models of simplicial quantum gravity and of topological field theory to algebraic geometry and symplectic topology raised new problems in the theory of integrable systems [Wit,D1]. In particular, one of the fundamental questions of the theory of Gromov - Witten (GW) invariants of smooth projective varieties and of compact symplectic manifolds is to understand their relationships with integrable hierarchies. The simplest examples of GW invariants and their descendents of a point is governed, due to Witten - Kontsevich, by the KdV hierarchy; for the projective line these invariants can be described by a tau-function of the Toda lattice hierarchy [OP, DZ4]. For the case of projective spaces the existence of an integrable hierarchy structure hidden behind GW invariants of other projective varieties follows from recent results of Dubrovin and Zhang [DZ1-DZ3] on the classification programme of integrable systems and the topological theorems of Givental [Gi] describing the structure of the higher genus GW invariants.

A classification scheme with respect to an appropriate extension of the group of Miura - Backlund transformations, of a wide class of spatially one-dimensional integrable evolutionary PDEs depending on a small parameter was successfully completed in [DZ3]. It concerns bihamiltonian PDEs that satisfy the following requirements:

i) they admit hydrodynamical limit;

ii) they admit a tau-function formulation

iii) they admit Virasoro symmetries linearly acting onto the tau function.

Due to these requirements, the classification scheme of [DZ3] has a wide but not exhaustive range of applicability. Although it covers a number of fundamental integrable PDEs, like Toda lattice, nonlinear Schroedinger equation, Boussinesq and, more generally, Drinfeld - Sokolov hierarchies associated with simply-laced simple Lie algebras, some physically important classes of integrable PDEs do not satisfy (all) the axioms. The celebrated Camassa - Holm (CH) equation discovered in the theory of shallow water waves [CH] does not possess a tau-function formulation. The physical importance of this equation as well as many remarkable geometric properties of it [ACFHM, KhM] urge to extend the classification scheme of [DZ3] in order to include a wider class of PDEs.

The small parameter appearing in the above evolutionary PDEs in many cases plays the role of small dispersion. The leading order system of the hydrodynamic type can be considered as the small dispersion limit of the original PDE. This suggests to use the techniques and ideas of the theory of Frobenius manifolds in the study of Whitham theory of nonlinear asymptotics for weakly dispersive evolutionary PDEs [Wh,LLV,DN]. These ideas led to the discovery of the variational formulation of the Cauchy problem for Whitham equations and to the proof of existence results for the solutions of the Cauchy problem [D2,GrT]. In general in presence of the oscillatory phases the formal expansion of the PDE in a neighborhood of a finite-dimensional invariant manifold do not admit a hydrodynamic limit. The problem of normal form of these evolutionary systems remains open, and demands a further understanding of the geometry of finite dimensional (bi)hamiltonian manifolds. Invariant tori for the KP equations have been further studied in [D4].
There is a quite small literature analyzing numerically asymptotic integrability and the behavior of solution of the PDEs in the highly oscillatory regime. The important part of setting a numerical code is to choose the best discretization. In this respect Lie group theory has shown to be a powerful tool for numerical methods for solving differential equations [Do],[LW]. The first problem that had to be faced is the mismatch between the continuous character of the Lie groups used and the discrete character of the equations studied. Recently [BCW] it was shown that numerical calculations using symmetry preserving discretization provide a much higher precision than standard schemes. Moreover, the symmetry preserving schemes make it possible to integrate around singularities of solutions. The extension of the umbral calculus technique to the nonlinear case, and the elaboration of a reductive perturbation method for nonlinear partial-difference equations are one of the tasks of the present research project.

Other geometric structures on the moduli space of Riemann surfaces are connected with two-dimensional quantum gravity and topological quantum field theory (TQFT), [Ko,ADJ,ACM]. Conformal field theory (CFT) and TQFT are based on algebraic structures parameterized by the moduli space of genus g Riemann surfaces with N punctures M(g;N). This parameterization is consistent with the operation of sewing any two such surfaces together, provided that we match the complex structure in the overlap and keep track of which puncture is ingoing and which is outgoing. This sewing gives to M(g;N) and its cell decomposition the structure of an operad. In such a setting a CFT leads to a natural algebra, over the cell decomposition of Riemann moduli space, which can be related to the algebra of physical space of states of the theory and to their dynamics. Both such descriptions are associated with the existence of an underlying integrable structure described by hydrodynamic type equations. Most likely, at the root of the connection between the CFT framework, the cellular structure of M(g,N), and the underlying integrability there is the common operadic structure mentioned above. Such a correspondence has provided one of the most successful tool for a rigorous mathematical interpretation of ideas in topological quantum field theory.
Recently, new applications of these ideas have made their impact in several guises also in Ricci flow [CK]. In dimension two, the Ricci flow resembles a (non-linear) heat flow equation on a surface due to the (weakly) parabolic character of the metric deformation in terms of the flow parameter t. Thus, it exhibits a dissipative behavior in t, but, as shown by I. Bakas [Ba], it possesses an underlying infinite dimensional Lie algebras structure which gives rise to a flow admitting a Toda field theoretic interpretation and turns out to be integrable in space. Since the two-dimensional Ricci flow provides an alternative proof of the standard uniformization theorem for surfaces one is led to discuss to what extent the Ricci flow considered as a dynamical system on the Riemann moduli space M(g) is connected to the above mentioned integrable hierarchies. This analysis relies on the study of the trajectories of the flow in regimes where the deformation parameter tends to certain limiting values and diffeomorphism solitons appear. We believe that characterizing this connection could be an important step forward in the study of the relation between the evolutive PDEs describing the two-dimensional Ricci flow and the dynamics of the integrable hierarchies characterizing the geometry of the moduli spaces M(g,N).

