Singular Integrals and Geometry

在数学中的许多自然问题中，当考虑经典意义时，会出现发散的积分，但由于潜在的抵消特性，它们可以被理解（非正式地，它们可以被理解，因为它们涉及到“正无穷”和“负无穷”的相加，它们相互抵消）。这些重要的对象被称为奇异积分。本课题拟研究几种相关的奇异积分。所提出的问题的一个统一方面是它们都与几何密切相关，本项目将研究如何使用这些潜在的几何来发展奇异积分的正确概念。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、流体在某些情况下的混合速率、几个复杂变量的新方向，以及具有潜在奇异积分的多线性算子的新方向。此外，考虑到奇异积分在过去的广泛应用，本项目开发的想法可能在物理和数学中有许多其他应用，而不仅仅是上面提到的。在这个项目中有五个主要的、相互关联的问题。第一个是利用首席研究员最近在多参数奇异积分上发展的思想，研究几个复杂变量的开放问题。在许多特殊情况下，几个复变量的算子是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，算子是某种不是Calderon-Zygmund型的奇异积分。这些算符通常具有潜在的多参数Carnot-Caratheodory几何，该几何最近由首席研究员以定量方式开发。下一个主题涉及振荡积分的新方向，振荡积分也有两个潜在的卡诺-卡拉多几何，并且适用于主要研究者关于这些几何的方法。第三个方向是由Christ和Journe提出的多线性奇异积分的推广，这是由Bressan的混合猜想推动的。这里的一个主要技术将是使用射影空间的几何来确定合适的算子类。第四个方向涉及到新的多重线性奇异积分，其中关键的工具将是利用半单李群的作用来研究它们的有界性。第五个方向在一条稍微不同的线上，并引入了一种新的微分方程，它是由伪微分算子和逆问题引起的。主要研究者对这个微分方程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.