Organisation de la journée :
9h30-10h30 : C. Sabbah » Structures de Hodge non commutatives et structures tt* » (partie I)
10h30 : pause café
11h-12h: C. Sabbah » Structures de Hodge non commutatives et structures tt* » (partie II)
14h-15h: P. Boalch « Transformation groups for isomonodromy equations (partie I)
15h30-16h30: P. Boalch » Transformation groups for isomonodromy equations » (partie II)
Résumé des exposés:
Claude Sabbah : » Structures de Hodge non commutatives et structures tt* »
Résumé:
Après avoir rappelé ce qu’est le problème de Birkhoff pour un système différentiel linéaire d’une variable et la notion de déformation isomonodromique pour un système à singularité irrégulière, je considérerai un problème analogue, mais qui fait intervenir la conjugaison complexe. Ceci conduit à la notion de structure de Hodge non commutative et à ses déformations, qui donnent une variante \og intégrable\fg de la notion de variation de structure de twisteur introduite par Carlos Simpson (prépublication Univ. Toulouse, 1997). Je mentionnerai enfin la relation avec Painlevé III.
Philip Boalch : » Transformation groups for isomonodromy equations »
Résumé : The best-known isomonodromy system, the system of Schlesinger equations, appeared in Schlesinger’s ICM talk in 1908. The next significant generalisation appeared in the work of Jimbo-Miwa-Mori-Sato (JMMS) in 1980, in relation to the quantum nonlinear Schrodinger equation. In 1994 Harnad showed that the JMMS system admits a symmetry, not shared by the Schlesinger equations, nor by the isomonodromy systems of Jimbo-Miwa-Ueno (1981). In this talk I will describe some of the theory of simply-laced isomonodromy systems and their automorphisms/isomorphisms. These systems generalise the JMMS system (and are not included in the JMU system), and the isomorphisms generalise Harnad’s duality. As a special case we recover the Okamoto symmetries of the fourth, fifth and sixth Painleve equations, and put these symmetries into the larger context of Weyl groups for not-necessarily-affine Kac-Moody root systems. On one hand this explains why there are such symmetries, via the Fourier-Laplace transform, and on the other hand it shows where such exotic root systems occur in nature. The appearance of such root systems and Weyl groups seems to distinguish this theory from earlier work on soliton equations. As a corollary we may attach an isomonodromy system to any complete k-partite graph (for any k), and more generally to any « supernova » graph (and some data on the graph). In particular the Painleve equations 4,5,6 are attached to the triangle, square and four legged star respectively, as suggested by Okamoto’s work (that the affine Weyl groups of these graphs give the symmetry groups of these Painleve equations). The bipartite case k=2 gives the JMMS system. Further, by considering hyperbolic (doubly extended) Dynkin graphs, this viewpoint yields a higher Painleve X system of order 2n, for each n=1,2,3,…, where X=I,II,…,VI is any of the Painleve equations (which appear when n=1). These higher Painleve systems are distinct from the so-called « Painleve hierarchies ». As part of this project we also (define and) solve many irregular additive Deligne-Simpson problems, in terms of the Kac-Moody root system determined by the graph, extending Crawley-Boevey’s results in the case of star-shaped graphs. Main references: Simply-laced isomonodromy systems Publ. Math. IHES 116, No. 1 (2012) 1-68 (arXiv:1107.0874) Irregular connections and Kac-Moody root systems arXiv:0806.1050 (June 2008) From Klein to Painlevé via Fourier, Laplace and Jimbo (Section 3) Proc. London Math. Soc. (3) 90 (2005) 167-208 (arXiv:math/0308221, Aug 2003)