Singular Integrals and Geometry

在数学中的许多自然问题中，当在经典意义上思考时，会出现发散的积分，但由于潜在的抵消性质，它们可以被理解（非正式地，它们可以被理解，因为它们涉及添加相互抵消的“正无穷大”和“负无穷大”）。这些重要的对象被称为奇异积分。本计画拟研究几种相关的奇异积分。所提出的问题的一个统一的方面是，它们都与几何密切相关，这个项目将研究如何使用这个基础几何来发展奇异积分的正确概念。所提出的问题有许多应用。 它们包括直接应用于医学成像，电阻抗断层扫描，地球物理勘探，在某些情况下，在流体混合的速度，在几个复杂的变量的新方向，和新的方向多线性算子，有潜在的奇异积分。 此外，鉴于奇异积分在过去已经发现的广泛应用，为该项目开发的想法可能在物理和数学中有许多其他应用，除了上面提到的那些。该项目中有五个主要的、相互关联的问题。首先是研究开放的问题，从几个复杂的变量使用的想法最近开发的主要研究人员对多参数奇异积分。在许多特殊情况下，多个复变量的算子是Calderon-Zygmund奇异积分算子，并且很容易理解。然而，在许多更一般的情况下，算子是某种奇异积分，不是Calderon-Zygmund型。这些运营商往往有一个潜在的多参数卡诺-Caratheodory几何，这是最近开发的定量方式的主要研究者。下一个主题涉及振荡积分的新方向，它也有两个基本的卡诺-Caratheodory几何，并服从有关这些几何的主要研究者的方法。 第三个方向是关于Christ和Cologne的多线性奇异积分的推广，这是由Bressan的混合猜想引起的。这里的一个主要技术将是使用射影空间的几何来确定正确的一类算子。第四个方向涉及新的多线性奇异积分，其中的关键工具将是使用半单李群的行动，以研究其有界性。第五个方向是在一个稍微不同的路线，并介绍了一种新的微分方程，这是从伪微分算子和反问题的问题的动机。主要研究者提出了一个关于该微分方程唯一性的猜想。

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.