Martingales and Painleve Equations

这个项目的激励思想是，函数具有一种迄今未知的概率结构：与函数的一阶偏导数的某些组合相关，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得到的值结束。找到这样的旋转场，结合Burkholder发展的用于鞅变换的技术，将使人们能够获得在复调和分析中占据中心位置的某些奇异积分算子的尖峰范数的猜想值：平面上的Beurling-Ahlfors变换及其在空间中的推广。首席调查员将致力于将他以前的结果推广到一般情况，表明在某些情况下存在这样的旋转和鞅。首席研究员还将研究第二和第四Painleve方程的解的增长顺序。六个Painleve微分方程组是二阶微分方程组的原型，没有可移去的奇点。它们最近在纯数学和应用数学以及科学和工程领域产生了越来越大的影响。在这一提议下进行的工作应该会导致将分析、概率理论和物理学联系在一起的新结构的发展：首席研究员发现，在三个维度上，鞅猜想有一个有趣的物理解释，即静电场和磁场之间的关系。它还将导致对重要的Painleve类非线性微分方程的更好的理解和具体结果，这类方程正被用于数学的其他领域以及物理和工程中的许多应用。Painleve方程被认为与非线性常微分方程和偏微分方程组的可积性概念有关。纯数学中的其他应用包括研究帽子曲面和随机矩阵。在跨学科数学和其他科学中，Painleve方程和先验知识的大量应用包括以下领域：物理中的伊辛模型；反铁磁体模型中的关联函数；量子场论和拓扑场理论；广义相对论和宇宙学；物理学中的超对称规范理论；浅水中的共振振荡；粘性流体中的Hele-Shaw问题；等离子体物理；超导；非线性光学和光纤光学；聚合物、聚电解质和胶体。

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.