# FELLESKOLLOKVIUM FREDAG 19. MAI 2006

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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

Kaffe og kjeks fra kl 1400

EB/HB/BJ/PAØ Hovedsiden