MIT PDE/Analysis Seminar, Fall 2011

At MIT in Building 2
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

  Abstract.pdf

• 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

  Abstract.pdf

• November 15: Monika Ludwig (Polytechnic Institute of New York University and Vienna University of Technology)

  Valuations on Sobolov Spaces

  Abstract.pdf

• 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 universal.

  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 prove stability and instability results for the solutions. The latter instability results are in sharp contrast with the picture in the subextremal case for which sharp stability results have been shown.