On more general grounds, the ``modern theory'' of finite dimensional integrable systems may in a sense be seen as an outgrowth of the theory of nonlinear integrable wave equations, as, e.g., testified by the theory of finite gap potentials in the KdV theory (see, e.g., [NMPZ]). Moreover, the interest in such Hamiltonian systems has also received a strong impact from the recent discovery of duality in certain models of supersymmetric Yang - Mills theory in four dimensions [SW]. The duality gives the possibility to express the effective action of the model in terms of action-angle variables for a suitable algebraically completely integrable system (ACIS). The case of Yang - Mills with certain particular type of matter fields has been shown to correspond to important classes of well known integrable systems, such as the periodic Toda, the Calogero-Moser-Ruijsenaars systems, and more general Hitchin systems. The study of the action-angle variables for Hitchin systems of higher genera has been initiated in [GTN,HK]. Some of the geometrical structures involved have been recently analyzed in the framework of special Kaehler geometry. In particular, it has been proved [DW], that special Kaehler geometry provides a natural bridge between supersymmetric gauge theories and integrable systems. Also the classical methods of analysis of finite-dimensional integrable systems based on the Hamilton - Jacobi theory and on separation of variables have been significantly developed in the last decade. The study of important features of algebraic integrability, initiated by Novikov and Veselov [VN] have been developed more recently (see, e.g., [AHP]). The relevance of these methods for quantum systems and spin models has been clearly pointed out by Sklyanin and his school [S1,KNS]. First steps towards understanding of the relationship between this framework and the geometry of bihamiltonian systems has been done, e.g., in [FP]. In particular, it has been shown how under certain geometrical hypotheses one can naturally associate with a Gelfand-Zakharevich [GZ1,GZ2] system a spectral curve, and thus an ACIS. As it is clearly explained in a fundamental paper by Semenov-Tianshanski [ST], in the realm of integrable finite-dimensional hamiltonian systems, Gaudin models play a distinguished role. The original Gaudin model was introduced in the 70's as a quantum model describing a one-dimensional (su(2)) spin chain with long-range interaction [G1,G2]. Its classical version has been first investigated by Sklyanin at the end of the 80's [S1,S2] by exploiting the Lax representation to find out separation variables in the rational and trigonometric case. In a more abstract language, classical Gaudin models are integrable systems defined on the Lie-Poisson manifold associated with the direct sum of N copies of a simple Lie algebra g. The Hamiltonians of the model are obtained as spectral invariants of a Lax matrix (linear in the coordinates functions on g*) satisfying a linear r-matrix formulation. Though in the present project we will be mainly concerned with classical hamiltonian systems, we have to stress the importance of its quantum version (and of its supersymmetric and q-deformed extensions), as a key model for understanding strongly correlated electron systems.


An intrinsic differential-geometric characterization of separable coordinate systems for natural Hamiltonians has been obtained in terms of Killing vectors and tensors [K,B1]. It is based on the geometrical notion of separable web [B2]. The basic starting point is that the property of separability actually depends only on a set of (local or global) geometrical requirements on the underlying Riemannian manifold. In this way both the search for separable coordinates and their classification should be carried out through the investigation of intrinsic geometric structures of the manifold itself. More recently, another intrinsic criterion of separability has been formulated, based on a suitable geometrical structure encoded in the notion of
"omega-N manifold'' [FP,MFP].
An omega-N manifold is a symplectic manifold endowed with a recursion operator satisfying suitable compatibility conditions. Given a Liouville integrable system described by n mutually commuting Integrals of the Motion H_1,..,H_n, the system is separable (in a suitable set of coordinates intrinsically defined by the tensor N) if and only if the Hamiltonian vector fields associated with such Hamilton functions form a distribution D which is invariant along N. This is equivalent to requiring that the Hamiltonian functions are commuting also with a second Poisson bracket defined by N or, in other words, to requiring that D be a `bilagrangian' distribution.

The Hamilton-Jacobi theory for integrable evolutionary PDEs is currently under construction. The interest in such a theory has been inspired by the progress in the so-called symplectic field theory [EGH]. This theory suggests a new link between GW invariants of symplectic manifolds and integrable hierarchies. The potential of GW invariants can be calculated as the value of the Hamilton-Jacobi generating functional in certain points of the Lagrangian subspace. The integrability of the evolutionary PDE suggests that also the corresponding Hamilton - Jacobi problem can be solved. The problem of implementing, in the infinite dimensional case, the analogues of the finite dimensional techniques mentioned above is still open.

[ACFHM] M.S. Alber, R. Camassa, Yu.N. Fedorov, D.D. Holm, J.E. Marsden, The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDEs of shallow water and Dym type, Comm. Math. Phys. 221 (2001) 197-227.
[AHP] M.R.Adams, J.Harnad, E.Previato, Isospectral flows in finite and infinite dimensions, Comm. Math. Phys. 117, 299-308, (1988).
[AvM] M.Adler, P.van Moerbeke, Compatible Poisson Structures and the Virasoro Algebra, Comm. Pure Appl. Math. XLVII (1994) 5-37.
[ADJ] J.Ambjorn, B.Durhuus, T.Jonsson, Quantization of Geometry, Cambridge University Press (1997).
[ACM] J.Ambjorn, M.Carfora, A.Marzuoli, "The geometry of dynamical triangulations". Lecture Notes in Physics 50, Springer (1997).
[Ba] I. Bakas "The algebraic structure of geometric flows in two dimensions", arXiv:hep-th/0507284.
[BCW] A. Bourlioux, C. Cyr-Gagnon and P. Winternitz, Difference schemes with point symmetries and their numerical tests, arXiv:math-ph/0602057.
[B1] S.Benenti, Intrinsic characterization of the variable separation in the Hamilton - Jacobi equation. J. Math. Phys. 38 (1997) 6578-6602.
[B2] S.Benenti, Orthogonal separable dynamical systems, Diff. Geom. and Its Appl. 1 (1993) 163-184.
[BK] E. Brezin and V.A. Kazakov, Exactly solvable field theories of closed strings, Phys. Lett. B236 (1990), 144.
[CD] F.Calogero, A.Degasperis, "Spectral Transforms and Solitons", North Holland (Amsterdam) (1982).
[CK] B. Chow, D. Knopf "The Ricci flow: an introduction" Mathematical Surveys and Monographs, Vol. 110, American Math.Soc. (2004).
[Do]V. A. Dorodnitsyn, Group Properties of Difference Equations, Moscow, Fizmatlit, 2001.
[DS] M. Douglas, S. Shenker, Strings in less than one dimension, Nucl. Phys. B335 (1990), 635.
[D1] B. Dubrovin, Geometry of 2D Topological Field Theories. In: Integrable systems and quantum groups, LNM 1620, M.Francaviglia and S.Greco eds.,Springer (Berlin), (1996) 120-348.
[D2] B. Dubrovin, Functionals of the Peierls-Fröhlich type and the variational principle for the Whitham equations. Solitons, geometry, and topology: on the crossroad, 35--44, Amer. Math. Soc. Transl. Ser. 2, 179, Amer. Math. Soc., Providence, RI, 1997.
[D4] B. Dubrovin, On analytic families of invariant tori for PDEs, Asterisque No. 297 (2004), 35--65.
[DN] B.Dubrovin, S.Novikov, Hydrodynamic of weakly deformed soliton lattices. Differential Geometry and Hamiltonian theory. Russ. Math. Surveys, 44, (1989) 36-124.
[DW] R.Donagi, E.Witten, Supersymmetric Yang-Mills theory and integrable systems. Nucl. Phys. B 460 (1996) 299-334.
[DZ1] B.Dubrovin, Y.Zhang, Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation, Commun. Math. Phys. 198 (1998) 311-361.
[DZ2] B.Dubrovin, Y.Zhang, Frobenius manifolds and Virasoro constraints, Sel. Math., New ser. 5, (1999) 423-466.
[DZ3] B.Dubrovin, Y.Zhang, Normal forms of integrable PDEs and Gromov-Wittien invariants, math/0108160, to appear (2003).
[DZ4] Dubrovin, Boris; Zhang, Youjin Virasoro symmetries of the extended Toda hierarchy. Comm. Math. Phys. 250 (2004), no. 1, 161--193.
[EGH] Y.Eliashberg, A.Givental, H.Hofer, Introduction to Symplectic Field Theory, Geom. Funct. Anal., Special Vol. Part II (2000) 560-673.
[FP] G.Falqui, M.Pedroni, Separation of variables for bihamiltonian systems, Math. Phys., Anal., Geom. 2003. 6, 139-179.
[FF] B.Fuchssteiner, A.Fokas, Symplectic structures, their Baecklund transformations and hereditary symmetries, Physica 4D (1981) 47-66.
[GGKM] C.S.Gardner, J.M.Greene, M.D.Kruskal, R.M.Miura, A method for solving the Korteweg deVries equations, Phys. Rev. Lett. 19, (1967) 1095-1097.
[G1] Gaudin, M.: Diagonalization d’une classe d’hamiltoniens de spin. J. de Physique 37, 1087–1098 (1976)
[G2] M. Gaudin, La Fonction d’Onde de Bethe (Paris: Masson, 1983).
[GTN] K.Gawedzki, P.Tran-Ngoc-Bich, Hitchin Systems at Low Genera, J.Math.Phys. 41 (2000) 4695-4712.
[GD] I.Gelfand, I.Dorfman, Schouten bracket and Hamiltonian operators. Functional Anal. Appl. 14 (1980)223--226.
[GZ1] I.M.Gelfand, I.Zakharevich, On the local geometry of a bihamiltonian structure, In: "The Gelfand mathematical Seminars: 1990-1992", pp. 51-112, Corwin, et. al. eds., Birkhauser, 1993.
[GZ2] I.M.Gelfand, I.Zakharevich, Webs, Lenard schemes, and the local geometry of bihamiltonian Toda and Lax structures Sel. Math., New ser. 6 (2000) 131-183.
[Gi] A.Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians, Moscow Math. J. 1 (2001) 551-558.
[GrT] T. Grava, Fei-Ran Tian, The generation, propagation and extinction of multiphases in the KdV Zero-Dispersion limit, Comm. Pure Appl. Math. 55 (2002), 1569-1639.
[GM] D.J. Gross and A.A. Migdal, A nonperturbative treatment of two dimensional quantum gravity, Phys. Rev. Lett. 64 (1990), 127.
[HK] J.Hurtubise, M.Kjiri, Separating coordinates for the generalized Hitchin systems and the classical r-matrices. Comm. Math. Phys. 210 (2000) 521-540.
[JM] Non-linear integrable systems: classical theory and quantum theory. Eds. M. Jimbo and T. Miwa, World Scientific (Singapore), 1983.
[K] E.G.Kalnins, Separation of variables on manifolds with constant curvature, Pitman Monographs and Surveys in Pure and Applied Math. 28, Longman (1986).
[KhM] B.Khesin, G.Misiolek, Euler equations on homogeneous spaces and Virasoro orbits. Adv. Math. 176 (2003) 116--144.
[Ko] M.Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 147 (1992) 1­23.
[KNS] V.Kuznetsov, F.Nijhoff, E.Sklyanin,Separation of variables for the Ruijsenaars system, Commun. Math. Phys. 189 (1997) 855-877.
[LLV] P.Lax, D.Levermore, S.Venakides, The Generation and Propagation of Oscillations in Dispersive Initial Value Problems and Their Limiting Behavior. In: Important Developments in Soliton Theory, Eds. A. S. Fokas, V. E. Zakharov, Springer Verlag (Berlin), (1993), pp. 205-241.
[LW] D. Levi and P. Winternitz, Continuous symmetries of difference equations J. Phys. A39 R1-R63, (2006).
[Mag] F.Magri, A simple model of the integrable Hamiltonian equation. J.Math. Phys. vol. 19, (1978) 1156-1162.
[MFP] F. Magri, G. Falqui, M. Pedroni, The method of Poisson pairs in the theory of nonlinear PDEs. Direct and inverse methods in nonlinear evolution equations, 85--136, Lecture Notes in Phys., 632, Springer, Berlin, 2003.
[MZ] Yu.Manin, P.Zograf, Invertible cohomological field theories and Weil-Petersson volumes, Ann. Inst. Fourier. 50 (2000) 519-535.
[NMPZ] Novikov, S.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E. Theory of solitons. The inverse scattering method. Translated from the Russian. Contemporary Soviet Mathematics.
[OP] A. Okounkov, R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517--560.
[SW] N.Seiberg, E.Witten, Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD, Nucl.Phys. B431 (1994) 484-550.
[ST] M. A. Semenov-Tian-Shansky, Quantum and classical integrable systems. Integrability of nonlinear systems (Pondicherry, 1996), 314--377, Lecture Notes in Phys., 495, Springer, Berlin, 1997.
[S1] E.Sklyanin, Separation of Variables: New trends. Progr. Theor. Phys. Suppl. 188 (1995), 35.
[S2] E K Sklyanin. Separation of variables in the Gaudin model. J. Soviet. Math.,47:2473–2488, 1989.
[VN] A.Veselov, S.P.Novikov, Poisson brackets and complex tori, Proc. Steklov Math. Inst. 1985, Issue 3, pp 52-65.
[Wh] G.Whitham, Linear and nonlinear waves, J. Wiley & Sons, New York, 1974.
[Wit] E.Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom. 1 (1991) 243-310.
[ZM] J.P.Zubelli, F. Magri, Differential Equations in the Spectral Parameter, Darboux Transformations, and a Hierarchy of Master Symmetries for KdV, Commun. Math. Phys, 141 (1991) 329-351.

