Advanced Studies Institute in Pluripotential Theory

OVERVIEW

Pluripotential theory, the study of plurisubharmonic functions in several complex variables, has been utilized in recent years in other areas of mathematics. In this minicourse, we will first develop some of the needed basic tools, e.g., extremal plurisubharmonic functions, complex Monge-Amperé measures, and generalized Vandermonde matrices. Here we will proceed by analogy to the univariate case, logarithmic potential theory in the complex plane C, instead of providing detailed proofs. Then we will use these tools and techniques to discuss recent results on asymptotic behavior of zero sets of sequences of random polynomials and random polynomial mappings in C^d, d > 1; on random point processes on compact sets K subset of C^^d and associated large deviation principles; and on applications to complex dynamics such as approximating the equilibrium measure K of certain compact sets K by a sequence of dynamically generated Brolin measures associated to a sequence of polynomials exhibiting a certain regular behavior on K.

Topics

(1) Basics of pluripotential theory; (2) Random polynomials and random polynomial
mappings; (3) Point processes and large deviation principles; (4) Applications to complex dynamics.