Hiroshi Umemura

Solutions of the third Painleve equation I

We classify transcendental classical solutions of the third Painleve equation. This result combined with the list of algebraic solutions in [11] gives a complete table of classical solutions of the third Painleve equation.

Solutions of the second and fourth Painlevé equations. I

A rigorous proof of the irreducibility of the second and fourth Painleve equations is given by applying Umemura’s theory on algebraic differential equations ([26], [27], [28]) to the two equations. The proof consists of two parts: to determine a necessary condition for the parameters of the existence of principal ideals invariant under the Hamiltonian vector field; to determine the principal invariant ideals for a parameter where the principal invariant ideals exist. Our method is released from complicated calculation, and applicable to the proof of the irreducibility of the third, fifth and sixth equation (e.g. [32]).

Special polynomials and the Hirota bilinear relations of the second and the fourth Painlevé equations

We give a purely algebraic proof that the rational functions Pn(t), Q n (t ) inductively defined by the recurrence relation (1), (2) respectively, are polynomials. The proof reveals the Hirota bilinear relations satisfied by the τ -functions.

Maximal algebraic subgroups of the Cremona group of three variables. Imprimitive algebraic subgroups of exceptional type

Enriques and Fano [4], [5] classified all the maximal connected algebraic subgroups of Cr3. Our aim is to give modern and rigorous proofs to their results. In [10], we studied the primitive subgroups. In this paper, we deal with exceptional imprimitive groups. The imprimitivity is an analytic notion. The natural translation of the imprimitivity in algebraic geometry is the de Jonquieres type operation (definition (2.1)). Every de Jonquieres type operation is imprimitive. However, the difference of these notions is subtle. The imprimitive algebraic operations in Cr3 which are not of de Jonquieres type are rather exceptional; there are only 3 such operations (theorem (3.26)). This paper together with [10] recovers all the results on Cr3 of Enriques and Fano [4]. It remains only to reconstruct Fanos classification [5] of the de Jonquieres type operations, which shall be done in our forthcoming paper. Our technique is rather old; the classification of 3 dimensional primitive operations, a very easy part of invariant theory which are of 19th century, combined with the theory of algebraic groups and transformation spaces of A. Weil. As the 4 dimensional primitive law chunks of analytic operations are classified, our method can be applied to the 4 dimensional Cremona group Cr4. We use the notations and the conventions of [10]. Therefore all manifolds, analytic groups, algebraic varieties, etc. are defined over C. The transformation spaces X of analytic or algebraic law chunk or operation (G, X) are connected. However, a differences lies in the language that we employ. Here is our French-English dictionary:

La dimension cohomologique des surfaces algébriques

En Geometrie Algebrique on a un critere pour qu’une surface moins une courbe soit affiine (Hartshorne (5)). Dans (5), on demande s’il existe un analogue analytique. Le but de cet article est de donner une condition numerique necessaire pour les surfaces complexes compactes (Theoreme 1) et une condition suffisante pour les surfaces reglees (Theoreme 2).

Invitation to Galois theory

We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic groups may be used to classify systems of second order linear differential equations.In [MV], some correspondences were defined between critical points of master functions associated to sl_{N+1} and subspaces of C[x] with given ramification properties. In this paper we show that these correspondences are in fact scheme theoretic isomorphisms of appropriate schemes. This gives relations between multiplicities of critical point loci of the relevant master functions and multiplicities in Schubert calculus.We give a description of the work of Andrey Bolibrukh on isomonodromic deformations and relate it to existing results in this domain. Résumé (Les travaux d’Andrei Bolibroukh sur les déformations isomonodromiques) Nous décrivons les travaux d’Andrëı Bolibroukh sur les déformations isomonodromiques en les situant dans le contexte des résultats existant dans ce domaine.Explicit solutions to the Riemann-Hilbert problem will be found realising some irreducible non-rigid local systems. The relation to isomonodromy and the sixth Painleve equation will be described. Keywords: Riemann-Hilbert problem, Painleve equations, algebraic solutions, Heun equations, tetrahedral/octahedral group, triangle groups, Belyi maps.

On the transformation group of the second Painlevé equation

We show that for the second Painleve equation y″ = 2y 3 + ty + α , the Backlund transformation group G , which is isomorphic to the extended affine Weyl group of type  1 , operates regularly on the natural projectification χ( c )/ℂ( c, t ) of the space of initial conditions, where c = α - 1/2. χ( c )/ℂ( c, t ) has a natural model χ[ c ]/ℂ( t )[ c ]. The group G does not operate, however, regularly on χ[ c ]/ℂ( t )[ c ]. To have a family of projective surfaces over ℂ( t )[ c ] on which G operates regularly, we have to blow up the model χ[ c ] along the projective lines corresponding to the Riccati type solutions.

Discrete Burgers’ Equation, Binomial Coefficients and Mandala

In discrete Burgers’ equation, time is discretized but space remains continuous. Reduction modulo 2 of discrete Burgers’ equation discretizes also space. This dynamical system generates an elegant pattern known since long time ago. We also propose to show that the Galois groups of particular solutions of Burgers’ equation are abelian. It would imply that the dynamical systems are integrable.