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The Painlevé Equations and Monodromy Problems

4th September 2006 to 29th September 2006

Organisers: Professor PP Boalch (ENS Paris), Professor PA Clarkson (Kent), Professor L Mason (Oxford), Professor Y Ohyama (Osaka)
Scientific Advisors: Professor B Dubrovin (SISSA), Professor AS Fokas (Cambridge), Professor B Malgrange (Grenoble), Dr M Mazzocco (Manchester) and Professor K Okamoto (Tokyo)

Programme Theme

The study of the Painlevé equations has progressed explosively in the last thirty years. At first, the Painlevé equations were studied mainly by the inverse scattering method. Isomonodromic deformations are one of the origins of the Painlevé equations, and the isomonodromy technique is a powerful method to investigate the Painlevé transcendents. Conversely, Painlevé analysis is useful for studying monodromy problems, especially the Riemann–Hilbert problem.

More recently, various new methods have been used to study the Painlevé equations, such as WKB methods, twistor theory, algebraic geometry of initial value spaces, affine Weyl groups, differential Galois groups, symplectic geometry of the moduli space of connections and so on. Equally, the Painlevé equations have many applications, for instance to soliton equations, random matrices, quantum field theory (especially topological field theory), differential geometry of self-dual metrics, etc. Moreover, the Painlevé equations themselves have been generalised to Schlesinger systems, discrete systems and q-difference systems.

H. Poincaré said of Painlevé, "Les mathématiqués constituent un continent solidment agencé, dont tous les pays sont bien reliés les uns aux autres; l'œuvre de Paul Painlevé est une île originale et splendide dans l'océan voisin," but Painlevé's work is now part of the continent and is related to other countries. In this programme, we expect that an understanding of different approaches will bring forth new developments in many fields of pure and applied mathematics. We will gather together experts with various backgrounds, and will share common knowledge.

Organisers Webpage

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons