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Mathematical Physics Seminars: January – December 2001

mardi 9 janvier 2001:16h00
Conférencier / Lecturer: Jorgen Rasmussen, Univ. of Lethbridge
Titre / Title: su(N) tensor product multiplicities and virtual Berenstein-Zelevinsky triangles


mardi 23 janvier 2001: 16h00
Conférencier / Lecturer: Luc Frappat, CNRS, LAPP
Titre / Title: Elliptic algebras, q-deformed W-algebras and Yangian limit


mardi 30 janvier 2001: 16h00
Conférencier / Lecturer: Bertrand Eynard, SPHT Saclay (France)
Titre / Title: Random matrices and (skew)-orthogonal polynomials
Resume: Many physical systems can be represented by a random matrixm, and they share some universal properties, The method of Orthogonal Polynomials was invented in order to understand universlity in random matrices *skew-orthogonal poynomials for non-hermitian matrices). I will briefly introduce the subject and show how one can derive some asymptotics for the skew-orthogonal polynomials.


mardi 6 février 2001: 16h00
Conférencier / Lecturer: J. Harnad, CRM & Université Concordia
Titre/Title: Multi-Hamiltonian structures, R-matrices and spectral separation of variables, I
Resume: A connection is made between: separation of variables in the spectral Darboux coordinates naturally associated to isospectral Lax matrix flows on finite dimensional Poisson subspaces of loop algebras in the rational R-matrix setting and: separation of variables in the « Nijenhuis coordinates » associated with multi-Hamiltonian systems. This is extended to multi-Hamiltonian structures related to trigonometric and elliptic R-matrices, and to the quadratic (Sklyanin) brackets on loop groups, viewed as Poisson Lie groups.


mardi 13 février 2001: 16h00
Conférencier / Lecturer: Bertrand Eynard, SPHT Saclay (France)
Titre / Title: O(n) Random matrix models.
Resume: The O(n) model is a famous toy model for 2D statistical physics. When put on a random lattice, the O(n) model is coupled to gravity, and the partition function can be represented by a matrix integral. The large n limit of that integral can be computed, and the results involve elliptic functions even in the one cut-case (because there is another « ghost » cut). The O(n) model is very rich because it interpolates all the possible (p,q) conformal minimal models, as well as non-rational cases.


mardi 20 février 2001: 16h00
Conférencier / Lecturer: Jacques Hurtubise, CRM & Univ. McGill
Titre / Title: Multi-Hamiltonian structures, R-matrices and spectral separation of variables, II


mardi 27 février 2001: 16h00
Conférencier / Lecturer: Oksana Yermolayeva, CRM & Concordia
Titre / Title: A review of the f-g method in orthogonal polynomials.


mardi 6 mars 2001: 16h00 – ANNULÉ

Conférencier / Lecturer: Oleg Bogoyavlenski, Queens University, Kingston
Titre/Title: Infinite dimensional Lie group of symmetries of equations of physical significance
Resume: An infinite dimensional Lie group $G$ of symmetries of the magnetohydrodynamics equilibrium equations is introduced that generates continuous families of new equilibrium solutions from any known ones. The Lie group $G$ depends upon the topology of the magnetic surfaces for a given equilibrium and is parametrized by arbitrary smooth functions on the corresponding graph $\Gamma$.

 


mardi, le 13 mars: 16h00
Conférencier / Lecturer : A. Zhedanov (Univ. Donetsk, CRM)
Titre/Title: « Integrable chains, algorithms and orthogonality I. & II »
Resume/abstract: We describe relations between Darboux transformations, numeric algorithms in linear algebra, integrable systems like Toda and relativistic Toda chains, and orthogonality properties of corresponding eigenfunctions. In particular, we present a new class of rational functions which are biorthogonal on elliptic grids.


mardi, le 20 mars: 16h00
Conférencier / Lecturer : A. Zhedanov (Univ. Donetsk, CRM)
Titre/Title: « Integrable chains, algorithms and orthogonality I. & II »
Resume/abstract:
We describe relations between Darboux transformations, numeric algorithms in linear algebra, integrable systems like Toda and relativistic Toda chains, and orthogonality properties of corresponding eigenfunctions. In particular, we present a new class of rational functions which are biorthogonal on elliptic grids.