The perturbative approach to the classification problem suggested in [DZ3] proved to be successful in classifying systems of spatially one-dimensional integrable evolutionary PDEs admitting hydrodynamic limit, a tau-function and Virasoro symmetry acting linearly on the tau-function [DZ1-DZ3]. The corresponding moduli space of n-components integrable PDE of this class (up to Miura transformations) can be identified [DZ3] with the moduli space of n-dimensional semi-simple Frobenius manifolds which are parameterized by n(n-1)/2 parameters [D1]. A certain class of quantum canonical transformations acts transitively on the moduli space. In the setting of the Gromov - Witten invariants of projective spaces, they were discovered by A. Givental in [Gi]. The quantum canonical transformations are obtained by quantizing the Riemann-Hilbert problem involved in the description [D1] of the moduli space of n-dimensional Frobenius manifolds.
The bihamiltonian structure of the integrable PDEs constructed in this way can be considered as a deformation of classical W-algebras. Virasoro symmetries of the integrable PDEs play an important role both in their construction as well as in the classification. We plan to study the full symmetry algebra of these PDEs in order to produce deformations of quantum W-algebras introduced by A. Zamolodchikov in his study of symmetries of conformal field theories with higher spin.
The classification scheme of [DZ3] has a wide but not exhaustive range of applicability. For example the celebrated Camassa - Holm (CH) equation discovered in the theory of shallow water waves [CH] does not possess a tau-function and it is not included in the above classification.
The general classification problem of systems of integrable PDEs remains open. The perturbative classification approach implemented in [DZ3] has been enlarged in [DLZ] to include all the integrable PDEs admitting a bihamiltonian hydrodynamic limit. In this framework equivalence classes (up to Miura transformations) of n-components bihamiltonian PDEs depend on n(n+1) arbitrary functions of one variable. The problem of proving the existence of such bihamiltonian structures for an arbitrary choice of the functional parameters. The CH equation is included in this classification programme. We remark that the CH hierarchy is transformed into the KdV negative one (i.e. the hierarchy generated by the second Casimir of the corresponding Poisson bracket), via a Miura type transformation followed by a reciprocal transformation. Then, naturally, another direction of investigation is to connect the family of PDEs classified in [DZ3] and the family classified in [DLZ] via reciprocal transformations, which act nonlinearly also on the independent variables. This research issue will require a preliminary study aimed at finding new integrable hierarchies of CH type, and at studying their properties.

