Related papers: Painlev\'e equations and deformations of rational …
We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with…
Nous montrons que les \'equations du rep\`ere mobile des surfaces de Bonnet conduisent \`a une paire de Lax matricielle isomonodromique d'ordre deux pour la sixi\`eme \'equation de Painlev\'e. We show that the moving frame equations of…
R.~Lawrence has conjectured that for rational homology spheres, the series of Ohtsuki's invariants converges p-adicly to the SO(3) Witten-Reshetikhin-Turaev invariant. We prove this conjecture for Seifert rational homology spheres. We also…
We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. We replace smooth K-oriented…
The theory of orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference…
This is a generalization of the procedure presented in [3] to construct semisimple bi-flat $F$-manifolds $(M,\nabla^{(1)},\nabla^{(2)},\circ,*,e,E)$ starting from homogeneous solutions of degree -1 of Darboux-Egorov-system. The Lam\'e…
The aim of this article is to study deformation theory of trianguline B-pairs for any p-adic field. For benign B-pairs, a special good class of trianguline B-pairs, we prove a main theorem concerning tangent spaces of these deformation…
The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions…
We are interested in an anisotropic singular diffusion equation in the plane and in its regularization. We establish existence, uniqueness and basic regularity of solutions to both equations. We construct explicit solutions showing the…
In this survey we present the interpretation of isomondromy preserving equations on Riemann surfaces with marked points as reduced Hamiltonian systems. The upstairs space is the space of smooth connections of GL(N) bundles with simple poles…
We construct analytic surface symplectomorphisms with unstable elliptic fixed points; this solves a problem of Birkhoff (1927). More precisely, we construct analytic symplectomorphisms of the sphere and of the disk which are transitive,…
We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer…
We provide a complete classification and an explicit representation of rational solutions to the fourth Painlev\'e equation PIV and its higher order generalizations known as the $A_{2n}$-Painlev\'e or Noumi-Yamada systems. The construction…
In this paper, we prove a result related to the deformation of complex submanifolds, modifying a result of Kodaira (Ann. Math, 75(1), 146-162, 1962).
Although the solutions of Painlev\'e equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions for specialized values of the equation parameters. A very…
We present a simple heuristic method to derive the Painlev\'e differential equations from the corresponding geometry of rational surafces. We also give a direct relationship between the cubic pencils and Seiberg-Witten curves.
The rational solutions of the Painlev\'e-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for…
The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by…
We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real…
By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the…