# Jerusalem Mathematics Colloquium

Thursday, 24th December 2009, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Barry Simon

(Caltech)

"Natural boundaries and spectral theory"

** Abstract: **

This talk describes joint work with Jonathan Breuer. The
last ten years has seen considerable understanding of the spectrum
of general Jacobi matrices in terms of its right limits due to work of
Last-Simon and especially Remling. We have discovered that analogs
of these ideas can be used to understand when a power series (with
bounded Taylor coefficients) has a natural boundary on the unit circle.
One recovers and (within the class of bounded coefficients) improves
many classical results. The main theorem depends on little more than
the notions of right limit and reflectionless double power series (that
we carry over from the theory of Jacobi matrices) and a clever lemma
proven by M. Riesz in 1916 (using the maximum principle). This will
be a colloquium-level talk that should be accessible to anyone with a
good complex variables course.

Light refreshments will be served in the faculty lounge at 3:30.

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