DISTINGUISHED VARIETIES: DETERMINANTAL REPRESENTATIONS AND BOUNDED EXTENSIONS
UNIVERSITY OF CALIFORNIA, IRVINE
Distinguished varieties are algebraic curves in C^2 that exit the unit bidisk through the distinguished boundary. We will discuss how these curves appear naturally in operator theory and function theory, and we will outline a connection between distinguished varieties and polynomials with no zeros on the bidisk (on the surface, two antithetical objects) that allows us to prove a determinantal representation and a "bounded analytic extension theorem" for distinguished varieties.
3:30 – 4:30 PM, Lecture #2:
HYPERGROUPS AND PROBABILITY THEORY
Hypergroups are locally compact spaces with a group-like structure for which the bounded measures convolve in a similar way to that of a locally compact group. Important examples of hypergroups are orbit spaces arising from groups.
There are fundamental constructions providing hypergroup structures on the nonnegative reals (Sturm - Liouville functions) and on the nonnegative integers (Jacobi polynomials).
In probability theory hypergroup convolutions admit for example the study of invariant Markov chains and Levy processes, prominent results being (local) central limit theorems and martingale characterizations respectively.
The method of carrying out the analysis of hypergroups and their applications is a generalization of the Fourier transform of measures defined on a dual object attached to the given hypergroup.
Dinner at a local restaurant will follow the concluding lecture.
For more information, please contact Professor Asuman Aksoy at: (909) 607-2769,