Martingales and Painleve Equations

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The motivating idea of this project is that functions have a hitherto unknown probabilistic structure: associated with certain combinations of first-order partial derivatives of functions, there are two fields of rotations, and two martingales that are martingale transforms of each other, starting from constants of equal modulus, and ending at what one obtains after rotating these combinations of the derivatives. Finding such rotation fields would, in combination with techniques developed by Burkholder for martingale transforms, enable one to obtain the conjectured values for the sharp norms of certain singular integral operators that occupy a central place in complex and harmonic analysis: the Beurling-Ahlfors transformation in the plane and its generalizations in space. The principal investigator will work on extending to the general case his previous results showing that in certain cases such rotations and martingales exist. The principal investigator will also study the order of growth of the solutions to the second and fourth Painleve equations. The six Painleve differential equations are prototypes of second-order differential equations that do not have removable singularities. They have recently had an increasing impact in pure and applied mathematics and in science and engineering. Work performed under this proposal should lead to the development of new structures tying together analysis, probability theory, and physics: the principal investigator has found that, in three dimensions, the martingale conjecture has an interesting physical interpretation as a relation between static electric and magnetic fields. It will also lead to a greater understanding of and concrete results for the important Painleve class of nonlinear differential equations, which is being used in numerous applications in other areas of mathematics as well as in physics and engineering. Painleve equations are considered to be related to the concept of integrability for nonlinear ordinary and partial differential equations. Other applications in pure mathematics include the study of Bonnet surfaces and random matrices. In interdisciplinary mathematics and other sciences, the numerous applications of the Painleve equations and transcendents include the following areas: the Ising model in physics; correlation functions in an antiferromagnet model; quantum field theory and topological field theory; general relativity and cosmology; supersymmetry gauge theories in physics; resonant oscillations in shallow water; Hele-Shaw problems in viscous fluids; plasma physics; superconductivity; nonlinear optics and fiber optics; polymers, polyelectrolytes, and colloids.