The principal investigator will continue working on his conjecture that functions have a new type of probabilistic structure: associated with certain combinations of first 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. The immediate focus is the study of the algorithm developed by the PI for creating the martingales; if it can be taken to its conclusion for each function, then that will prove the conjecture. The conjecture is motivated by the problem of the sharp norm of the Beurling-Ahlfors transformation in the plane, and its generalizations to space, but it has a wider scope and implies even stronger inequalities and provides further insight into the geometric behavior of functions. The principal investigator has found that in three dimensions, the conjecture has a new physical interpretation as a relation between the static electric and magnetic fields: they also would be related by rotations and a pair of martingales. Secondly, the principal investigator proposes to study the order of growth and other global properties of the solutions to the Painleve differential equations.
The first part of this project strives to create new structures tying together analysis and probability theory, with a connection to the physics of the static electromagnetic field in dimension three. Work in the second part of the proposal addresses the properties of Painleve transcendents, an important class of nonlinear special functions. Painleve equations appear in numerous ways in current work on other parts of mathematics, such as differential geometry and random matrices. They are being used in ongoing work by a large number of scientists in physics, astronomy, and engineering, for example in the following areas: bubble break-off and other Hele-Shaw problems in incompressible viscous fluids, resonant oscillations in shallow water, general relativity and cosmology, plasma physics, superconductivity, non-linear optics and fiber optics, polymers, polyelectrolytes, and colloids, the Ising model in physics, correlations functions in an antiferromagnet model, quantum field theory and topological field theory. Painleve equations unify many fields as they provide a connection to integrability for non-linear ordinary and partial differential equations. This project contributes to increasing our knowledge of the behavior of the Painleve transcendents and should benefit those applying these equations. The graduate students of the PI will be involved in both parts of the project.
The research performed under the grant by the principal investigator and his collaborators has increased our understanding of the properties of the solutions of the Painleve differential equations. The Painleve differential equations are those second order non-linear ordinary differential equations whose solutions have no movable singularities other than poles and whose general solutions cannot be expressed in terms of elementary functions or classical special functions. Their essentially new solutions are now viewed as non-linear analogues of the classical special functions. On the one hand, in pure mathematics, the Painleve equations are related to the concept of integrability for non-linear ordinary and partial differential equations. Other applications in pure mathematics include particularly random matrix theory and differential geometry. On the other hand, in interdisciplinary mathematics, science, and engineering, the Painleve equations have numerous applications, such as the following: statistical phenomena including the Ising model in physics, statistical mechanics in elasticity, superconductivity, nonlinear and fiber optics, stimulated Raman scattering, polymers, and polyelectrolytes. Thus the reason for the strong explosion in Painleve-related research is the emerging high impact of these equations on both pure and applied mathematics and on applications to many sciences. The work under the grant has focused on the growth properties of the solutions of the second and fourth Painleve equations, and has yielded rigorous proofs for certain growth properties. While this work is part of pure mathematics, the information obtained is then also available for applications.
