Analysis Seminar: The Corona Theorem for Slice Hyperholomorphic Functions

Abstract: This talk will discuss the Corona problem for slice hyperholomorphic functions for a single quaternionic variable. While the Corona problem is well-understood in the context of one complex variable, it remains highly challenging in the case of several complex variables. The extension of the theory of one complex variable to several complex variables is not the only possible extension to multi-dimensional complex analysis. Instead of holomorphic functions in each variable separately, we will consider functions in some hypercomplex algebras, in particular in the algebra of quaternions. 

Previously, the Corona problem had not been studied within the hypercomplex framework because of challenges posed by pointwise multiplication, which is not closed for hypercomplex-valued analytic functions. Alternative notions of multiplication that are closed often compromise other desirable properties, further complicating the analysis. 

In this talk we will discuss the proof of the Corona problem within the quaternionic slice hyperholomorphic setting. Our approach involves reformulating the quaternionic Bezout equation with respect to the appropriate multiplication into a new system of Bezout equations on the unit disc. We solve this system by adapting Wolff's proof of the Corona theorem for bounded analytic functions. As the number of generators increases, the associated algebra grows increasingly intricate.

This talk is based on joint work with Fabrizio Colombo, Elodie Pozzi and Irene Sabadini.

Host: Walton Green