Martingales and Painleve Equations

Martingales 和 Painleve 方程

• 批准号: 0758226
• 负责人: Aimo Hinkkanen
• 金额: $ 15万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-15 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0758226&HistoricalAwards=false
• 关键词: Martingales; Painleve; Equations

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.

这个项目的动机思想是函数有一个迄今为止未知的概率结构：与函数的一阶偏导数的某些组合相关联，有两个旋转域，以及两个鞅，它们是彼此的鞅变换，从相等模的常数开始，并在旋转这些导数组合后得到的结果结束。找到这样的旋转场，结合伯克霍尔德为鞅变换开发的技术，使人们能够获得某些奇异积分算子的尖锐范数的约束值，这些奇异积分算子在复分析和调和分析中占据中心位置：平面中的Beurling-Ahlfors变换及其在空间中的推广。首席研究员将致力于将他之前的结果扩展到一般情况，表明在某些情况下存在这种旋转和鞅。首席研究员还将研究第二和第四Painleve方程的解的增长顺序。这六个Painleve微分方程是不具有可移除奇点的二阶微分方程的原型。它们最近在纯数学和应用数学以及科学和工程方面产生了越来越大的影响。根据这一建议进行的工作应该导致新结构的发展，将分析，概率论和物理学联系在一起：主要研究者发现，在三维空间中，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。它还将导致更好地理解和具体的结果，为重要的Painleve类的非线性微分方程，这是在许多应用中使用的其他领域的数学以及在物理和工程。Painleve方程被认为与非线性常微分方程和偏微分方程的可积性概念有关。在纯数学中的其他应用包括研究Bonnet曲面和随机矩阵。在交叉学科数学和其他科学中，Painleve方程和超越的众多应用包括以下领域：物理学中的伊辛模型;反铁磁模型中的相关函数;量子场论和拓扑场论;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;等离子体物理学;超导性;非线性光学和纤维光学;聚合物、聚电解质和胶体。