Report of the Conference: Riemann-Hilbert Problems, Asymptotics, and Integrability
(Joint General Meeting of MISGAM and the Marie Curie FP6 RTN ENIGMA)
Venue: Sissa, Trieste (Italy), September 20- 25, 2005.
Scientific and Organizing Committee: B. Dubrovin, G. Falqui, T. Grava
Web-page: http://misgam.sissa.it/RHPIA05
Summary
The conference has been the occasion for the general meeting of the participants of the MISGAM programme and the Marie Curie FP6 RTN Enigma.
The conference took place in SISSA, Trieste from September 20-25. It attracted 75 registered participants from Europe (mainly), USA, Russia and Japan. The conference was also attended by scientists from the nearby International Center of Theoretical Physics and the University of Trieste.
There have been 26 plenary talks of 45 minutes and 22 talks of 30 minutes delivered in parallel sessions. The conference was a training ground for young scientists. Indeed, six post-docs and six students delivered their lectures during the conference. The plenary talks, given mainly by senior scientists, but also by a student and a post-doc, aimed at establishing the state of the art of the conference research topics and the related subjects of the MISGAM Programme and ENIGMA RTN. The conference enabled the contacts between the various nodes of the MISGAM programme and ENIGMA network and the interaction between different research lines. For many of the participants the conference was the occasion for their first personal meeting.
Scientific content
The main research lines presented during the conference were the following:
1. Random Matrices, Riemann-Hilbert problems and Conformal maps (T. Claeys, E. Daem, P. Elbau, V. Kazakov, A. Kuijlaars, M. Vanlessen, P. van Moerbeke, C. Tracy).
2. Dispersionless equations and conformal maps (G. Carlet, D. Crowdy, E. Ferapontov, J. Gibbons, B. Konopelchenko, P. Lorenzoni, M. Mañas, J. Marshall, L. Martinez Alonso, A. Moro).
3. Riemann-Hilbert problems, isomonodromic equations and discrete geometry (A. Bobenko, G. Khimshiashvili, V. Novokshenov, K. Takemura, Masa-Hiko Saito, Y. Suris).
4. Geometry, singularity theory and Frobenius manifolds (B. Dubrovin, C. Hertling, V. Shramchenko, I. Strachan, A. Veselov).
5. Explicit integration of ODES and PDES (J. Arnlind, P. Garcia Estevez, A. Fokas, A. Hone, C. Klein, M. Hallnas, I. Hitzazis).
In addition, other topics were covered, e.g. in the talks by A. Cattaneo, M. Dunajski, J. Hoppe, R. Ivanov, L. Mason, S.P. Novikov, L. Samaj, P. Santini, F. Smirnov.
We briefly describe some of the talks.
The talk by A. Kuijlaars (joint work with A. Aptekarev and P. Bleher) described the large N limit of a random matrix model with an external source. It is well known that the eigenvalues of a Gaussian matrix model with external source have an interpretation in terms of non-intersecting Brownian motion. Using the steepest descent method applied to the Riemann-Hilbert problem associated with the corresponding matrix model A. Kuijlaars showed that the probability distribution of the of the Brownian motion is described, in different regimes, in terms of the Airy and Pearcy kernels.
P. van Moerbeke (joint work with M. Adler) introduced a series of PDEs related to the Airy and Pearcey processes and studied some of their properties. These processes arise in random matrix ensemble with external source, where the support of the equilibrium measure consists of one interval, with a gap in the middle. A fine tuning of the external source leads to a closure of the gap at the origin. The statistics of the eigenvalues of the problem near this critical point is described by the Fredholm determinant of the Pearcey kernel. An explicit fourth order non-linear PDE for
the probability that an interval contains no eigenvalues was presented.
C. Tracy (joint work with H. Widom) described the Painlevé representation for the distribution functions of next largest, next-next largest eigenvalues in the Gaussian unitary, orthogonal and symplectic ensembles. He gave a very elegant and complete surveys of the results obtained for the unitary and symplectic ensembles and pointed out the problems that remain still open for orthogonal ensembles. The distribution functions of next largest eigenvalue in the Gaussian orthogonal ensemble is strictly related to the problem of the statistic of a sample covariance matrix.