jeudi, le 22 mars: 15h30 (session supplémentaire)
Conférencier C. Klein, Institut für Theoretische Physik Eberhard-Karls-Universität, Tübingen
Titre/Title : Relativistic dust disks and hyperelliptic Riemann surfaces
Resume: Infinitesimally thin disks of pressureless matter, so called dust, are discussed in astrophysics as models for certain galaxies and the matter in accretion disks around black holes. Since the vacuum Einstein equations in the stationary axisymmetric case are equivalent to the completely integrable Ernst equation, global spacetimes can be constructed for these models. The matter in the disk leads to a boundary value problem for the Ernst equation which can be treated with Riemann-Hilbert techniques. In the scalar case this leads to the Poisson integral. The matrix Riemann-Hilbert problem can be gauge transformed to a scalar problem on a Riemann surface. In the case of rational jump data, this surface is compact and the corresponding solutions to the Ernst equation form a subclass of Korotkin’s hyperelliptic solutions. Within this class one can study which boundary value problems can be solved on a given Riemann surface. As an example we discuss a family of disks made up of two counterrotating dust components. The complete metric is given explicitly in terms of hyperelliptic functions which are evaluated numerically.


mardi, le 27 mars : 16h00
Conférencier / Lecturer : P Bracken (CRM)
Titre/Title : Symmetries, Integrability and MultiSoliton Solutions of the Generalized Weierstrass System.
Resume/Abstract:

The Generalized Weierstrass system for inducing minimal surfaces in R^3 as proposed by B Konopelchenko will be introduced. The integrability of the system has been studied, in particular, by using a specific transformation, the initial system can be transformed into the completely integrable two-dimensional Euclidean nonlinear sigma model. The group invariant solutions of the sigma model system have been classified, and we briefly outline how this is done. Of more interest is that these results lead to very complicated new multisoliton solutions. It is shown how conditional symmetries lead to an Auto-Backlund for the system, from which the Theorem of Permutability can be formulated. Finally, we outline how this work can be extended to surfaces immersed in R^4, and give some new multisoliton solutions, and discuss a class of vortex solution.


mardi, le 3 avril : 16h00
ATTENTION! Lieu: (exceptionellement) Concordia Library Building LB 450 (1400 de Maisonneuve O.)
Conférencier/Speaker: Chongying Dong, University of California, Santa Cruz
Titre / Title: Monster, Moonshine and Vertex (Operator) Algebras
Resume / Abstract: The Monster is the largest sporadic finite simple group. Moonshine is the relationship between the monster and modular functions. Vertex operator algebras are a new class of algebraic structures which have recently arisen in mathematics and physics. In this talk we will review the Mckay-Thompson-Conway-Norton moonshine conjecture and discuss how Borcherds proved the conjecture for the Frenkel-Lepowsky-Meurman’s moonshine vertex operator algebra by using the monster Lie algebra. We will also present recent developments in orbifold conformal field theory and Norton’s generalized Moonshine conjecture.


mardi, le 10 avril : 16h00
Conférencier/Speaker: Dmitri Korotkin (Univ. Concordia, CRM)
Titre/Title: Isomonodromic deformations and Hurwitz spaces: tau-function and determinant of Laplacian operator
Resume: We discuss a class of isomonodromic deformations associated to Hurwitz spaces. A solution of the associated Riemann-Hilbert problem is given in terms of a Szego reproducing kernel. Calculation of isomonodromic tau-function of Jimbo, Miwa et al reveals a close link with the determinant of the Laplacian and Cauchy-Riemann operators.



mardi, le 17 avril : 16h00
Conférencier/Speaker : Dmitri Korotkin (Univ. Concordia, CRM)
Titre/Title: Isomonodromic deformations and Hurwitz spaces: tau-function and determinant of Laplacian operator II.
Resume / Abstract: We discuss a class of isomonodromic deformations associated to Hurwitz spaces. A solution of the associated Riemann-Hilbert problem is given in terms of a Szego reproducing kernel. Calculation of isomonodromic tau-function of Jimbo, Miwa et al reveals a close link with the determinant of the Laplacian and Cauchy-Riemann operators.



