Distinguished Lecture Series 2017: Maciej Zworski19 Oct 2016, by Events in
27 – 29 March 2017
Organised in collaboration with the School of Mathematics, University of Bristol, UK
Maciej Zworski, University of California, Berkeley, USA
Scattering by the Sphere
Next year brings the 100th anniversary of Watson’s classic paper “The diffraction of electric waves by the earth” and I would like to use this as a welcome excuse to review various results about scattering by spherical obstacles. Is the sphere determined by its scattering resonances among other obstacles? Do waves decay faster or slower for other obstacles of the same diameter? What about resonances of hollow spheres, that is, of Helmholtz resonators? What about spheres in hyperbolic space?
A Discrete View of Correlations
Resonances introduced in the first lecture can be seen in long time asymptotics of correlations for the wave equation. Many other physical systems are described using evolution of states. Observations are then based on correlations, that is on measuring an evolved state against another state. The time representation can be replaced by the frequency representation (by taking the Fourier transform) which produces the power spectrum. Riemann first proposed, in a different language, that fine features of correlations can be understood by continuing the power spectrum into complex frequencies. Bounds on imaginary parts of singularities of that continuation can be of interest in different settings and I will present various recent results.
Pollicott–Ruelle Resonances from a Scattering Theory Point of View
Following the insights of Faure-Sjoestrand and Tsujii, microlocal/semiclassical methods have proved themselves useful in the study of closed and open smooth hyperbolic systems. I will explain how they give a simple proof of the meromorphy of Ruelle zeta functions for Anosov flows (and indicate the ideas behind the Axiom A case), show stochastic stability of Pollicott–Ruelle resonances and prove that there are always infinitely many of these resonances.