Embeddability and Transversality in CR Geometry

The study of CR geometry dates back to the pioneering works of Poincaré and Élie Cartan. A major turning point came with Hans Lewys discovery of a simple partial differential equation (PDE) that has no solution, which challenged the prevailing belief that all PDEs could be solved under suitable conditions. Our research focuses on a large class of PDEs known as the tangential Cauchy- Riemann equations, whose solutions carry deep geometric meaning. In this setting, the PDE corresponds to an abstract geometric object known as a CR manifold, and its solvability (or integrability) reflects whether the CR structure can be realized in a concrete Euclidean space through an embedding. This project has two main goals. The primary objective is to study the stability of embeddings of compact three-dimensional CR manifolds and to develop the theory of semiclassical kernel asymptotics, particularly in the context of weakly pseudoconvex cases. The second is to investigate the transversality of CR mappings between manifolds of different dimensions, with particular attention to mappings between Levi-nondegenerate hypersurfaces.