mardi, le 24 avril : 16h00
Conférencier/Speaker: Anna Krasowska (Univ. Concordia)
Titre/Title: Wigner functions for semidirect product groups R^n \rtimes H
Resume / Abstract: In this talk we consider semidirect product groups R^n \rtimes H admitting open free H-orbits in \hat R^n (dual to R^n). We give a classification of such groups in dim n=3. Their square -integrable representations give a basis for a construction of general Wigner functions, a useful tool in signal analysis and quantum optics. We also discuss the relation between wavelets and Wigner functions.



mardi, le 1er mai : 16h00
Conferencier/Speaker: Marco Bertola (CRM, Univ. Concordia)
Titre/Title: Duality in Random Matrices and Biorthogonal Polynomials
Resume/Abstract : Correlation functions and spacing distributions in two-matrix models may be computed as determinants involving « integrable » Fredholm kernels. These may be expressed in the case of 2-matrix models by a generalized Darboux-Christoffel fomula consisting of finite sums over biorthogonal sequences of quasi-polynomials and their Fourier-Laplace transforms. These in turn give rise to representations of the Heisenberg commutation relations for the shift operators which in the case of measures that are exponentials of polynomials, are finite band semi-infinite matrices of band sizes equal to the degrees of the polynomials defining the measure. These representations may be « folded » and used to determine « dual pairs » of covariant derivative operators involving matrices having the size of the band in one of the shift operators, with entries that are polynomials of degree equal to the size of the band of the dual operator. Interchanging the two, it is shown that the resulting characteristic polynomials are identical. Deforming the measure within this class gives rise to commuting flows that preserve the monodromies of both the operators.

 

Date                                                       Le mardi 18 septembre 2001
Heure/Time:                                            16h00
Conférencier / Lecturer:                      Ahmed Sebbar , Univ. Bordeaux, France
Titre / Title:                                              Capacities, Jacobi Matrices, & Jacobi Forms
Résumé/Abstract:

Dans cet exposé, nous nous intéréssons au calcul de la capacité d’une union finie $\Sigma$ d’intervalles réels. Dans un  premier temps on considère une matrice de Jacobi J, périodique et à l’aide d’un analogue du théorème de Burchnall et  Chaundy nous montrons que le spectre $\sigma(J)$ de J est en général une union finie d’intervalles dont nous  donnons la capacité et une sous suite de polynômes de Tchebychev . Ceci rejoint agréablement certaines idées de R.  Robinson. Inversement, à une union finie $\Sigma$ d’intervalles, on associe une matrice de Jacobi dont dont le spectre est $\Sigma$. Une partie de cette construction suit les idée de Date, Tanaka, Krichever etc….mais l’ingrédient principal est  la considération des points critiques de la fonction de Green avec pôle à l’infini du complémentaire de $\Sigma$.

 L’analyse de ces points nécéssite l’introduction de la fonction thêta de Riemann d’une courbe hyperellptique associée à  $\Sigma$ . Enfin, on étudie la dynamique de ces points d’équilibre lorsqu’on permet au pôle de la fonction de Green de bouger. Dans le cas où $\Sigma$ est une union de deux intervalles, on montre que le point d’équilibre varie entre deux  positions limites liés à l’étude de l’équation $\wp(z,\tau) = -\frac{\pi^2}{3} E_2(\tau)$. La solution de cette équation adapte les idées d’Eichler et Zagier sur les formes de Jacobi et emploie des fonctions introduites par Ramanujan.


Date                                                       Le mardi 25 septembre 2001
Heure/Time:                                            16h00
Conférencier / Lecturer :                     Misha B. Sheftel, St. Petersburg et Istanbul
Titre/Title:                                               « Method of group foliation, non-invariant solutions of the heavenly
                                                                   equation and heavenly metrics« 
Résumé/Abstract:

Using the heavenly equation as an example, we propose the method of group foliation as a tool for obtaining non-invariant solutions of PDEs with infinite-dimensional symmetry groups. The method involves the study of compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup and therefore selects exactly one orbit of solutions. By studying the integrability conditions of this automorphic system, {\it i.e.} the resolving equations, one can provide an explicit foliation of the entire solution manifold into separate orbits. The new important feature of the method is the extensive use of the operators of invariant differentiation and their commutator algebra for the derivation of the resolving equations and for obtaining their particular solutions. Applying this method we obtain exact analytical solutions of the heavenly equation, non-invariant under any subgroup of the symmetry group of the equation. They generate exact solutions of the Einstein field equations with only one rotational Killing vector. These metrics belong to the general category of gravitational instantons with Euclidean, or ultra-hyperbolic signature. They are derived from non-trivial solutions of the heavenly equation.