A related direction of development is to consider formal perturbation of diagonal hyperbolic systems possessing a complete family of commuting flows (the so-called semi-hamiltonian systems in Tsarev sense [T]). While in the classification of bihamiltonian hydrodynamic equations is well understood [DN], the problem of classification of semi-hamiltonian systems is still open. A fortiori, so is the classification problem of their perturbations.

An important part of the present project is the study of the asymptotic behavior of the solutions of these PDEs. Let us consider bihamiltonian perturbations of the Riemann equation (with one dependent variable). The formal perturbative solution of the corresponding Cauchy problem with monotone initial data can be expressed in terms of the unperturbed solution [DLZ]. This feature is known as quasi-triviality property that permits to express the perturbed solution as a series that depends rationally on the unperturbed solution and its derivatives. This series becomes meaningless at the point of gradient catastrophe of the unperturbed solution and a double scale expansion is needed. One of our tasks is to prove the universality property conjectured in [D5] about the generic solution to a Hamiltonian perturbation of an hyperbolic PDE near the point of gradient catastrophe of the unperturbed equation. Up to shifts and rescalings this behavior does not depend upon the choice of the initial data nor on the choice of the equation and it is described by a particular solution of an ODE of fourth order that belongs to the family of Painleve' equations. Existence of such a particular solution has been proved recently [CV]. Numerical investigations of this conjecture will be performed as well. In this framework we plan to rigorously study the small dispersion limit of the KdV equation at the time of gradient catastrophe. This limit has been extensively investigated in [GP,LLV, DVZ], but only before or after the time of gradient catastrophe. We plan to extend the conjecture formulated for the one component case in [D5], to the multi-component case. Double scaling expansions in matrix models suggest that the conjecture formulated in [D5] remains true for the two-component Toda lattice equations.

After the gradient catastrophe the solution to the PDE becomes oscillatory [ZK, GP]. To study this regime, we are going to develop a PDE analogue of the Krylov - Bogoliubov - Mitropolskii asymptotic method, or nonlinear WKB method, well known in the theory of nonlinear oscillations [Wh,DN,LLV]. In this respect we plan to study the deformation of the Whitham system as well as the corresponding bihamiltonian structure of hydrodynamic type. This corresponds to determine all orders of the nonlinear WKB formal asymptotic solution of the integrable PDE in the neighborhood of a finite dimensional invariant manifold fibered into Liouville tori. A natural geometric structure induced on such invariant manifolds by a Hamiltonian structure of the PDE consists of a symplectic foliation equipped with a flat metric on the base of such a foliation. We plan to develop a classification programme of such structures and to apply it to describing normal forms of more general systems of 1+1 evolutionary PDEs not admitting a hydrodynamic limit.
Next we plan to study the asymptotic solution near singular points (namely the turning points in the linear WKB theory) and the corresponding matching of the asymptotic solutions in the oscillatory and non-oscillatory regimes. A detailed analysis of the development of the various regimes for a given initial data has been obtained in [GrT]. Preliminary numerical tests suggest that the matching of solutions in the various regimes is described by Painleve' transcendents not only at the point of gradient catastrophe but also after few oscillations are formed [GK].

