MIT PDE/Analysis Seminar, Fall 2011
At MIT in
Room 2-142 at 4:00 PM on Tuesdays, unless otherwise noted.
September 20: Michael Singer (Edinburgh)
Partial Bergman kernels and stability
Abstract: The Bergman kernel on a compact Kaehler manifold is the Schwartz kernel of the L^2 projection onto the space of holomorphic sections of some ample line bundle L. Much is known about the asymptotic behaviour of the Bergman kernel for high powers of the bundle L. In this talk, we will consider the `partial' projection onto the subspace of sections vanishing to order lk along a submanifold. Restricting to the toric case, we will prove the existence of a distributional asymptotic expansion for the corresponding partial density function (the value of the kernel on the diagonal), at least to leading order, and indicate an application to the study of stability of the polarized manifold (M,L).
October 4: Daniel Grieser (Oldenburg)
Special time 3-4PM and Room 2-132
The exponential map at a cuspidal singularity
October 11: No talk
October 18: Hamid Hezari (MIT)
Spectral rigidity of the ellipse
Abstract: This is a joint work with Steve Zelditch. We prove that ellipses
are infinitesimally spectrally rigid among smooth domains
with the symmetries of the ellipse. Spectral rigidity of the ellipse has
been expected for a long time and is a kind of model
problem in inverse spectral theory. Ellipses are special since their
billiard flows and maps are completely integrable.
It was conjectured by G. D. Birkhoff that the ellipse is the only convex
smooth plane domain with completely integrable billiards.
Our results are somewhat analogous to the spectral rigidity of flat tori or
the sphere in the Riemannian setting.
The main step in the proof is the Hadamard variational formula for the wave
trace. It is of independent interest and it might have
applications to spectral rigidity beyond the setting of ellipses. The main
advance over prior results is that the domains
are allowed to be smooth rather than real analytic. Our proof also uses many
techniques developed by Duistermaat-Guillemin and Guillemin-Melrose in
closely related problems.
October 25: Irene Gamba (U. Texas and ICERM)
Convolution estimates in the regularity of the Boltzmann equation
Abstract: The Boltzmann integral in strong form can be view as a double
randon transform via the Carleman representation. Also, in weak
formulation can be represented in weighted convolutional structure.
We have shown that these representations, depending on the collisional
kernels yield either a Young's inequality for hard potentials, or Hardy
Littlewood Sobolev (HSL) inequality for soft ones, where using
symmetrization techniques, these inequalities were developed with exact
constants (in collaboration with R. Alonso and E. Carneiro).
In addition, these estimates are shown to be a powerful tool to show a
gain of integrability that implies the propagation of regularity for the
solution to the space homogeneous Boltzmann equation, as well as estimates
for the convergence of spectral approximations to this solution. In
addition the HSL inequality is used to prove the propagation of L^p
regularity and stability for the space inhomogeneous problem for soft
potentials with initial data near local Maxwellian states (In
collaboration with Ricardo Alonso).
October 27: Alex Iosevich (Rochester)
Special Day, time 11AM and Room 2-102
Distribution of lattice points near families of surfaces
November 1: Robert Strain (U. Penn.)
Optimal large-time decay rates for collisional kinetic equations in the whole space
November 2: Sergiu Klainerman (Princeton)
Note special Day, time 3PM and Room 2-136
On the rigidity of black holes
November 8: Jared Speck (MIT)
Global stability results for relativistic fluids in expanding spacetimes
November 15: Monika Ludwig (Polytechnic Institute of New York University
and Vienna University of Technology)
Valuations on Sobolov Spaces
November 22: Semyon Dyatlov (Berkeley)
Quasi-normal modes for Kerr-de Sitter black holes>
Abstract: Quasinormal modes of black holes are supposed to describe oscillations and
decay of gravitational waves produced by the black hole interacting with
another object. They have been extensively studied by physicists, most
recently in the context of string theory.
We provide a rigorous definition of quasinormal modes of slowly rotating
Kerr-de Sitter black holes, resonance expansions of linear waves in terms
of these modes (and in particular exponential decay), and a semiclassical
description of quasinormal modes, which matches numerical results already
at low energies.
November 29: Daniela Tonon (ICERM, Brown)
SBV regularity results for Hamilton-Jacobi equations
Abstract: We present two results on the regularity of viscosity solutions of
Hamilton-Jacobi equations obtained in collaboration with Professor
Stefano Bianchini. When the Hamiltonian is strictly convex viscosity
solutions are semiconcave, hence their gradient is BV. First we prove
the SBV regularity of the gradient of a viscosity solution of the
Hamilton-Jacobi equation u_t+ H(t,x,D_x u)=0 in an open set of
R^(n+1), under the hypothesis of uniform convexity of the Hamiltonian
H in the last variable. Secondly we remove the uniform convexity
hypothesis on the Hamiltonian, considering a viscosity solution u of
the Hamilton-Jacobi equation u_t+ H(D_x u)=0 in an open set of
R^(n+1) where H is smooth and convex. In this case the viscosity
solution is only locally Lipschitz. However when the vector field
d(t,x):=H_p(D_xu(t,x)), here H_p is the gradient of H, is BV for all t
in [0,T] and suitable hypotheses on the Lagrangian L hold, the
divergence of d(t, ) can have Cantor part only for a countable number
of t's in [0,T]. These results extend a result of Bianchini, De Lellis
and Robyr for a uniformly convex Hamiltonian which depends only on
the spatial gradient of the solution.
December 6: Antti Knowles (Harvard)
Finite-rank deformations of Wigner matrices
Abstract: The spectral statistics of large Wigner matrices are by now
well-understood. They exhibit the striking phenomenon of universality:
under very general assumptions on the matrix entries, the limiting
spectral statistics coincide with those of a Gaussian matrix ensemble.
I shall talk about Wigner matrices that have been perturbed by a
finite-rank matrix. By Weyl's interlacing inequalities, this
perturbation does not affect the large-scale statistics of the
spectrum. However, it may affect eigenvalues near the spectral edge,
causing them to break free from the bulk spectrum. In a series of
seminal papers, Baik, Ben Arous, and Peche (2005) and Peche (2006)
established a sharp phase transition in the statistics of the extremal
eigenvalues of perturbed Gaussian matrices. At the BBP transition, an
eigenvalue detaches itself from the bulk and becomes an outlier.
I shall report on recent joint work with Jun Yin. We consider an NxN
Wigner matrix H perturbed by an arbitrary deterministic finite-rank
matrix A. We allow the eigenvalues of A to depend on N. Under optimal
(up to factors of log N) conditions on the eigenvalues of A, we
identify the limiting distribution of the outliers. We also prove that
the remaining eigenvalues "stick" to eigenvalues of H, thus
establishing the edge universality of H + A. On the other hand, our
results show that the distribution of the outliers is not universal,
but depends on the distribution of H and on the geometry of the
eigenvectors of A. As the outliers approach the bulk spectrum, this
dependence is washed out and the distribution of the outliers becomes
December 13: Stefanos Aretakis (Cambridge)
Stability and instability of extremal black holes
Abstract: We consider the wave equation on the exterior region of extremal
black holes, mainly extremal Reissner-Nordstrom and extremal Kerr. We
stability and instability results for the solutions. The latter
results are in sharp contrast with the picture in the subextremal case for
which sharp stability results have been shown.