 

Série spéciale de 3 conférences
par le professeur A. N. Tyurin
(exceptionnellement les mercredis)

« Theta functions and applications in physics »

1.
Date                                                          Le mercredi 26 septembre 2001
Heure/Time:                                            15h00
Conférencier / Lecturer:                      Prof. A. Tyurin, Steklov Mathematical Institute, Russian Academy of sciences, Moscow
Titre / Title:                                              Quantization and theta-functions
Résumé/Abstract:

Geometric Quantization links holomorphic geometry with real geometry, a relation that is a prototype for the modern development of mirror symmetry. We show how to use this treatment to construct a special basis in every space of conformal blocks. This is a direct generalization of the basis of theta functions with characteristics in every complete linear system on an Abelian variety. We also discuss the geometry behind these constructions.

2.
Date                                                          Le mercredi 3 octobre 2001
Heure/Time:                                            16h00
Conférencier / Lecturer:                      Prof. A. Tyurin, Steklov Mathematical Institute, Russian Academy of sciences, Moscow
Titre / Title:                                              Symplectic geometry of moduli spaces of vector bundles
Résumé/Abstract:

This lecture combines algebraic and Lagrangian geometry to construct a special basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We can apply these bases to compare the Hitchin connection with the KZ connection defined by the monodromy of the Knizhnik-Zamolodchikov equation in combinatorial theory.

3.
Date                                                          Le mercredi 10 octobre 2001
Heure/Time:                                            16h00
Conférencier / Lecturer:                      Prof. A. Tyurin, Steklov Mathematical Institute, Russian Academy of sciences, Moscow
Titre / Title:                                              Non-abelian theta-functions
Résumé/Abstract:

In this lecture we explain how to use the Coherent States Transform (CST) for the analytical construction of non-abelian theta-functions.


• Conférence Spéciale •
Date                                                       Le jeudi 27 septembre 2001
Heure/Time:                                            15h30
Conférencier/Lecturer:                        Peter E. Hydon, University of Surrey, UK
Titre/Title:                                              « Direct construction of conservation laws for difference equations« 
Résumé/Abstract:
Conservation laws for differential equations can be constructed directly with the aid of the variational complex, which is a generalization of the well-known de Rham complex. Unlike Noether’s theorem and its variants, the direct method makes no explicit use of symmetries. Furthermore, it does not require the existence of an underlying Lagrangian or Hamiltonian structure. Therefore the direct method is able to yield conservation laws for a large class of differential equations. This talk introduces an analogue of the direct method for difference equations. It is based on a variational complex in which the independent variables are integer-valued. This new direct method differs substantially from the method for differential equations (for topological reasons). Nevertheless, with the aid of computer algebra, the new method can be used to construct conservation laws of difference equations in a systematic manner.


Date                                                      Le mardi 2 octobre 2001
Heure/ Time :                                          16h00
Lieu                                                           E.G.Kalnins, University of Waikato, New Zealand
Titre/Title:                                                « Perturbations of Black holes and special functions »
Résumé/Abstract:
It is known that perturbations of black holes for which not all defining parameters (i.e.mass,angular momentum and charge) are nonzero can be computed using Debye potentials which are special functions of confluent Heun type.There is however no scheme for the corresponding solution of the perturbation problem for a massive rotating black hole.In this talk we discuss how this problem may be solved using the idea of a symmetry operator and an integral equation formulation.In addition we give a summary of the geometric features of black hole space times which account for some of their remarkable properties.


Date                                                      Le mardi 9 octobre 2001
Heure/Time:                                            16h00
Conférencier/Lecturer:                        Michel Grundland
Titre/Title:                                               « Sur Certains Aspects Géométriques des Applications du Model Sigma CP2. »

Résumé/Abstract:
Nous donnons une généralisation du sysème d’équations de Weierstrass qui correspond aux applications harmoniques CP2. Cette généralisation nous permet d’étudier les surfaces à deux dimensions plongées dans un espace R8 avec une métrique euclidienne. Nous utilisons ce système pour suggérer une interprétation géometrique des applications CP2 harmoniques.


