Alumni Association
Professor Lionel Mason (University of Oxford)
18 November 2011
Joint Research Project
Project title: The Painleve Equations and Monodromy Problems
Japanese Lead Scientist: Professor Yousuke Ohyama, Graduate School of Information Science and Technology, Osaka University
UK Counterpart: Professor Lionel Mason Mathematical Institute, University of Oxford
Project Duration: September 2006
The Painlevé Equations and Monodromy Problems
4-29 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)
Basic theme and Background
The Painlevé equations, and their solutions, the Painlevé transcendants arise in many disparate parts of pure and applied mathematics and theoretical physics. Painlevé transcendents arise as partition functions in string theories, correlation functions in statistical mechanics, important solutions of differential equations from fluids and general relativity through to Einstein manifolds and monopole moduli spaces in differential geometry, and they arise as generating functions for the topology of moduli spaces of Riemann surfaces and for enumerative problems in algebraic geometry. The Painlevé equations are an integrable system, and therefore have much underlying structure, but, despite their integrability, much of the general theory is still in a somewhat embryonic stage. Indeed the solutions of the Painlevé equations themselves (the so-called Painlevé transcendents) are still some way from being understood as well as the more classical special functions.
Some of the outstanding problems in the field are to fully understand the asymptotics and solve all the corresponding connection problems of the Painlevé transcendents: for example if one has a Painlevé transcendent with a certain behaviour at zero, then can we say how it behaves at infinity? More generally, the Painlevé equations can be viewed as the simplest cases of equations controlling monodromy preserving deformations of linear differential operators on the Riemann sphere and one can ask the same questions for any such (nonlinear) isomonodromy equations.
Some less concrete problems arise in understanding the various applications the Painlevé equations have found, for example in Random matrix ensembles or the Tracey-Widom distribution controlling the largest increasing subsequences of random permutations. One can try for example to see directly an isomonodromic deformation in the original problem (which would explain clearly why we expected to find a Painlevé solution in the answer).
※ Click here to view the full report (pdf).
