Gérard–Levelt membranes

We present an unexpected application of tropical convexity to the determination of invariants for linear systems of differential equations. We show that the classical Gérard–Levelt lattice saturation procedure can be geometrically understood in terms of a projection on the tropical linear space attached to a subset of the local affine Bruhat–Tits building, which we call the Gérard–Levelt membrane. This provides a way to compute the true Poincaré rank, but also the Katz rank of a meromorphic connection without having to perform either gauge transforms or ramifications of the variable. We finally present an efficient algorithm to compute this tropical projection map, generalising Ardila’s method for Bergman fans to the case of the tight-span of a valuated matroid.     

Mikhail Nikolaevich Lagutinskii (1871–1915): Un Mathématicien Méconnu

Dans cette note biographique nous rapportons, à quelques détails près, tout ce qui est connu de la vie de M. N. Lagutinskii. (Certains faits supplémentaires concernant M. N. Lagutinskii seront publiés dans la version russe de ce travail (en préparation).) Ce mathématicien russe, dans une série de travaux importants mais complètement méconnus, a développé substantiellement la méthode de Darboux de recherche, en termes finis, des intégrales premières de systèmes d'équations différentielles ordinaires polynomiales. Il a aussi développé la théorie de l'intégrabilité en termes finis de tels systèmes d'équations. Nous fournissons sa bibliographie complète. L'analyse détaillée du contenu mathématique de son œuvre est le sujet de la seconde partie ([30], en préparation) de ce travail. Elle paraı̂tra ultérieurement.Copyright 1998 Academic Press.In this biographical note, we report virtually all that is known about the life of the Russian mathematician, M. N. Lagutinskii. (Some additional facts concerning Lagutinskii will be published in the Russian version of this paper (to appear).) In a series of important but completely unrecognised works, Lagutinskii developed the Darboux method of determination in finite terms of the first integrals of systems of polynomial, ordinary differential equations. He also developed the theory of integrability in finite terms of such systems of equations. We provide his complete bibliography. A detailed analysis of the mathematical contents of his works is the subject of the second part [30] of the present study. It will appear at a later date.Copyright 1998 Academic Press.[formula]Copyright 1998 Academic Press.AMS 1991 subject classifications: 01A60, 01A70, 01A73, 34–03     

Géométrie Orthogonale Non Symétrique et Congruences Quadratiques (Nonsymmetric Orthogonal Geometry and Quadratic Congruences)

This paper is devoted to the study of quadratic congruences, which appear in the theory of exceptional bundles on projective spaces. A quadratic congruence associates to each point P of a projective space a quadratic in containing P. We study several geometric constructions of quadratic congruences and try to classify them.     

Rkk (X)-nucléarité (d‘après; G. Skandalis)

We define a notion of nuclearity for C(X)-C^*-algebras yielding the same results of semi-exactness, for Kasparovs bifunctor RKK(X), as those obtained by Skandalis in the case X=. We connect explicitly this new notion of nuclearity with those already known.