Analysis and
Partial Differential Equations Seminar, Fall 2009
Mondays at 4PM in Krieger 304
|
Date |
|
Speaker |
Title |
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September 14 |
|
Chris
Sogge |
|
|
September 21 |
|
Matei Machedon |
Quantum
Dynamics of Many-Body Systems, the Hartree
Equation, and Beyond |
|
October 5 |
|
|
Counterexamples to the Strichartz estimates for
the wave equation in domains |
|
October 12 |
|
Christina Sormani |
|
|
November 2 |
|
Fabrice
Planchon |
|
|
November 9 |
|
Dmitry Jakobson |
Estimates from below: spectral
function, remainder in Weyl's law and resonances |
|
November 16 |
|
George Daskalopoulos |
|
|
November 23 |
|
Jason Metcalfe |
Long-time existence for
quasilinear wave equations with small data in exterior domains |
|
November 30 |
|
Kate Okikiolu
|
TBA |
|
December 7 |
|
Yng-Ing Lee |
Eternal solutions to Lagrangian Brakke flow |
|
December 7 |
|
Hans Lindblad |
The weak null condition, global
existence and the asymptotic behavior of solutions to Einstein's equations |
Abstracts
September 14, Chris Sogge,
On any compact Riemannian
manifold $(M, g)$ of dimension $n$, the
$L^2$-normalized eigenfunctions ${\phi_{\lambda}}$
satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$
where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp
in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics
on the standard $n$-sphere $S^n$. But of course, it
is not sharp for many Riemannian manifolds, e.g. flat tori
$\R^n/\Gamma$. We say that $S^n$, but not $\R^n/\Gamma$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this
paper is to determine the $(M, g)$ with maximal eigenfunction growth. In an earlier work, two of us showed
that such an $(M, g)$ must have a point $x$ where the
set ${\mathcal L}_x$ of geodesic loops at $x$ has
positive measure in $S^*_x M$. We strengthen this result here by showing that
such a manifold must have a point where the set ${\mathcal
R}_x$ of recurrent directions for the geodesic flow
through x satisfies $|{\mathcal R}_x|>0$. We also
show that if there are no such points, $L^2$-normalized quasimodes
have sup-norms that are $o(\lambda^{n-1)/2})$, and, in the other extreme, we
show that if there is a point blow-down $x$ at which the first return map for
the flow is the identity, then there is a sequence of quasi-modes with $L^\infty$-norms that are $\Omega(\lambda^{(n-1)/2})$.
September 21, Matei Machedon,
This is joint work with M. Grillakis and D. Margetis. I will
review how tensor products of solutions of the Hartree
equation (or NLS) in 3+1 dimensions approximate the solution to a many-body Schr\"odinger equation in 3N +1 dimensions. This was
implemented in several papers by Elgart, Erd\"os, Schlein and Yau.
Then I will introduce a new, non-local Schr\"odinger
equation in 6+1 dimensions, which provides a better approximations (at least
for some potentials). This will go through the formalism of the second
quantization (in the spirit of Hepp, Ginibre and Velo, Rodnianski and Schlein), and then
use an algebraic trick related to the Segal-Shale-Weil representaion,
whose application to this type of problem is new. The resulting second order
correction is inspired by (but different from) that of the physicist Wu.
October 5, Oana Ivanovici, Johns Hopkins University: Counterexamples
to the Strichartz estimates for the wave equation in domains
We prove that the Strichartz estimates
for the wave equation inside a strictly convex domain \Omega of dimension 2
suffer losses when compared to the usual case \mathbb{R}^2,
(at least for a subset of the usual range of indices) and this is due to microlocal phenomena such as caustics generated in arbitrarIly small time near the boundary.
October 12, Christina Sormani,
We define a new distance between oriented Riemannian manifolds
that we call the intrinsic flat distance based upon Ambrosio-Kirchheim's theory of integral currents on metric
spaces. Limits of sequence of manifolds with a uniform upper bound on their
volume and diameter are countably Hm rectifiable
metric spaces with an orientation and multiplicity that we call integral
current spaces.
In general the Gromov-Hausdorff and
intrinsic flat limits do not agree. Intrinsic flat
convergence is a weaker notion. We show that they do agree when the sequence of
manifolds has nonnegative Ricci curvature and a uniform lower bound on volume
and also when the sequence of manifolds has a uniform linear local geometric
contractibility function. These results are proven using work of
Greene-Petersen, Gromov, Cheeger-Colding
and Perelman.
This is joint work with S. Wenger.
November 2, Fabrice Planchon, Universitˇ
Paris 13: On uniqueness for the Cauchy
problem in general relativity
Local (in time) existence for the Einstein vacuum equations
has been known since the 50's and the pioneering work of Y. Choquet-Bruhat.
Uniqueness, however, only holds for the reduced system of equations in wave
coordinates, unless one is willing to allow more regularity on the data. We
will present an elementary argument which allows to solve
this gauge invariance problem, relying on a geometrical derivation and the
cancellation properties of Ricci flat metrics. This is joint work with Igor Rodnianski.
November 9, Dmitry
Jakobson,
This is joint work with I. Polterovich, J. Toth and F. Naud.
We obtain asymptotic lower bounds for the spectral
function of the Laplacian on compact manifolds. In
the negatively curved case,
thermodynamic formalism for hyperbolic flows is
applied to improve the estimates.
Our results can be considered pointwise
versions (on a general manifold) of lower bounds (due to Hardy and Landau) for
the error term in the
Gauss circle problem. We next discuss lower bounds
for the remainder in WeylÕs law on negatively-curved
surfaces. Our approach works in variable negative curvature, and is based on
wave trace asymptotics for long times, thermodynamic
formalism for hyperbolic flows, and small-scale microlocalization.
At the end, we shall discuss how to obtain
logarithmic lower bound for the local density of resonances for infinite area,
geometrically
finite surfaces, and how to improve them to
polynomial lower bound for infinite index subgroups of arithmetic groups.
November 16, George Daskalopoulos,
We prove rank one and higher
rank superrigidity for the isometry
groups of a class of complexes which includes hyperbolic buildings as a special
case. Our method uses harmonic maps to singular spaces.
November 23, Jason Metcalfe,
We explore long time existence for
quasilinear wave equations with small data in exterior domains. In
particular, we
explore nonlinearities which are permitted to
depend on the solution not just its first and second derivatives. The
primary
new tool is a weighted Strichartz estimate which was developed for use in other contexts.