Singular Integrals and Geometry

在数学中的许多自然问题中，存在着积分，这些积分在经典意义上被认为是发散的，但由于潜在的抵消性质(非正式地，它们可以被理解为有意义，因为它们涉及到相互抵消的“正无穷大”和“负无穷大”)。这些重要的对象被称为奇异积分。本课题拟研究几类相关的奇异积分。提出的问题的一个统一方面是，它们都与一个几何密切相关，这个项目将研究如何利用这个基础几何来发展奇异积分的正确概念。提出的问题有很多应用。它们包括在医学成像、电阻抗断层扫描、地球物理勘探、流体在某些情况下混合的速度、几个复变量的新方向以及具有潜在奇异积分的多线性算子的新方向上的直接应用。此外，考虑到奇异积分在过去发现的广泛应用，为这个项目开发的思想可能在上述之外的许多其他物理和数学应用中。这个项目有五个主要的、相互关联的问题。第一种是利用多参数奇异积分的主要研究人员最近提出的思想来研究几个复变量的开放问题。在许多特殊情况下，来自多个复变量的算子是Calderon-Zygmund奇异积分算子，这是众所周知的。然而，在更一般的情况下，算子是某种不属于Calderon-Zygmund类型的奇异积分。这些算子通常具有潜在的多参数Carnot-Caratheodory几何，该几何是最近由首席研究人员以定量方式发展出来的。下一个主题涉及振荡积分的新方向，它也有两个潜在的Carnot-Caratheodory几何，并且服从主要研究者关于这些几何的方法。第三个方向是关于基督和乔恩的多线性奇异积分的推广，这是由布雷桑的混合猜想引起的。这里的一个主要技术将是利用射影空间的几何来确定正确的算子类。第四个方向涉及新类型的多线性奇异积分，其中的关键工具将是利用半单李群的作用来研究它们的有界性。第五个方向的方向略有不同，它引入了一种新的微分方程，它是由伪微分算子和反问题的问题引起的。主要研究者对这个微分方程解的唯一性提出了一个猜想。

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.