ABSTRACTS FOR ANALYSIS TALKS - SPRING 2005
Friday, February 11, 2005, 4:00 p.m., VV B239
COLLOQUIUM
Carlos Kenig (University of Chicago)
Recent developments on the well-posedness of nonlinear dispersive equations
Abstract: I will review some of the work on nonlinear dispersive equations
done in the 90's and then discuss a class of interesting problems were these
techniques do not apply and the recent partial progress in them.
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Friday, March 4, 2005, 1:20 p.m., VV B115
Stephen Yau (University of Illinois, Chicago)
Holomorphic De Rham cohomology of strongly pseudoconvex CR manifolds.
Abstract: In this talk we shall discuss the holomorphic DeRham cohomology
of a compact strongly pseudoconvex CR manifold X. Holomorphic De Rham
cohomology is derived from the Kohn-Rossi cohomology and is particularly
interesting when X is real dimension 3 and the Kohn-Rossi cohomology is
infinite dimesional. We shall use holomorphic De Rham cohomology to study
the complex Plateau problem for a 3 real dimension compact strongly
pseudoconvex CR manifold sitting in C^3. In general, for compact strongly
pseudoconvex CR manifolds with trasversal holomorhpic S^1 action, we can
relate the holomorphic De Rham cohomology to the puntured local
holomorphic De Rham cohomology at the singularity in the variety V which X
bounds.
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Friday, March 4, 2005, 4:00 p.m., VV B239
COLLOQUIUM
Stephen Yau (University of Illinois, Chicago)
CR manifolds and isolated singularities
Abstract: CR manifolds are boundaries of complex manifolds or complex
varities. In this talk we shall discuss the CR equivalent problem among
CR manifolds, i.e. given two CR manifolds, how can we know whether they
are CR biholomorphically equivalent. Recently we have introduced a new
nonnegative Bergman function for any strongly pseudoconvex complex
manifold. This Bergman function is invariant under biholomorphic maps and
vanishes precisely on the exceptional set of the strongly pseudoconvex
complex manifold. We shall show that it can be used to study the
variations of CR structure of strongly pseudoconvex CR manifolds lying
within an variety. In the joint work with H.S. Luk and X.J. Huang, we
study the simultaneous embedding and filling problems for a CR family of
CR manifolds. As a corollary, we showed that Buchweitz-Milson-Miyajima
theorem is true even for singularities with dimension 3. Thus if (V,0) is
a normal isolated singularity with dimension and depth at least 3, then
the Kuranishi family of the link of (V,0) is realized as a real
hypersurface of the versal family of deformation of (V,0).
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Svetlana Jitomirskaya (UC Irvine) will give two talks on
Friday, March 11, 2005:
Analysis Seminar:
Friday, March 11. 2005, 1:20 p.m., VV B115
Title: The ten martini problem.
Abstract: we will discuss the recent proof of Cantor spectrum for the
almost Mathieu operator for all conjectured values of the parameters.
COLLOQUIUM:
Friday, March 11, 2005, 4:00 p.m., VV B239
Title: Spectral properties of quasiperiodic operators: the
competition between order and chaos.
Abstract:
Up until the mid 70s the kind of spectra most people had in mind in the
context of theory of Schrodinger operators were spectra occurring for
periodic potentials and for atomic and molecular Hamiltonians. Then
evidence started to build up that "exotic" spectral phenomena such as
singular continuous, Cantor, and dense point spectrum do occur in
mathematical models that are of substantial interest to theoretical
physics. One area where such exotic phenomena are particularly abundant is
quasiperiodic operators. They feature a competition between
randomness (ergodicity) and order (periodicity), which is often resolved
on a deep arithmetic level. Mathematically, the methods involved include a
mixture of ergodic theory, dynamical systems, probability, functional and
harmonic analysis. The interest in those models was enhanced by strong
connections with some major discoveries in physics, such as integer
quantum Hall effect, experimental quasicrystals, and quantum chaos theory.
Quasiperiodic operators provide central or important models for all three.
The talk will consist of an overview of certain topics related to this
subject and of a glimpse into how one can fight small denominators
in this setting.
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Friday, April 1, 2005, 4:00 p.m., VV B239
COLLOQUIUM
Renormalization group approach to spectral problems and theory of radiation
I.M. Sigal (Toronto and Notre Dame)
Abstract: In this talk I will describe some analytical problems in
Quantum Field Theory, viewed as Quantum Mechanics of infinitely many
degrees of freedom or of extended objects (e.g. curves, surfaces, etc).
I will present some recent results on the problem of radiation and
describe a novel renormalization group technique used in proving
these results. I will not assume any prior knowledge of the subject.