Geometric Analysis: a Conference in Honor of Richard Melrose
Department of Mathematics
Massachusetts Institute of Technology
SaturdayMonday, March 2325, 2002
This was a conference in Geometric Analysis in honor of Richard
Melrose, on the occasion of his 25th anniversary at MIT.
The following distinguished mathematicians spoke at
the meeting:
Nicolas Burq 
Charles Epstein 
Charles Fefferman 
Mitsuru Ikawa 
Victor Ivrii 
Peter Lax 
Vesselin Petkov 
Peter Sarnak 
Johannes Sjöstrand 
Terry Tao 
Daniel Tataru 
Michael Taylor 

Gunther Uhlmann 
Steve Zelditch 
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.
The talks will take place in
Building 34,
room 101.
This building is located at 50 Vassar Street.
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 semiclassical parametrix approach. Most
of these results were obtained in collaboration with P. Gerard and N. Tzvetkov
for University ParisSud Orsay.
Charles Epstein: Indices and relative indices in contact and CRgeometry
Charles Fefferman: QCurvature and the Poincare Metric
Abstract: The talk explains recent joint work with Robin Graham, giving
a simple construction for Tom Branson's Qcurvature 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 $O(\bigl\log xy \bigr^{1})$
and derive semiclassical spectral asymptotics with sharp remainder estimate
$O(h^{1d})$; for operators with continuity modulus $o(\bigl\log xy\bigr^{1})$
we derive semiclassical spectral asymptotics with the remainder estimate
$o(h^{ld})$ 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 LaxWendroff with upwind. Joint work with XuDong 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 BreitWigner 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 Lfunctions on the critical line. In particular this
leads to a proof of the nonnegativity of these special values.
Johannes Sjöstrand: Spectrum of nonselfadjoint operators in dimension
2
Abstract: In a work with A. Melin we found natural stable classes of nonselfadjoint
semiclassical operators in dimension 2, for which the spectrum can be
determined by a BohrSommerfeld condition in some region of the complex
plane which does not shrink when the semiclassical parameter tends to
zero. We review that result and discuss more recent results (still in progress)
on small nonselfadjoint perturbations of selfadjoint 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 wellposedness 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 BernsteinKouchnirenko 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.
MIT has arranged for a special rate of $109.00/night plus tax, including
breakfast, at the
Marriott
Residence Inn in Cambridge, a few blocks from campus, for the nights
of March 22, 23 and 24. To make a reservation, please call the central
reservation number of Marriott, 18003313131, and mention that you are
participating in the MIT mathematics conference. You may also contact Aaron
Katz at the Marriott at 6175772545 or by email at
aaron.katz@marriott.com.
If you are sharing a room with someone, mention that a double room is required
(at $119/night+tax). Please let the organizers know if you are experiencing
any problems.
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.