We study the following problem: can a classical sln Toda field theory be represented by means of free bosonic oscillators through a Drinfeld–Sokolov construction? We answer affirmatively in the case of a cylindrical space–time and for real hyperbolic solutions of the Toda field equations. We establish in fact a one–to–one correspondence between such solutions and the space of free left and righ...

[UMSt] www.math.uci.edu/ ̃mfried/paplist-cov/UMStory.html. Algebraic equations in separated variables: (*) f(x)−g(y) = 0. Defines a projective nonsingular algebraic curve Xf,g with two projections to the (Riemann sphere) z-line Pz = C ∪ {∞}: f : Px → Pz and g : Py → Pz. Two problems from the 60s (Davenport’s and Shinzel’s) solved by the monodromy method . I explain that and the associated Genus ...

It is well known that the sixth Painlevé equation PVI admits a group of Bäcklund transformations which is isomorphic to the affine Weyl group of type D (1) 4 . Although various aspects of this unexpectedly large symmetry have been discussed by many authors, there still remains a basic problem yet to be considered, that is, the problem of characterizing the Bäcklund transformations in terms of R...

A spherical quadrilateral is a bordered surface homeomorphic to a closed disk, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in ...

Braid monodromy is an important tool for computing invariants of curves and surfaces. In this paper, the rectangular braid diagram (RBD) method is proposed to compute the braid monodromy of a completely reducible n-gonal curve, i.e. the curves in the form (y−y1(x))...(y−yn(x)) = 0 where n ∈ Z+ and yi ∈ C[x]. Also, an algorithm is presented to compute the Alexander polynomial of these curve comp...

We show that there exists a Lamé operator Ln with projective octahedral monodromy for each n ∈ 1 2 (N + 1 2 ) ∪ 1 3 (N + 1 2 ), and with projective icosahedral monodromy for each n ∈ 1 3 (N+ 1 2 )∪ 1 5 (N+ 1 2 ). To this end, we construct Grothendieck’s dessins d’enfants corresponding to the Belyi morphisms which pull-back hypergeometric operators into Lamé operators Ln with the desired monodro...

We study the interplay between discrete quantum symmetries at certain points in the moduli space of Calabi-Yau compactifications, and the associated identities that the geometric realization of D-brane monodromies must satisfy. We show that in a wide class of examples, both local and compact, the monodromy identities in question always follow from a single mathematical statement. One of the sim...

Given a closed hyperbolic 3-manifold Tψ that fibers over the circle with monodromy ψ : S → S, the monodromy ψ determines an isometry of Teichmüller space with its Weil-Petersson metric whose translation distance ‖ψ‖WP is positive. We show there is a constant K ≥ 1 depending only on the topology of S so that the volume of Tψ satisfies ‖ψ‖WP/K ≤ vol(Tψ) ≤ K‖ψ‖WP.

This article is based on a talk given in Dijon for the 2000 summer school “Topological and Geometrical Methods: Applications to Dynamical Systems”. The standard definitions of the monodromy invariant for completely integrable classical systems are reviewed, and the link to the quantum monodromy observed in the joint spectrum of commuting operators is explained. The mathematical treatment relies...

We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination in order to study properties of the braid monodromy of the image of curves under a given rational transformation. A description of the general method is give...