Upper and lower bounds on resonances for manifolds hyperbolic near infinity

For a conformally compact manifold that is hyperbolic near infinity and of dimension $n+1$, we complete the proof of the optimal $O(r^{n+1})$ upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an $r^{n+1}$ lower bound on the counting function for scattering poles.Comment: 29 pages, minor corrections, added one figur

Scattering theory for conformally compact metrics with variable curvature at infinity

We develop the scattering theory of general conformally compact metrics. For low frequencies, the domain of the scattering matrix is shown to be frequency dependent. In particular, generalized eigenfunctions exhibit L^2 decay in directions where the asymptotic curvature is sufficiently negative. The scattering matrix is shown to be a pseudodifferential operator. For generic frequency in this part of the continuous spectrum, we give an explicit construction of the resolvent kernel.Comment: AMS-LaTeX, 43 pages, 5 figure

Scattering theory and deformations of asymptotically hyperbolic metrics

For an asymptotically hyperbolic metric on the interior of a compact manifold with boundary, we prove that the resolvent and scattering operators are continuous functions of the metric in the appropriate topologies.Comment: 21 pages, AMS-LaTe

Inverse scattering results for manifolds hyperbolic near infinity

We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.Comment: 25 pages. v3: Minor corrections, references adde

Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?

About a year ago, a team of physicists reported in Science that they had observed "evidence for E8 symmetry" in the laboratory. This expository article is aimed at mathematicians and explains the chain of reasoning connecting measurements on a quasi-1-dimensional magnet with a 248-dimensional Lie algebra.Comment: v2: Added some material clarifying the word "evidence" and the precise role of E8 in the theory // v3: many small changes to the expositio