FELLESKOLLOKVIUM FREDAG 19. MAI 2006

Peter W. Jones:
"The Corona Theorem: From Lennart Carleson's Proof to Later Developments"

Abstract:

We will discuss the history of the corona theorem and the influence this has had on analysis for almost the past half century. The statement is deceptively simple. Suppose we are given a finite number of bounded holomorphic functions on the unit disk, F1,...,FN. Can we find bounded holomorphic functions G1,...,GN such that F1G1+...+FNGN=1?

The positive answer was given by Lennart Carleson in his famous work of the early 1960's. The proof, however, was by no means easy and the technical tools Carleson introduced have had a deep impact on the development of harmonic analysis. We will discuss the methods that have grown out of that proof as well as their impact on other areas of mathematics, notably Fourier analysis. In addition we will outline the ideas (though not the difficult technicalities) of Carleson's proof. One of these ideas is to perform clever bookkeeping on a geometric construction (the "corona construction") that keeps all constants under control. This can be explained philosophically by a simple sequence of pictures. There are now other methods of proof (notably one due to Tom Wolff) and extensions to settings other than the unit disk. However, despite about 45 years of research we still do not know if the Corona Theorem is valid for arbitrary planar domains. We will attempt to explain some positive results and remaining difficulties for the general case, and will discuss relations to recent work of X. Tolsa on analytic capacity.

We do not assume that the audience has seen more than a basic course in complex analysis. While not used in the rest of the talk, some background in Banach algebras is required to understand why this result is called the Corona Theorem: There is a functional analytic analogy to the solar corona. An amusing aspect of the Corona Theorem is that, expressed in this language, "there is no corona."  

1430-1530 i pausearealet i 7. etage
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