ABSTRACTS FOR SOME ANALYSIS TALKS - Fall 2005



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Tuesday, November 1, 2005, 4:00 p.m., VV B139
Jonathan Breuer (Hebrew University of Jerusalem, and Princeton University)
Upper Bounds on the Dynamical Spreading of Wavepackets



Let $H=\Delta+V$, where $\Delta$ is the discrete Laplacian on the 
half-line and
$V$ is some real bounded potential defined
there. Then $H$ is self-adjoint and $U(t)=e^{-itH}$, 
$t \in \mathbbR$ is a
unitary group.
A wavepacket, initially localized at the origin, 
tends to `spread out' under
the action of $U(t)$. The determination of the
rate of this spreading from properties of $H$ 
(spectral and others), has been
the focus of numerous research papers in recent
years.

After reviewing some relevant notions and results, 
the talk will discuss the
relation between some properties of the finite
dimensional approximants to $H$, and this spreading-rate. 
In particular, we show that an upper bound is implied by
sufficient `clustering' of the levels of these approximants.
This is joint work with Y. Last and Y. Strauss.

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Xianghong Gong (UW)
Differential inequalities of continuous functions and removing singularities of Rado type for J-holomorphic maps, 

Part I: Geometric Analysis   Seminar, Monday, October 10.
Part II:  Analysis   Seminar, Thursday, October 13.


Abstract of Part I: 
A classical removable singularity theorem for J-holomorphic curves
with finite energy deals with isolated singularities.
In this talk, we will prove a removable singularity theorem for a continuous
map on the disc that is J-holomorphic off a closed polar set.
Here, the removal of singularities is obtained by estimating an L^p norm
of the Laplacian of the curves.


Abstract of Part II: 
We will consider a continuous function f on a domain in C^n such
that d-bar f is bounded by f off its zero set. The main conclusion is that the
zero set of f is a complex variety. 
For n=1, this is an easy consequence of Rado's
theorem. The proof for higher dimension case is still based on Rado's
theorem, but one needs to study carefully about the d-bar closedness of (d-bar f)/f
on the domain. The case for vector-valued f is largely open, except for a real
analytic case.
There is a unifying theme of differential inequality in Parts I and II. This is
 joint work with J.-P. Rosay.

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Tuesday, September 20, 2005, 4:00 p.m., VV B139
Han Peters (UW)
Non-autonomous dynamical systems and Fatou-Bieberbach domains. 

We will try to prove the following generalization of a Theorem of
Rosay and Rudin: A basin of attraction of a (in some sense
uniformly bounded) sequence of automorphism of C^k is
always biholomorphically equivalent to C^k. This question has relations to
autonomous holomorphic dynamics (i.e. iteration of holomorphic mappings)
and classical complex analysis.

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Tuesday, September 13, 2005, 4:00 p.m., VV B139,
Serguei Denissov (UW),
Absolutely continuous spectrum for multidimensional Schrödinger operators.

Abstract: The approximation theory proved to be an excellent framework for
understanding one-dimensional scattering theory in mathematical physics. In
this talk we present recent developments in PDE (multidimensional Schrodinger
operator).


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Wednesday, August 3, 4:00 p.m., VV 901,
Patrick Speissegger (McMaster University),
O-minimality for correspondance maps of planar analytic vector 
fields near a non-resonant hyperbolic saddle point.


Abstract: (Joint work with Tobias Kaiser and Jean-Philippe
      Rolin)A major motivation for the study of o-minimal
      structures is the suspected connection to Dulac's problem,
      which states that the number of limit cycles of a planar real
      structures is the suspected connection to Dulac's problem,
      which states that the number of limit cycles of a planar real
      analytic vector field is finite. Many of the recent
      constructions of o-minimal structures were done mainly to
      develop techniques for such constructions, and they only
      relate to Dulac's problem in a few very special cases.
      However, only some managable modifications of these
      techniques are needed to include the first interesting class
      of functions appearing in Dulac's problem, namely the
      correspondance maps of planar analytic vector fields near a
      non-resonant hyperbolic saddle point.