Geometric Analysis on CR manifolds

In mathematics, Partial Differential Equations (PDE) are used for modeling most natural phenomena, like the spread of heat or the motion of fluids. Until the 1950s, it was commonly believed that every PDE could be solved with appropriate assumptions. However, Hans Lewy provided an example of an "aberrant" PDE that cannot be solved, even though it is a very nice and simple PDE. This example sparked an investigation into the solvability of PDEs and their relationship to regularity theory. Our research project focuses on a situation where the solutions of a large class of PDEs, known as tangential CR equations, have a geometric interpretation. In this context, the PDE corresponds to an abstract geometric object, a CR manifold, and its solvability (integrab ility) is interpreted as the ability to realize this geometric object in a concrete Euclidean space through an embedding. There are local as well as global problems in our context, both of them posing their own set of questions. Most of the global solutions dealing with CR manifolds always require strong assumptions on the structure, and few optimal results are known. Our project has two primary goals. The first is to explore the "nice" assumptions necessary for CR embeddability. To achieve this, we will investigate the spectral theory and kernel theory of the Kohn Laplacian, which is a fundamental Laplace operator used in CR manifolds. These theories are closely related to the CR embedding problem. The second goal of our project is to explore CR transversality, which is another important property that arises in the mapping between two CR manifolds. It presents a distinct yet related problem to the embedding problem. Specifically, we aim to find good geometric conditions for CR transversality when the source and target CR manifolds exist in complex spaces of different dimensions.