The geometric and analytic foundations of the averaging methods rely upon the study of behaviour of trajectories of integrable systems in the neighborhood of a finite dimensional invariant manifolds. Since the pioneering works of the 70's and 80's, the study of finite dimensional invariant submanifolds of integrable PDEs has been connected with the theory of algebraic completely integrable systems and the theory of isomonodromic deformations, as well as with the classical counterparts of quantum integrable models.
Various features of the integrable PDE structure, such as the bihamiltonian recursion procedure, the action-angle variables, etc. induce a very rich geometry on suitable finite dimensional invariant submanifolds. We plan to study some of the outstanding questions that are still open in this field, as well as their links with the other main themes of the research project. In particular we plan to use the so called Gel'fand - Zakharevich (see [GZ1,GZ2]) scheme, which suggests the possibility of constructing integrable systems from geometric data. An instance of this situation, connected with non linear wave equations has been described in the case of stationary reductions of the KdV and Boussinesq hierarchies (see. e.g., [FMPZ,FMT]).
We also plan to develop an alternative construction of the families of these PDEs 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 Lax operator will be obtained by a suitable quantization of the symbol of the Lax representation constructed in [D1].

The Hamilton-Jacobi theory for integrable evolutionary PDEs is currently under construction. The interest in such a theory has been inspired by progress in the so-called symplectic field theory [EGH]. This theory suggests a new link between GW invariants of symplectic manifolds and integrable hierarchies. The potential of the genus zero GW invariants can be calculated as the value of the Hamilton-Jacobi generating functional in certain points of the Lagrangian subspace. The corresponding Hamiltonian flow is a system of hydrodynamic type in the space of periodic functions which turns out to be integrable.
The research line we will follow in the present project in order to construct an extension of the HJ method to soliton equations is based on the bihamiltonian setup for integrable systems, rather than on the symplectic field theory. To describe it, we will refer to a specific class of bihamiltonian manifolds, called omega-N manifolds. On these (symplectic) manifolds there exists a torsionless recursion operator N which can be used to induce iterative processes, called "generalized Lenard chains", since the chains originally introduced under such a denomination in the theory of integrable PDEs are particular examples of such structures. Indeed, (generalized) Lenard chains are nothing but special distributions of vector fields on phase space. Under suitable completeness hypotheses, such distributions give rise to bilagrangian foliations. In the finite dimensional case, it has been proved (see. e.g. [FP,MFP]) that the generating function of such foliations satisfies a separable HJ equation. In other words, the intimate relation between bihamiltonian geometry and separable HJ equations has, in the finite dimensional setting, been established. The primary goal of this line of research is to prove similar results in the case of the recursion operators involved in the theory of non-linear evolutionary PDEs of solitonic type. It is indeed natural to conjecture that the integrability properties of such PDEs might be reformulated in the language of the intrinsic theory of separable systems, and that the inverse scattering method might be thought of as a particularly clever way of implementation of the techniques for the integration of the classical separable systems.
In this framework we plan to prove that the Whitham equations are equations that describe a particular deformation scheme of associative algebras, called "isotropic deformation". Such kind of deformations have already been preliminarily studied, and an unexpected link between them and the "tau function" (and hence with the Witten, Dijkgraaf, Verlinde, Verlinde associativity conditions) has been detected. We plan to bring to the light the meaning of this deformation process in the framework of these quasi-linear PDEs. We also want to show that the algebraic curves associated with the Whitham equations are a hallmark of particular "separation equations" that ultimately provide the possibility of solving them via the hodograph method.
More in general, we will try to extend the theory of bilagrangian foliations to Whitham equations, and, more in general, transplant what we know about the finite dimensional case to integrable PDEs. We have two main instruments at our disposal: the theory of generalized Lenard sequences and the theory of systems of Gel'fand-Zakharevich type. It is fair to say that both tools are under control in the finite dimensional case, while for the infinite dimensional case further efforts are required to understand their meaning and power.

A parallel step in our research concerning methods of solutions for integrable systems considers finite dimensional reductions of infinite dimensional systems as well as so Heisenberg and Gaudin spin chains. To this problems we will devote an intensive research. Indeed, suitable algebraic procedures performed on N-site Gaudin models associated with a simple Lie algebra g and with "any" (i.e. rational, trigonometric, elliptic) dependence on the spectral parameter, have recently enabled our group to construct a hierarchy of new integrable models, called "extended" Lagrange tops, sharing the linear r-matrix structure proper of the ancestor models [MPR]. A straightforward direct sum procedure yields chains of long-range interacting bodies, with rational, trigonometric and elliptic r-matrices, denoted in the simplest case (g=su(2),rational) Lagrange chains.
There are a couple of issues on this subject deserving special investigation:
1a. The bihamiltonian structure of the constructed systems, that certainly can be inherited from that of the ancestor Gaudin model, but whose explicit reduction is expected to be a nontrivial task.
2a. The problem of integrable time-discretizations. So far, integrable discretizations have been obtained "a' la Sklyanin", through Backlund transformations respectively for the Lagrange top, its first extension and the rational Lagrange chain. An alternative discretization approach, proposed by Moser and Veselov [MV] and developed by other authors [Su] turned out to be an effective tool to construct integrable maps when the basic manifold carries the additional structure of a Lie group. Our starting point is an integrable discretization [BSu] for the Lagrange top (N=2 in our framework), providing a discrete Lax representation of the map (with a "deformed" Lax matrix) and a discrete Lagrangian formulation. In the same spirit we have proposed a discrete map for the whole su(2) hierarchy (i.e., for any N), reducing to those derived in [Bsu] for N=2. An evidence for the existence of a complete set of invariant functions has been obtained with the help of symbolic computation programs, but a lot of questions are yet to be solved.

