The aim of this course is to provide an introduction to CR (Cauchy Riemann/Complex Real) geometry, which is broadly the study of the structure(s) inherited by real submanifolds in complex spaces. We will first give a parallel introduction to the fundamental objects of SCV and CR geometry. These include holomorphic functions in several variables, CR manifolds (embedded and abstract) and CR functions. Next, we will cover some examples, results, and techniques from the following range of topics.
a) embeddability of abstract CR structures;
b) holomorphic extendability of CR functions;
c) CR singularities.
Wherever possible (and time permitting), we will highlight the connections of this field to other areas of analysis and geometry. For instance, abstract CR structures will be discussed in the broader context of involutive structures on smooth manifolds.