Martingales and Painleve Equations

主要研究者将继续研究他的猜想，即函数具有一种新型的概率结构：与函数的一阶偏导数的某些组合相关联，有两个旋转场，以及两个相互之间是鞅变换的鞅，从等模常数开始，并在旋转这些导数组合后得到的结果结束。当前的重点是研究PI为创建鞅而开发的算法;如果它可以对每个函数得出结论，那么这将证明猜想。这个猜想的动机是平面上Beurling-Ahlfors变换的尖锐范数问题，以及它在空间上的推广，但它有更广泛的范围，包含更强的不等式，并提供了对函数几何行为的进一步了解。首席研究员发现，在三维空间中，这个猜想有一个新的物理解释，即静电场和磁场之间的关系：它们也可以通过旋转和一对鞅联系起来。其次，主要研究者提出研究Painleve微分方程解的增长阶和其他全局性质。该项目的第一部分致力于创建新的结构，将分析和概率论结合在一起，并与三维静态电磁场的物理学相联系。工作的第二部分的建议地址的性质Painleve超越，一类重要的非线性特殊功能。Painleve方程以多种方式出现在当前数学其他部分的工作中，如微分几何和随机矩阵。它们正在被物理学、天文学和工程学领域的大量科学家用于正在进行的工作，例如在以下领域：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，等离子体物理学，超导性，非线性光学和光纤光学，聚合物，聚电解质和胶体，物理学中的伊辛模型，反铁磁模型、量子场论和拓扑场论中的关联函数。Painleve方程统一了许多领域，因为它们为非线性常微分方程和偏微分方程提供了可积性。该项目有助于增加我们对Painleve超越行为的了解，并应使那些应用这些方程的人受益。PI的研究生将参与项目的两个部分。

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.