From Wave Propagation to K-theory: a Conference in Honour of the 60th Birthday of Richard Melrose

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.

Schedule

(Still subject to change!)

The talks will be in Room 380-C in the basement of the Department of Mathematics.

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 "theorem-conjecture" of Shnirelman-Zelditch-Colin de Verdiere asserts that if geodesic flow on the cotangent bundle of X is ergodic, the spectral measures associated with the norm-squares 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 Navier-Stokes equations, with no-slip 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 Ray-Singer, 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 T-dual circle bundles with closed 3-form 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 non-normality 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).