Date                                                      Le mardi 16 octobre 2001
Heure/Time:                                            15h30
Conférencier/Lecturer:                       Aleksander Strasburger, Université de Bialystok
Title\ title :                                               « On the ordering problem in QM and its connection with
                                                                   certain classes of Orthogonal Polynomials« 
     in collaboration with Ewa Gnatowska
Résumé/ Abstract :
We present an application of the duality technique of R. Howe to study the structure of the Weyl algebra. We introduce a one-parameter family of  « ordering maps », where by an ordering map we understand a vector space isomorphism of the polynomial algebra onR2d with the Weyl algebra generated by creation and annihilation operators a1,…,ad, a1+,…, ad+. Corresponding to these orderings, we construct a one-parameter family of  sl2 actions on the Weyl algebra, what enables us to define and study certain subspaces of the Weyl algebra — the space of Weyl spherical harmonics and the space of  « radial polynomials ». For the latter we generalize results of Biedenharn and Louck, Bender et al., and Koornwinder describing the radial elements in terms of continuous Hahn polynomials of the number operator.


Date                                                       Le mardi 23 octobre 2001
Heure/Time:                                            15h30
Conférencier/Lecturer:                        David Calderbank, University of Edinburgh
Titre/Title:                                              « Integrable background geometries in dimensions one to four« 


Date                                                       Le jeudi le 1er novembre 2001
Heure/Time:                                            16h00
Conférencier/Lecturer:                        Henrik Aratyn, University of Illinois
Titre/Title:                                                « Symmetries of Integrable Models and Applications
                                                                   to the Witten–Dijkgraaf–Verlinde–Verlinde Equations »

Résumé/Abstract:
The Riemann-Hilbert approach to integrable models is used to describe the symmetry structure of a class of integrable models. The symmetry flows obtained via this approach provide canonical coordinates for the Darboux-Egoroff system related to the Witten–Dijkgraaf–Verlinde–Verlinde equations of topological field theory. The construction involves dressing matrices expressed in terms of the tau function. The Connection between integrable systems and topological field theories will be used to impose the Virasoro constraints on the partition functions of the topological field theory.


Date                                                       Le vendredi 2 novembre 2001
Heure/Time:                                            15h30
Lieu:                                                          CRM – Pavillon Aisenstatd salle 6214
                                                                   1400 de Maisonneuve O., Library Building, salle LB-510
Conférencier/Lecturer:                        Chris Woodward, Rutgers Univ.
Titre/Title:                                                « Eigenvalues of products of unitary matrices,
                                                                   fusion polytopes, and quantum Schubert calculus »

Responsable:                                       Chris Cummins


Le mardi 6 novembre 2001

Pendant l’atelier de l’année thématique du CRM sur les Groupes de Lie de dimension infinie (2-6 novembre), le séminaire de physique mathématique fera relâche et sera remplacé par les conférences présentées lors de l’atelier qui auront lieu dans la salle 6214 du Pavillon A. Aisenstadt. À la date usuelle de cette semaine, la conférence présentée sera la suivante:

Date                                                       Le mardi 6 novembre 2001
Heure/Time:                                            15h30
Conférencier/Lecturer:                        V. Ovsienko, CPT, CNRS Luminy, Univ. d’Aix-en-Provence III
Titre/Title:                                                « Differential operators on tensor densities »

Programme (les personnes intéressées sont invitées à s’y inscrire)


Date                                                       Le mardi 13 novembre 2001
Heure/Time:                                            15h30
Conférencier/Lecturer:                        Decio Levi, Universita Roma Tre
Titre/Title:                                                « Multiscale Reduction for Differential Difference equations and Integrability. »

Résumé/ Abstract:
Leon and Manna proposed in 1999 a multiscale procedure to reduce nonlinear differential-difference equations. Applying this procedure to the Toda lattice equation they obtained a new nonlinear differential- difference equation which, in the continuous limit goes over to the Nonlinear Schroedinger equation (NLS). By analyzying a general class of coupled nonlinear differential difference real equations, which contains the new obtained discrete NLS, we prove that it has no local generalized symmetries of sufficiently high order. This shows that this equation has not the same integrability level as the discrete NLS derived by Ablowitz and Ladik.