Still within the theory of spin chains, we will study universal graded contractions of rational N-site Gaudin and new integrable models. In [MPR] it has been shown that applying a prescribed pole coalescence procedure to the Lax matrix of a N-site Gaudin model and a related Inonu-Wigner contraction to its Poisson tensor, one obtains a new Lax matrix satisfying the same linear r-matrix formulation of the ancestor model with respect to the contracted Poisson brackets. The procedure works whatever be the functional dependence of the Gaudin Lax matrix on the spectral parameter: rational, trigonometric or elliptic. The particular Inonu-Wigner contraction considered in [MPR] maps G in a new Lie algebra that turns out to be an extension of g.

More in general, we will use Gaudin systems, as well as other models, as an effective tool to enable us to provide an answer to the following problems:
1b. To better understand and fully describe the reduction process of bihamiltonian structures from a PDE infinite dimensional phase space, to its finite dimensional invariant submanifolds.
2b. To establish a full comparison with Sklyanin's method of the poles of the Baker-Akhiezer function, for those bihamitonian systems admitting a Lax representation with spectral parameter.
A parallel direction, equally important towards a full-fledged comprehension of the theory, will concern the study of the relations of the points outlined above with the classical HJ theory. As it is known, in its historical development, the classical theory of separability is based on methods of Riemannian geometry. The full comprehension of the relations between these Riemannian methods and those based on the recursion operator technique is an interesting subject. It will also be helpful to extend such techniques in the infinite dimensional situation. This problem will be studied, starting from the simplest examples of the HJ theory for KdV and Toda, where applications to the theory of GW invariants are envisaged.

Finally, a third related research direction concerns the theory of algebraically completely integrable systems. Most - if not all - of the finite dimensional reductions of infinite dimensional integrable systems can be solved by means of algebro-geometrical techniques. (see, e.g., [BBEIM]) An important specific issue we plan to consider in the present proposal can be described as follows.
The relation of special Kaehler geometry to integrable models (especially Toda and Calogero-type) has been established within the Seiberg-Witten theory of 4D supersymmetric gauge theory [SW,DW]. In his description of moduli spaces of complex lagrangian submanifolds, Hitchin has provided a new characterization of special Kaehler manifolds as those admitting a bilagrangian local immersion in their cotangent spaces. On the other hand, the notion of ``bilagrangian" distribution is crucial in the intrinsic study of the Hamilton - Jacobi equations.
So, it is natural to explore the relationship between these two points of view and extend the first results available in this direction.
Namely, in connection with the existence of a special Kaehler structure on the base of hypersymplectic integrable systems it is important to find out explicit examples, possibly among classical hamiltonian systems. In this framework we intend to study the question of duality of integrable systems (according to the definition proposed by Fock et al., [FGNR]) by means of suitably defined relative Fourier-Mukai transforms [BBHM]. Also, we want to assess whether the equation of the spectral curve associated with a separable systems defined on a regular bihamiltonian manifold can be derived in a purely geometrical fashion from the separation coordinates (i.e. the so-called Darboux-Nijenhuis coordinates). We expect that the equation of the Poisson surface underlying the associated Jacobian fibration can be derived from the separability conditions as well.

Within applicative study of the the theory of nonlinear waves, we plan to address the problem of the transition from continuos to discrete models, and namely study the problem of umbral calculus and symmetries.
In the construction of symmetries of difference equations the umbral calculus technique is very helpful as it allows to get results which are independent from the discretization scheme used. Up to now no effective application of the umbral calculus to nonlinear equation has been devised. Overcoming such limitation is one of the scopes of the present project.
To be more specific, we plan to extend the umbral-calculus technique from the case of linear equations to that of nonlinear ones by solving the nonlinear equations through a "nonstandard" perturbation scheme. Typically, in the usual perturbation scheme for nonlinear equations one expands the seeked solution in terms of solutions of their linearized version. Here we will instead consider our system as a perturbation to a nonlinear system in different variables, namely a discrete nonlinear system which is the umbral correspondent to the differential nonlinear equation. We plan to apply our results in the study of asymptotic symmetries. Though in physical applications the model equations can be quite complicated, relevant informations are usually carried by a properly defined asymptotic behaviour of the system. In the asymptotic regime the system will be simpler and more symmetries will appear.
We plan to carry out a classification scheme of the asymptotic symmetries of physically interesting equations appearing in the description of nonlinear optical lattices, taking also into account the umbral correspondence with differential equations.

