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.
