Selected abstracts for the fall semester 2006:
Tuesday, September 19, 2006, 4:00-5:00 p.m., VV B139
Jean-Pierre Rosay (UW Madison)
Extension of holomorphic bundles (and Serre's
problem on Stein bundles)
Abstract: I'll explain an extremely simple technique to extend
holomorphic bundles with fiber C^n and base an open set in C to
holomorphic bundles on C. In particular it applies to the Skoda bundle
and so it gives a very transparent construction of non trivial
holomorphic bundles on the unit disc. It also gives new examples
with transition automorphisms that are polynomials (the question has
been open for many years).
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Tuesday, October 3, 2006, 4:00-5:00 p.m., VV B139
Han Peters (UW)
Attracting basins of volume preserving automorphisms of C^k.
Joint work with Liz Raquel Vivas and Erlend Fornaess Wold.
We study the size of attracting sets for automorphisms of complex Euclidean
space, and we are particularly interested in volume preserving
automorphisms. For an automorphism with a fixed point at the origin that is
not attracting (i.e. at least one eigenvalue of the derivative is larger
than or equal to 1), we ask the following two questions:
- Can the basin contain a neighborhood of the origin?
- Can the basin be a dense set?
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Tuesday, October 10, 2006, 4:00-5:00 p.m., VV B139
Serguei Denissov (UW)
Scattering for multidimensional Schrödinger operator with slowly decaying
potential.
Abstract:
Motivated by a conjecture of Simon, we try to obtain a multidimensional
analog of the classical Szegö
Theorem in the approximation theory. The
conjecture has a positive solution on the Caley tree and can be justified
for Euclidean three-dimensional case when the potential is assumed to be
sparse. We will conclude with showing that the scattering process can be
very complicated and is controlled by certain evolution equation that
generalizes the usual WKB correction. This talk might be interesting for
those working in analysis, PDE, mathematical physics.
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COLLOQUIUM
Friday, October 27, 2006, 4:00 p.m., VV B239
Gunther Uhlmann (University of Washington)
Invisibility for electromagnetic waves via singular Riemannian metrics
Abstract:
There has been considerable interest in the possibility, both theoretical and practical, of shielding
(or "cloaking") a region or object from detection via electromagnetic waves. We will discuss several
results describing how to cloak an object or region using singular Riemannian metrics.
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Tuesday, November 7, 2006, 4:00-5:00 p.m., VV B139
Alina Andrei (UW)
On the dynamics of a family of quadratic polynomial maps in two
dimensions.
Abstract: In one dimensional complex dynamics, the study of the parameter
space of the
family p_c=z^2 +c has led to the discovery of the well-known Mandelbrot set.
In higher dimensions, the parameter space is not well understood and no
analog of the Mandelbrot set is known. In this talk, we
investigate the parameter space of a quadratic
family of polynomial maps of C^2 that exhibits interesting dynamics. Two
distinct subsets of the parameter space are studied as
appropriate analogs of the one-dimensional Mandelbrot set and
some of their properties are proved by using Lyapunov exponents.
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Tuesday, November 28, 2006, 4:00-5:00 p.m., VV B139
Allan Greenleaf (University of Rochester)
A Fourier integral operator calculus for marine seismic imaging.
Abstract: I will describe a class of degenerate FIOs
that arises in offshore oil exploration in the presence of
caustics (multipathing of sound waves) of the simplest type,
namely fold caustics. The resulting normal operators
$F^*F$ belong to the same general class as Hilbert transforms
along the cubic $(t,t^3)$, and we obtain estimates for such operators,
exhibiting an interesting loss of derivatives.
This is joint work with Raluca Felea and
Malabika Pramanik.TBA