The research concerning Topological Field theory and 2D-Quantum Gravity is prompted by the geometrical analysis of the properties of two-dimensional simplicial quantum gravity briefly recalled in the state of the art above. Particularly important are the connections between the (microcanonical) entropy of the set of distinct dynamical triangulations and the Weil-Petersson measure of the moduli space M(g,N) of N-pointed stable curves. Recently Manin and Zograf have characterized the generating functions for the volume of such moduli spaces as the tau-function of a particular solution to KdV. By exploiting the large N asymptotics of the Weil-Petersson volume, it is then possible to show that the anomalous scaling properties of the counting measure of dynamical triangulations are due to the modular degrees of freedom which parametrizes in M(g;N) the vertices of the triangulations.
Recently, new applications of these ideas have made their impact in several guises also in Ricci flow theory and even in quantum computing. Ricci flow [CK] is an evolutionary system of PDEs of geometric origin defined by deforming a given Riemannian metric in the direction of its Ricci tensor. It arises in a variety of non-linear problems in physics, typically as the renormalization group flow for non-linear sigma-model [Fr], wheras in Riemannian geometry has been introduced by R. Hamilton [Ha] in his attempt to a proof of the Thurston geometrization conjecture. Since the two-dimensional Ricci flow provides an alternative proof of the standard uniformization theorem for surfaces one is led to discuss to what extent the Ricci flow considered as a dynamical system on the Riemann moduli space M(g) is connected to the above mentioned integrable hierarchies. This analysis relies on the study of the trajectories of the flow in regimes where the deformation parameter tends to certain limiting values and diffeomorphism solitons appear.
We believe that characterizing this connection could be an important step forward in the study of the relation between the evolutive PDEs describing the two-dimensional Ricci flow and the dynamics of the integrable hierarchies characterizing the geometry of the moduli spaces M(g,N). The project will address some of the issues of interest in this area, with particular emphasis on the roles of Perelman's shrinker entropy functionals [Per], [Car] and the construction of explicit examples in terms of the Egorof metrics which naturally appear in the study of Gromov-Witten invariants.
Related research topics to be pursued in the present proposal are three-dimensional hyperbolic geometry and two-dimensional singular Euclidean structures, Chern-Simons theory and cellular decomposition of the moduli space M(g,N) and pplications to problems in quantum computing knots invariants. This latter line of research is prompted by the geometrical analysis briefly discussed above and stems from observing that metrically triangulated surfaces can be naturally associated with three-dimensional hyperbolic manifolds triangulated by means of ideal hyperbolic tetrahedra (this being a consequence of the Ahlfors-Bers theorem according to which the 3D hyperbolic structure in a manifold is characterized by the conformal structure on the boundary of the manifold).

Further References:
[BBHM] Bartocci C., Bruzzo U., Hernandez Ruiperez D., Munoz Porras J., Realtively stable bundles over elliptic fibrations, Math. Nachr. 238 (2002), 23-36.
[BSu] A.I. Bobenko, Y. Suris, A discrete time Lagrange top and discrete elastic curves. Amer. Math. Soc. Transl. 201, 39--62 (2000).
[Car] M. Carfora, Fokker-Planck dynamics and entropies for the
renormalized Ricci flow. arXiv:math.DG/0507309 v2.
[CV] T. Claeys, M. Vanlessen, The existence of a real pole-free solution of the fourth order analogue of the Painleve I equation. arXiv:math-ph/0604046.
[DVZ] Deift, P.; Venakides, S.; Zhou, X. New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems. Internat. Math. Res. Notices 1997, no. 6, 286--299.
[D5] B. Dubrovin, On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour, xxx.lanl.gov//math-phys/0510032.
[DLZ] B. Dubrovin, Si-Qi Liu, Y. Zhang (2006). On Hamiltonian perturbations of hyperbolic systems of conservation laws. I. Quasi-triviality of bihamiltonian perturbations. Comm. Pure Appl. Math. vol. 59 pp. 559-615.
[FMT] G. Falqui, F. Magri, G.Tondo. Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy. Theor. Math. Phys., 122, 212-230, (2000).
[Fr] D. H. Friedan, Nonlinear models in 2+epsilon dimensions. Ann. Physics 163 (1985), no. 2, 318--419.
[FGNR] Fock, V., Gorsky, A., Nekrasov, N., Rubtsov, V., Duality in integrable systems and gauge theories. J. High Energy Phys. 2000, no. 7, Paper 28, 40 pp.
[GP] A.G. Gurevich, L.P. Pitaevskii, Non stationary structure of a collisionless shock waves, JEPT Letters, 17:193-195 (1973).
[Ha] R. Hamilton Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982), 255-306.
[MV] J. Moser, A.P. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials. Comm. Math. Phys. 139, 217--243 (1991).
[MSS] A.Mikhailov, A.Shabat, V.Sokolov, The symmetry approach to classification of integrable equations, in "What is
integrability?", V.Zakharov ed., pp. 115-184, Springer-Verlag (Berlin) 1990.
[MPR] F. Musso, M. Petrera, O. Ragnisco, Algebraic extensions of Gaudin models. Jour. Nonlin. Math. Phys. 12, Suppl. 1, 482--498 (2005).
[Per] G. Perelman The entropy formula for the Ricci flow and its geometric applications. math.DG/0211159.
[Su] Y. Suris, The problem of integrable discretization: Hamiltonian approach, Birkhauser Verlag, Basel, 2003.