Singular Integrals and Geometry

奇异积分和几何

• 批准号: 1401671
• 负责人: Brian Street
• 金额: $ 14.7万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401671&HistoricalAwards=false
• 关键词: Singular; Integrals; Geometry

In many natural questions in mathematics, there arise integrals that diverge when thought of in a classical sense, but which can be made sense of due to an underlying cancellation property (informally, they can be made sense of because they involve adding a "positive infinity" and a "negative infinity" that cancel each other out). These important objects are known as singular integrals. This project proposes to study several related kinds of singular integrals. A unifying aspect of the proposed questions is that they are all intimately connected to a geometry, and this project will study how this underlying geometry can be used to develop the correct notions of singular integrals. The questions proposed have many applications. They include direct applications to medical imaging, electrical impedance tomography, geophysical prospection, the rate at which fluids mix in certain situations, new directions in several complex variables, and new directions for multilinear operators that have underlying singular integrals. Furthermore, given the wide range of applications that singular integrals have found in the past, it is possible that the ideas developed for this project may have many other applications in physics and mathematics beyond those mentioned above.There are five main, interrelated questions in this project. The first is to study open questions from several complex variables using ideas recently developed by the principal investigator on multiparameter singular integrals. In many special cases, operators from several complex variables are Calderon-Zygmund singular integral operators and are well understood. However, in many more general cases, the operators are some sort of singular integral that is not of Calderon-Zygmund type. These operators often have an underlying multiparameter Carnot-Caratheodory geometry, which was recently developed in a quantitative way by the principal investigator. The next topic concerns new directions in oscillatory integrals, which also have two underlying Carnot-Caratheodory geometries and are amenable to the prinicipal investigator's methods concerning these geometries. The third direction concerns a generalization of multilinear singular integrals due to Christ and Journe, which was motivated by Bressan's Mixing Conjecture. Here a main technique will be to use the geometry of projective space to determine the right class of operators. The fourth direction involves new kinds of multilinear singular integrals where the key tool will be to use actions of semisimple Lie groups to study their boundedness properties. The fifth direction lies in a slightly different line, and introduces a new kind of differential equation that is motivated by questions from pseudodifferential operators and inverse problems. The prinicipal investigator offers a conjecture as to the uniqueness properties of this differential equation.

在数学上的许多自然问题中，在经典意义上考虑时会出现积分，但是由于基本的取消属性，可以理解这一点（非正式地，它们可以理解它们，因为它们涉及添加“正无限”和“负面无穷大”，而“负面无穷大”互相取消）。这些重要的对象称为单数积分。该项目建议研究几种相关的单数积分。提出的问题的一个统一方面是它们都与几何形状密切相关，并且该项目将研究如何使用这种潜在的几何形状来开发奇异积分的正确概念。提出的问题有许多申请。 其中包括直接应用医学成像，电阻抗断层扫描，地球物理假期，流体在某些情况下混合的速率，几个复杂变量的新方向以及具有潜在单数积分的多线性操作员的新方向。 此外，鉴于过去发现了奇异积分的广泛应用，因此为该项目开发的想法可能还可能在上面提到的物理和数学方面具有许多其他应用。首先是使用主要研究者在多组分奇异积分方面开发的思想来研究几个复杂变量的开放问题。在许多特殊情况下，来自几个复杂变量的操作员是Calderon-Zygmund单数积分运算符，并且对其进行了充分的了解。但是，在许多一般情况下，操作员是某种单数积分，而不是Calderon-Zygmund类型。这些操作员通常具有潜在的多参数Carnot-Caratheodory几何形状，该几何形状最近是由主要研究人员定量开发的。下一个主题涉及振荡积分的新方向，它们还具有两个基本的Carnot-Caratheodory几何形状，并且适合有关这些几何形状的Prinicipal研究者的方法。 第三个方向是由于基督和乔恩（Journe）引起的多线性奇异积分的概括，这是由布雷桑（Bressan）的混合猜想所激发的。在这里，主要技术将是使用投影空间的几何形状来确定正确的操作员类别。第四个方向涉及新型的多线性奇异积分，其中关键工具将使用半圣母谎言组的动作来研究其界限。第五个方向在略有不同的线上，并引入了一种新型的微分方程，该方程是由假反相差操作员和反问题的问题所激发的。 Prinicipal研究者对这种微分方程的唯一性能提出了猜想。