473 research outputs found
Upper and lower bounds on resonances for manifolds hyperbolic near infinity
For a conformally compact manifold that is hyperbolic near infinity and of
dimension , we complete the proof of the optimal 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 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
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