From Wave Propagation to Ktheory: a Conference in Honour of the 60th
Birthday of Richard Melrose
SaturdaySunday, October 2526, 2008
This will be a conference on the wide range of fields of interest of Richard
Melrose, honouring him on the occasion of his 60th birthday.
The following distinguished mathematicians will speak at
the meeting:
Gilles Carron (Nantes) 
Victor Guillemin (MIT) 
Gilles Lebeau (Nice) 
Mathai Varghese (Adelaide) 
Michael Taylor (North Carolina) 
Gunther Uhlmann (Washington) 
Jared Wunsch (Northwestern) 
Maciej Zworski (Berkeley) 
The schedule and abstracts are given below.
There will be a banquet on Saturday at 7pm at
MacArthur Park Restaurant.
It is right by the Palo Alto Caltrain stop, on the Stanford side. See
the restaurant web site for details.
Please register for the conference!
Travel information is available
here.
Further information will be posted as it becomes available. The available
information may be more up to date on the copy of this web page
here.
Schedule
(Still subject to change!)
The talks will be in Room 380C in the basement of the Department of
Mathematics.
Saturday:  
10:0010:45  Refreshments 
10:45  11:45  Victor Guillemin 
Lunch break  
1:30  2:30  Gilles Lebeau 
2:45  3:45  Jared Wunsch 
Coffee break  
4:15  5:15  Michael Taylor 
7:00  Banquet 
 
Sunday:  
9:009:45  Refreshments 
9:45  10:45  Maciej Zworski 
11:00  12:00  Mathai Varghese 
Lunch break  
2:00  3:00  Gilles Carron 
3:15  4:15  Gunther Uhlmann 
4:45  Bus for Berkeley 
Titles and abstracts

Gilles Carron
"QALE metrics on the Hilbert Scheme of n points on C^2"
Abstract : We show that the Hyperkahler metric introduced by H. Nakajima on
the Hilbert Scheme of n points on C^2
is QALE (Quasi Asymptotically Locally Euclidean).

Victor Guillemin
"Asymptotic properties of spectral measures"
Abstract: Let X be a Riemannian manifold and dx its volume form.
The quantum ergodicity "theoremconjecture" of ShnirelmanZelditchColin
de Verdiere asserts that if geodesic flow on the cotangent bundle of X is
ergodic, the spectral measures associated with the normsquares of the
eigenfunctions of the Laplace operator tend to dx as n tends to infinity.
In this talk we will discuss some versions of this result involving
"packets" of eigenfuntions and some analogues of this result in Kaehler
geometry.

Gilles Lebeau
"Experimental study of the HUM control operator for linear waves"
(with Maelle Nodet)
Abstract : We consider the problem of the numerical approximation of the
linear controllability of waves. All experiments are done in a bounded
domain $\Omega$ of the plane, with Dirichlet boundary conditions and
internal control. We use a Galerkin approximation of the optimal control
operator of the continuous model, based on the spectral theory of the
Laplace operator in $\Omega$. This allows us to obtain surprisingly good
illustrations of the main theoretical results available on the
controllability of waves, and
to formulate some questions for the future analysis of optimal control
theory of waves.

Michael Taylor
"Vanishing viscosity limits for a class of 3D flows,
and related singular perturbation problems"
Abstract. We study special classes of solutions to the 3D
NavierStokes equations, with noslip boundary conditions,
for which it is possible to establish convergence in the limit
of vanishing viscosity to solutions to the Euler equations.
Examples include certain classes of channel flows and of circular
pipe flows. Carrying out the analysis also leads to consideration of
related singular perturbation problems on bounded domains.

Gunther Uhlmann
"The scattering relation and the broken scattering relation"
Abstract: We review in this talk some recent results on the inverse problems
of determining a Riemannian metric by known the scattering relation (or lens
relation) or the broken scattering relation. The scattering relation
measures the exit points of a medium and directions of ray paths (geodesics)
of waves, given its entrance point and entrance direction as well as the
travel times. The broken scattering relation measures the exit points and
directions of once broken (reflected) ray paths and the travel times. This
is joint work with L. Pestov and P. Stefanov in the case of the scattering
relation and Y. Kurylev and M. Lassas in the case of the broken scattering
relation.

Varghese Mathai
"Analytic torsion for twisted de Rham complexes"
Abstract:
We define analytic torsion for the twisted de Rham complex, consisting of
differential forms on a compact Riemannian manifold X with coefficients in a
flat vector bundle E, with a differential given by a flat connection on E
plus a closed odd degree differential form on X. The definition in our case
is more complicated than in the case discussed by RaySinger, as it uses
pseudodifferential operators. We show that this analytic torsion is
independent of the choice of metrics on X and E, establish some basic
functorial properties, and compute it in many examples. We also establish
the relationship of an invariant version of analytic torsion for Tdual
circle bundles with closed 3form flux. This is joint work with Siye Wu.

Jared Wunsch
"Diffraction of wavefronts on manifolds with corners"
I will discuss some recent joint work with Melrose and Vasy on the
microlocal regularity of solutions to the wave equation on manifolds with
corners. We show that under certain hypotheses, the singularities
`diffracted' by the corner are weaker than those incident upon it.

Maciej Zworski
"Breathing patterns in nonlinear relaxation"
Abstract: In numerical experiments involving nonlinear solitary waves
propagating through nonhomogeneous media one observes "breathing" in the
sense of the amplitude of the wave going up and down on a much faster scale
than the motion of the wave. In this paper we investigate this phenomenon in
the simplest case of stationary waves in which the evolution corresponds to
relaxation to a nonlinear ground state. The particular model is the popular
$\delta_0$ impurity in the cubic nonlinear Schrödinger equation on the
line. We give asymptotics of the amplitude on a finite but relevant time
interval and show their remarkable agreement with numerical experiments. We
stress the nonlinear origin of the "breathing patterns" caused by the
selection of the ground state depending on the initial data, and by the
nonnormality of the linearized operator.
One dimensional scattering theory which I learned from Richard Melrose as
an undergraduate plays a crucial rôle in this analysis. It is joint
work with Justin Holmer.
This conference is partially supported by a Chambers Fellowship and by the Stanford Department
of Mathematics.
Some travel support is available. Please contact one of the organizers
for details.
With Regards from the Organizing Committee:

Rafe Mazzeo (Stanford),

András Vasy (Stanford).