P. Elbau described his joint work with G. Felder about the asympotics of NxN normal matrix model. The relevant result is the proof of the existence of an equilibrium measure in the large N limit. Furthermore he showed that when the support of the equilibrium measure is a polynomial curve, there exists a diffeomorphism between the exterior harmonic moments associated to the polynomial curve and the coefficients of the polynomial exponential weight of the matrix model. The talk of V. Kazakov (joint work with N. Gromov) was focused on the emergence of non trivial algebraic curves in the thermodynamical limit of Bethe Ansatz equations, working out in details the case of the XXX Heisenberg chain. He pointed out the remarkable similarities with the large N limit in matrix models, such as the double scaling at the spectrum edge and the asymptotics of Baker-Akhiezer functions. In particular, a regular way to generate finite volume corrections to Bethe equations in the thermodynamical limit was discussed. Also he showed that the behaviour near the edge of distribution of Bethe roots displays universality of Airy type. Finally various possible generalizations (e.g., to the multicut and non-hyperelliptic cases of Bethe Ansatz Equations) were pointed out.
The talk by L. Martinez Alonso was devoted to "String equations and Whitham hierarchies in conformal map dynamics". The speaker presented a class of solvable string equations for Whitham hierarchies which underly integrable deformations of simply-connected quadrature domains in the
complex plane. These hierarchies are defined on a suitable space of Sato functions, that is, local coordinates near suitable punctures in the complex plane. A theorem that expresses solutions of the Whitham equations in terms of solutions of the string equations having prescribed polar singularities was discussed. Applications to the evolution of supports of eigenvalues for normal random matrix ensembles in the large N limit, as well as other types of conformal map dynamics, were provided.
In a related talk by M. Manas, the dressing scheme, string equations, and additional symmetries for Whitham hierarchies were discussed. A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations was
provided. A class of string equations which extends those previously considered by Krichever and by Takasaki and Takebe was presented. This scheme was applied for a convenient derivation of additional symmetries.
In the talk by K. Takasaki some aspects of dispersionless integrable hierarchies, as well as recent progress on the dispersionless limit of q-analogues and multi-component systems, were reported. In particular, the speaker discussed the route from the KP to the dispersionless KP hierarchy, as well as that from the Toda Hierarchy to its dispersionless counterpart. Then he reviewed the WKB ansatz for the wave function and, finally, provided a new example from q-difference equations, and addressed the problem of the Whitham hierarchy associated with multi-component KP hierarchies. Applications of dispersionless equations to physical problems were illustrated in the talk of A. Moro. He introduced a nonlocal nonlinear Schroedinger equation to describe the propagation of a laser beam in a medium which exhibits different degrees of nonlocality. He showed that in the high frequency limit, under suitable phenomenological assumptions, this system is described by the dispersionless Veselov-Novikov hierarchy. Each equation of the hierarchy corresponds to a specific degree of nonlocality. The problem of compatibility with a general class of nonlinear response of physical interest was also discussed. A reduction method based on symmetry constraints was used to construct hydrodynamic type reductions which are compatible with the class of nonlinear responses considered.
The talks by J. Gibbons, D. Crowdy, and J. Marshall were devoted to various aspects of the theory of reductions of the Benney equations. Namely, J. Gibbons described reductions of the Benney hierarchy to a system with N Riemann invariants, showing that it can be described by an N-parameter family of conformal maps to domains with N slits. He also showed that some of such reductions can be explicitly constructed as Schwartz-Christoffel mappings.
The starting point of D. Crowdy's talk was the mathematical connections between Laplacian growth problems of planar domains (such as the evolution of blobs of fluid in a Hele-Shaw cell), the planar Dirichlet problem and its associated Green's function and dispersionless integrable hierarchies. In the talk, the connection between the planar Dirichlet problem and integrability was shown to extend to the case of the hierarchy of Benney moment equations. Moreover, the way in which a special class of genus-N reductions (due to Gibbons and Tsarev) can be parametrized, in a natural way, using modified Green's functions associated with reflectionally-symmetric multiply connected planar domains was described. Various applications of these ideas to the uniformization problem of quadrature domains were presented in the talk by J. Marshall.
