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Speaker
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Title/Abstract
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Date and Time
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Week July 5-11 |
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S. Klainerman |
On the Causal Structure in General Relativity I/II |
Tuesday, July 6, 15:00
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Abstract: I will give a general
introduction and discuss some recent results in collaboration with
I. Rodnianski concerning the radius of injectivity of null
hypersurfaces with only a bound on the curvature flux through the
hypersurface. |
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Week July 12-18 |
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| M. Dafermos |
A proof of Price's law for the
collapse of a self-gravitating scalar field |
Monday, July 12, 14:00 |
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Abstract: An open problem in general
relativity, dating back 1972, has been to
prove Price's law for an appropriate model of gravitational
collapse. This law postulates inverse-power decay rates for the
gravitational radiation flux on the event horizon and null
infinity with respect to appropriately normalized advanced and
retarded time coordinates. It is intimately related both to
astrophysical observations of black holes and to the fate of
observers who dare cross the event horizon.
In this talk I shall outline a proof of (the upper bound half
of) Price's law for the collapse of a self-gravitating spherically
symmetric scalar field. This is joint work with Igor Rodnianski. |
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| M. I. Sigal
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Soliton dynamics in nonlinear
Schroedinger equation |
Tuesday, July 13, 15:00 |
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Abstract: I review some recent results
on long-time and asymptotic dynamics of
solitons in nonlinear Schroedinger equation with a potential
(the Gross-Pitaevskii equation). |
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| A. Rendall |
Analogies between spacetime
singularities and inflationary late-time asymptotics |
Wednesday, July 14, 15:00 |
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Abstract: This talk will discuss
similarities and differences in the asymptotic behavior of
solutions of the Einstein equations near Big Bang singularities
and in an inflationary phase. In particular, the strengths and
weaknesses of Fuchsian techniques as applied to these problems
will be discussed. Special attention will be devoted to the
generation of non-uniform structure (spikes). |
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| L. Andersson |
BKL and asymptotic silence |
Thursday, July 15, 15:00 |
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Abstract: I will use the
example of cosmologies with two commuting Killing fields,
, to illustrate the BKL conjecture. The singularity in
this case is oscillatory. We identify an asymptotic dynamical
system which governs the dynamics near the singularity and
illustrate this by numerical experiments. |
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Week July 19-25 |
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J.J. Velazquez |
Singular behaviors for the Keller-Segel model |
Tuesday, July 20, 15:00
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Week July 26-August 1 |
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| V. Moncrief |
Progress towards light cone
estimates for Einstein's equations |
Tuesday, July 27, 14:00 |
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Abstract: We discuss recent
progress on a project to develop light cone estimates for the
curvature of a 4-dimensional spacetime satisfying Einstein's
equations. The aim is to extend the scope of methods that have
been used successfully for the Yang-Mills (-Higgs) equations in
flat and curved backgrounds to the treatment of curvature
propagation in general relativity. Some of the techniques under
study may be useful for understanding the development of
singularities by the focusing of energy in higher dimensional
nonlinear problems. |
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| M. Fila |
Continuation beyond blow-up
for superctitical parabolic equations |
Wednesday, July 28, 11:00 |
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Abstract: We shall discuss
solutions of semilinear parabolic equations which blow up in
finite time in the sup-norm but continue to exist as weak
solutions globally in time. We first describe a class of initial
data from which solutions of this kind emanate and then we study
the regularity of the minimal continuation beyond blow-up. |
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| P. Bizon |
On convergence towards a
self-similar solution for some semilinear wave equations |
Wednesday, July 28, 15:00 |
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Abstract: We consider a problem
of asymptotic stability of a self-similar attractor for some
simple semilinear radial wave equations which arise in the study
of equivariant wave maps and the Yang-Mills equations. Our
analysis consists of two steps. In the first step we determine the
spectrum of linearized perturbations about the attractor in a
semi-analytic manner using a method of continued fractions. In the
second step we demonstrate numerically that the resulting
eigensystem provides an accurate description of the dynamics of
convergence towards the attractor. |
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| J.M. Martin-Garcia |
The global structure of the
Choptuik spacetime |
Thursday, July 29, 11:00 |
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Abstract: At the threshold of
black hole formation in the gravitational collapse of a scalar
field a naked singularity is formed through a universal
criticalsolution. We study the global spacetime structure of this
solution. It is spherically symmetric, discretely self-similar,
regular at the center to the past of the singularity, and regular
at the past lightcone of the singularity. At the future lightcone
of the singularity, which is also a Cauchy horizon, the curvature
is finite and continuous but not differentiable. To the future of
the Cauchy horizon the solution is not unique, but depends on a
free function (the null data coming out of the nked singularity).
There is unique continuation with a regular center (which is
self-similar). All other self-similar continuations have a central
timelike singularity with negative mass. |
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Week August 2-8 |
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| H. Ringstroem |
Strong cosmic censorship in T^3
Gowdy spacetimes |
Thursday, August 5, 11:00 |
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Abstract: I will discuss a
result stating that for an
open and dense set of initial data, in the class of
initial data corresponding to T^3 Gowdy spacetimes, the
corresponding spacetimes exhibit curvature blow up
everywhere on the singularity. |
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Week August 9-15 |
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| M. Struwe |
Uniqueness for nonlinear wave
equations |
Monday, August 9, 14:00 |
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| J.Frauendiener |
On stable propagation of
constraints |
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Abstract: In all the free
evolution codes for the Einstein equations the divergence of the
constraints is a major problem. I discuss a method for getting
insight into the cause of the divergence based on the example of
the spin-2 equation for relativity. |
Tuesday, August 10, 15:00 |
| J.F. Williams |
Adaptive numerical methods for
singular PDEs |
Wednesday, August 11, 11:00 |
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Abstract: This talk will
describe adaptive strategies in both time and space for the
reliable resolution of finite-time singularities. Both strategies
are based on preserving the underlying symmetries of the physical
PDE. Examples in one and two dimensions will be presented. |
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