Probability, Geometry and Integrable Systems

The three main themes of this book, probability theory, differential geometry, and the theory of integrable systems, reflect the broad range of mathematical interests of Henry McKean, to whom it is dedicated. Written by experts in probability, geometry, integrable systems, turbulence, and percolation, the seventeen papers included here demonstrate a wide variety of techniques that have been developed to solve various mathematical problems in these areas. The topics are often combined in an unusual and interesting fashion to give solutions outside of the standard methods. The papers contain some exciting results and offer a guide to the contemporary literature on these subjects. "The three main themes of this book<-->probability theory, differential geometry, and the theory of integrable systems<-->reflect the broad range of mathematical interests of Henry McKean, to whom it is dedicated. Written by experts in probability, geometry, integrable systems, turbulence, and percolation, the 17 papers included here demonstrate a variety of techniques that have been developed to solve various mathematical problems in these areas. The topics are often combined in an unusual fashion to give solutions outside of the standard methods. A few specific topics explored are stochastic evolution of inviscid Burger fluid, singular solutions for geodesic flows of Vlasov moments, and reality problems in soliton theory." --Book News1. Direct and inverse problems for systems of differential equations Damir Arov and Harry Dym; 2. Turbulence of a unidirectional flow Bjorn Birnir; 3. Riemann-Hilbert problem in the inverse scattering for the Camassa-Holm equation on the line Anne Boutet de Monvel and Dimtry Shepelsky; 4. The Riccati map in random Schrodinger and matrix theory Santiago Cambronero, Jose Ramirez and Brian Rider; 5. SLE6 and CLE6 from critical percolation Federico Camia and Charles M. Newman; 6. Global optimization, the gaussian ensemble and universal ensemble equivalence Marius Costeniuc, Richard S. Ellis, Hugo Touchette and Bruce Turkington; 7. Stochastic evolution of inviscid Burger fluid Paul Malliavin and Ana Bela Cruzeiro; 8. A quick derivation of the loop equations for random matrices N. M. Ercolani and K. D. T.-R. McLaughlin; 9. Singular solutions for geodesic flows of Vlasov moments J. Gibbons, D. D. Holm and C. Tronci; 10. Reality problems in soliton theory Petr G. Grinevich and Sergei P. Novikov; 11. Random walks and orthogonal polynomials; some challenges F. Alberto Grunbaum; 12. Integration of pair flows of the Camassa-Holm hierarchy Enrique Loubet; 13. Landen survey Dante V. Manna and Victor H. Moll; 13. Lines on abelian varieties Emma Previato; 14. Integrable models of waves in shallow water Harvey Segur; 15. Nonintersecting brownian motions, integrable systems and orthogonal polynomials Pierre Van Moerbeke; 16. Homogenization of random Hamilton-Jacobi-Bellman equations S. R. S. Varadhan.