Martingales and Painleve Equations

主要研究人员将继续研究他的猜想，即函数具有一种新的概率结构：与函数的一阶偏导数的某些组合相关联，有两个旋转场，以及两个彼此的鞅变换，从等模常数开始，到旋转这些导数的组合后获得的值结束。眼下的焦点是研究PI开发的用于创建鞅的算法；如果对每个函数都能得出结论，那么就证明了这个猜想。这个猜想的动机是平面上的Beurling-Ahlfors变换的锐范数问题，以及它在空间中的推广，但它的范围更广，蕴含着更强的不等式，并提供了对函数几何行为的进一步洞察。首席研究人员发现，在三维空间，这个猜想有了新的物理解释，即静电场和磁场之间的关系：它们也将通过旋转和一对鞅联系在一起。其次，主要研究者建议研究Painleve微分方程解的增长阶和其他整体性质。这个项目的第一部分致力于创建将分析和概率理论结合在一起的新结构，并与三维静态电磁场的物理联系起来。提案第二部分的工作解决了Painleve超越函数的性质，这是一类重要的非线性特殊函数。Painleve方程以多种方式出现在当前关于数学的其他部分的工作中，例如微分几何和随机矩阵。大量物理、天文和工程科学家正在进行的工作中使用它们，例如在下列领域：不可压缩粘性流体中的气泡破裂和其他Hele-Shaw问题，浅水中的共振振荡，广义相对论和宇宙学，等离子体物理，超导，非线性光学和光纤光学，聚合物，聚电解质，胶体，物理中的伊辛模型，反铁磁模型中的关联函数，量子场论和拓扑场理论。Painleve方程统一了许多领域，因为它们提供了与非线性常微分方程和偏微分方程可积性的联系。这个项目有助于增加我们对Painleve超越者行为的了解，并将使那些应用这些方程式的人受益。国际和平研究所的研究生将参与该项目的两个部分。

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.