Pseudoconvexity plays a central role in the theory of several complex variables, since pseudoconvex domains are the most natural domains for the study of holomorphic functions in several variables. However, there are many gradations of pseudoconvexity, with some more amenable to study than others. I will begin my talk by characterizing the various types of pseudoconvex domains, with some hints as to the problems posed by "weaker" forms of pseudoconvexity.
The use of PDEs to study pseudoconvex domains leads naturally into the study of subelliptic operators. Roughly speaking, these are differential operators where the solution is more regular than the data, but without the best regularity that one could hope for (as in the case of elliptic operators). In the second part of my talk, I will introduce elliptic operators and subelliptic operators in more detail, and talk about the relationship between pseudoconvexity and subellipticity for the d-bar Neumann problem.
To volunteer to give a talk, or for any other questions regarding this schedule, contact Wesley Calvert