Geometric Analysis: a Conference in Honor of Richard Melrose

Organizing Committee:

Victor Guillemin (MIT),

Rafe Mazzeo (Stanford),

András Vasy (MIT),

Maciej Zworski (University of California, Berkeley).

This is the official poster. And this is an unofficial one.

This conference is partially supported by NSF and by the MIT Department of Mathematics.

Some travel support is available. Please contact one of the organizers for details.

Pictures of the meeting, taken by Victor Ivrii, may be accessed here.

The schedule is now available.

Titles/abstracts:

Nicolas Burq: Non linear Schrodinger equations on manifolds

Abstract: We present some recent results on well posedness and stability (or unstability) for the non linear Schrodinger equation on manifolds. These results are deduced from Strichartz inequalities with loss of derivatives, which are in turn obtained by a semi-classical parametrix approach. Most of these results were obtained in collaboration with P. Gerard and N. Tzvetkov for University Paris-Sud Orsay.

Charles Epstein: Indices and relative indices in contact- and CR-geometry

Charles Fefferman: Q-Curvature and the Poincare Metric

Abstract: The talk explains recent joint work with Robin Graham, giving a simple construction for Tom Branson's Q-curvature in higher (even) dimensions, and simple proofs of its basic properties.

Mitsuru Ikawa: On scattering by several convex bodies

Abstract: Concerning the scattering by several convex bodies, there must exist deep relationships between scattering matrices and zeta functions of the classical mechanics. But we know actually a little about them. In general, we do not know whether the zeta function has a singularity or not. We shall consider the case of three convex bodies, and look for an explicit representation of the zeta function in order to find a pole.

Victor Ivrii: Sharp spectral asymptotics for operators with irregular coefficients

Abstract: We consider operators with coefficients, first derivatives of which are continuous with continuity modulus $O(\bigl|\log |x-y |\bigr|^{-1})$ and derive semiclassical spectral asymptotics with sharp remainder estimate $O(h^{1-d})$; for operators with continuity modulus $o(\bigl|\log |x-y|\bigr|^{-1})$ we derive semiclassical spectral asymptotics with the remainder estimate $o(h^{l-d})$ under standard condition to Hamiltonian flow. Some further developments are discussed as well.

Peter Lax: Positive schemes

Abstract: A theorem of Friedrichs on positive difference schemes for solving the initial value problem for symmetric hyperbolic systems in any number of space dimensions is used to design second order accurate schemes for solving hyperbolic systems of conservation laws with entropy in any number of space dimensions.These schemes are hybrids of second order schemes such as Lax-Wendroff with upwind. Joint work with Xu-Dong Liu.

Vesselin Petkov: Spectral shift function and resonances

Abstract: In a work with V. Bruneau we obtained a representation of the derivative of the spectral shift function as a sum of harmonic and delta measures and a remainder which is a harmonic function. This representation holds in the general case of long range perturbations when the existence of a scattering operator is far from apparent. We survey recent results concerning the Weyl type asymptotics, the Breit-Wigner approximation and the connection between the distribution of the eigenvalues and the resonances.

Peter Sarnak: The first term in Shnirelman's asymptotics for arithmetic surfaces

Abstract: The theorem of Shnirelman, Colin de Verdiere and Zelditch gives a nontrivial upper bound for the squares of the equidistribution of the Wigner measures for eigenfunctions of the Laplacian on a manifold whose geodesic flow is ergodic.For the case of an arithmetic hyperbolic surface we obtain the first term in the asymptotics of these quantities. This turns out to be quite intricate and leads to a nonnegative operator whose eigenvalues are special values of L-functions on the critical line. In particular this leads to a proof of the nonnegativity of these special values.

Johannes Sjöstrand: Spectrum of non-self-adjoint operators in dimension 2

Abstract: In a work with A. Melin we found natural stable classes of non-self-adjoint semi-classical operators in dimension 2, for which the spectrum can be determined by a Bohr-Sommerfeld condition in some region of the complex plane which does not shrink when the semi-classical parameter tends to zero. We review that result and discuss more recent results (still in progress) on small non-selfadjoint perturbations of self-adjoint operators (some of which in collaboration with M. Hitrik), as well as related questions and results about resonances.

Terence Tao: A new Morawetz inequality for nonlinear Schrodinger equations

Abstract: The classical Morawetz inequality prevents the solution to a defocusing nonlinear Schrodinger equation from concentrating at a fixed point in space for an extended period of time, and has had many applications to the study of this equation, especially in the radially symmetric case. In this talk we present a variant of this inequality, sharing some features in common with the Glimm interaction potential method, which prevents concentration at variable points in space for extended periods of time. In particular, we are able to remove the radial restriction on some scattering results for the nonlinear Schrodinger equation. This is joint work with Jim Colliander, Mark Keel, Gigliola Staffilani, and Hideo Takaoka.

Daniel Tataru: The nonlinear wave equation

Abstract: The aim of this talk is to describe some recent work concerning the local well-posedness of second order nonliner hyperbolic equations with rough initial data.

Michael Taylor: PDE on Rough Domains

Abstract: Since the breakthrough initiated by A.P. Calderon and completed by Coifman, McIntosh, and Meyer, it has been possible to tackle various elliptic PDE on Lipschitz domains via layer potential techniques. Initial papers concentrated on PDE with constant coefficients, but the desire to remove apparently artificial topological restrictions on these domains leads one naturally to consider PDE with variable coefficients, and then the desire to let those coefficients be as rough as possible naturally arises, and leads to a multitude of interesting analytical problems.
The talk will discuss progress on some of these problems, and a conjecture or two that might lead to further progress.

Gunther Uhlmann: Travel Time Tomography and Boundary Rigidity

Abstract: We survey recent results on the inverse kinematic problem The question is whether one can determine the sound speed (index of refraction) of a medium by measuring the travel times of the corresponding ray paths. This inverse problem arose in geophysics in an attempt to determine the substructure of the Earth by measuring at the surface of the Earth the travel times of seismic waves. An early success of this inverse method was the estimate by Herglotz and Wiechert and Zoeppritz of the structure of the Earth in the case of a spherically symmetric index of refraction. This problem can be reformulated in more general geometric terms as to whether given a compact Riemannian manifold with boundary one can determine the Riemannian metric in the interior knowing the lengths of geodesics joining points on the boundary. This is a problem that also appears naturally in rigidity questions in Riemannian geometry. It is known as the boundary rigidity problem. In this talk we will discuss what is known about the boundary rigidity problem and formulate several open problems.

Steven Zelditch: Statistical Patterns in polynomials with fixed Newton polytope

Abstract: The Bernstein-Kouchnirenko theorem states that the number of joint zeros of a system of m (generic holomorphic) polynomials in m complex variables with fixed Newton polytope P equals Vol(P) m! My talk, based on recent joint work with B. Shiffman, explains how P further influences the mass density of such polynomials and the positions of their zeros and critical points. The basic idea is that P defines a `classically allowed region' in which the mass, joint zeros and critical points concentrate. Only an expontially small amount `tunnels' into the complement. These results are statistical and asymptotic as P is dilated (i.e. the degree tends to infinity). Graphics are included to help vizualize the results.
 
 

There is a banquet on Sunday, probably starting around 7pm, at Ashdown House on the MIT campus. This is across Massachusetts Avenue from the main building, on Memorial Drive, facing the river. The banquet is free of charge for the conference participants. However, please note the general voluntary donation suggested below.

 There will be coffee breaks every morning and afternoon. In order to support the refreshment, etc., expenses, we would like to encourage voluntary donations of $20/person.

 Further information will appear as it becomes available.