B.Dubrovin in his talk discussed the properties of Hamiltonian perturbations of hyperbolic systems. This research, as a part of investigation of the relationships between Frobenius manifolds and
integrable PDEs, unravels remarkable properties of universality of behaviour of solutions of an arbitrary Hamiltonian perturbations near the point of gradient catastrophe of the unperturbed equation.
C.Hertling presented a novel approach to the problem of characterization of the Riemann - Hilbert problem data associated with the quantum cohomology of smooth projective varieties. His specific interest was focused on the problem of computation of the so-called central connection matrix involved in the reconstruction of the associated Frobenius manifolds. For certain important classes of varieties he found an unexpected connection between the theory of central connection matrices and number theory.
V.Shramchenko gave an overview of her recent results on construction of Frobenius structures on Hurwitz spaces of algebraic curves. These constructions, unexpected from the theory of integrable systems, might be helpful in computation of oscillatory asymptotics of nonlinear waves.
A. Fokas showed how to integrate initial boundary valued-problems associated with tomography. He reduced the problem to a Riemann-Hilbert problem in the complex plane. This talk showed how fundamental techniques in the theory of integrable systems can be used in concrete applied problems.
C. Klein (joint work with D. Korotkin) studied the Riemann-Hilbert problem for the Ernst equation which is equivalent to the stationary asymmetric Einstein equations in vacuum. Using a gauge freedom of the associated linear system he showed that the Riemann-Hilbert problem can be solved explicitly in terms of hyperelliptic theta functions for rational jump data.
M.-S. Saito (joint work with M. Inaba and K. Iwasaki) constructed the moduli space M of stable parabolic connections of any rank on a compact Riemann surface of any genus with regular singular points as a non-singular algebraic variety, and constructed the moduli space R of representations of the fundamental group of the punctured curve. The Riemann-Hilbert correspondence RH : M -> R can be defined naturally and he showed that this gives a symplectic resolution of singularities of R. Isomonodromic flows on M which are the pull-backs of the constant flows on R define an algebraic vector field on M, which he called equations of Painlevé type. In this geometric setting, his results clarify most of the geometric propeties of the equations of Painlevé type, like Hamiltonian, symmetry (Bäcklund transformations), tau-functions, as well as Painlevé property of these equations.
V. Novokshenov constructed a uniform asymptotics in the complex plane for the third Painlevé transcendent. The leading term asymptotics for large |z| is given by an elliptic function with its modulus depending on the argument of z. The phase shift in the elliptic function depends on the initial data, and it is calculated with the help of the isomonodromic deformation method in the same way as Bolibrukh-Its-Kapaev proved a similar asymptotic description of PII transcendent.
K. Takemura generalized several results on Heun's equation to a certain class of Fuchsian differential equations. He obtained integral representations of solutions and develop Hermite-Krichever Ansatz on them. In particular, he investigated linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtained solutions of the sixth Painlevé equation which include Hitchin's solution.
A. Bobenko (joint work with Y. Suris and B. Springborn) studied circle packings (that is, patterns of circles on possibly singular surfaces, where it is given an embedded graph (usually a triangulation) of incidence; the vertices correspond to circles, edges correspond to intersections among circles, and each edge is equipped with a real number which gives the angle of intersection between the two circles. The question of existence and uniqueness of circle packings in various geometries (Euclidean, hyperbolic, elliptic) with given combinatorial data has been considered by a number of authors, in view of the beautiful connections this theory has to seemingly disparate areas of low dimension geometry and topology. Several descriptions of circle patterns were provided and each of these descriptions leads to a certain integrable system on a graph. He then showed various computational realization of circle packings.
Y. Suris (joint work wit A. Bobenko and C. Mercat) considered two discretizations, linear and nonlinear, of basic notions of the complex analysis. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. He clarified the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable . He then extended both theories in d dimensions, where d is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, he gave an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions. He introduced the d-dimensional discrete logarithmic function which is a generalization of Kenyon's discrete Green's function, and showed that it is an isomonodromic solution of the discrete Cauchy-Riemann equations.
S.P. Novikov presented his new ideas about the topology of generic Hamiltonian dynamical systems on Riemann surfaces. The dynamics is given by the real part of generic holomorphic 1-forms. His approach is based on the notion of transversal canonical basis of cycles. This approach allows him to present a convenient combinatorial model of the whole topology of the flow, especially effective for g=2.