Date                                                      Le mardi 20 novembre 2001
Heure/Time:                                            15 h 30
Conférencier/Lecturer:                       
Aleksander Orlov , Institute of Oceanology, Moscow and CRM
Titre/Title:                                                « Hypergeometrical tau-functionsty. »

Résumé/ Abstract:
We shall present a certain class of tau-functions which we call hypergeometrical tau-functions. They include hypergeometrical functions of matrix argument and their q-deformed versions, considered by Milne. These functions can be viewed as a result of the action of P-infinity symmetries on the vacuum solution of the KP hierarchy. They can also be obtained as certain 2-cocycles evaluated on a pair of pseudodifferential operators on a circle. We shall demonstrate links with isomonodromy problems. Linear equations, integral representations and determinant representations can be obtained via an explicit fermionic representation of this class of tau-functions. Links with matrix models and Painleve equations will be presented. The work mainly based on the joint papers with D.Scherbin, and also on discussions with J.Harnad.


Date:                                                       Le jeudi 22 novembre 2001
Heure/Time:                                            14h00-15h30
Conférencier/Lecturer:                       Aleksander Orlov , Institute of Oceanology, Moscow and CRM
Titre/Title:                                                « Hypergeometrical tau-functions. II »

Résumé/Abstract:
This is a continuation of the talk given on Tues., Nov. 20. Special constructions of both KP and 2-Toda tau functions leading to different forms of generalized hypergeometric type functions in many variables (e.g., Milne’s hypergeometric series) will be discussed, within the context of free fermionic field operators. Applications to integrable systems and related problems will be demonstrated.


Date:                                                       Le mardi 27 novembre 2001
Heure/Time:                                            15h30
Conférencier/Lecturer:                        Eyal Markman

Titre/ Title:                                                Elliptic Sklyanin integrable systems for arbitrary reductive groups.

Resume/ Abstract:
We present the analogue, for an arbitrary complex reductive group G, of the elliptic integrable systems of Sklyanin (Joint work with J. Hurtubise). The Sklyanin integrable systems were constructed on symplectic leaves of a quadratic Poisson structure on a loop group of type A. Etingof and Varchenko constructed poisson structures on loop groupoids, using solutions of the Classical Dynamical Yang-Baxter Equation. Our construction involves an algebro-geometric analogue of the Etingof-Varchenko poisson structure. The phase space of the integrable systems is the moduli space of pairs (P,f), where P is a principal G-bundle on an elliptic curve E, and f is a meromorphic section of the adjoint group bundle. The Sklyanin system is the special case, when G is PGL(n), and the bundle is rigid.

 


Date:                                                       Le mardi le 4 décembre 2001
Heure/Time:                                            15 h 30

Conférencier/Lecturer:                       
Ray McLenaghan, University of Waterloo
Titre/Title:                                                « Group invariant classification of separable Hamiltonian systems in the Euclidean plane« .

Résumé/Abstract:
An apparently new and effective method of determining separable coordinate systems for natural Hamiltonians in the Euclidean plane is presented. The method is based on intrinsic properties of the associated Killing tensors and their invariants under the group of rigid motions. Applications to the O(4)-symmetric Yang-Mills theories are considered.


Date:                                                       Le mardi 11 décembre 2001
Heure/Time:                                            15 h 30
Conférencier/Lecturer:                        Zuzana Masakova, CRM
,
Titre/Title:                                                « Combinatorial properties of cut and project sequences »
Resume/Abstract:

We study certain combinatorial properties of infinite binary and ternary words associated to cut-and-project sequences. We show that the infinite words have linear complexity, i.e. number of blocks of length n is linear in n. We study combinatorial equivalence of the infinite words and determine conditions under which the words are invariant under a morphism.


Date:                                                       Le mardi 18 décembre 2001
Heure/Time:                                            15 h 30
Conférencier/Lecturer:                       Dmitri Korotkin Concordia et CRM

Titre/Title:                                             « New integrable systems on spaces of rational maps »
Resume/Abstract:

We obtain a class of new integrable systems of partial differential equations associated to each stratum of the space of rational maps. These systems are natural generalization of the Ernst equation from general relativity. the relationship to Riemann-Hilbert problems, isomonodromic deformations and Frobenius manifolds is discussed, together with a possible generalization to Hurwitz spaces of higher genus.

 

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