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SUMMARY:Acousto-gravitational spectra\, singular geometry and inverse prob
 lems on gas giants - Maarten de Hoop (Rice)
DTSTART:20260402T140000Z
DTEND:20260402T150000Z
UID:TALK246001@talks.cam.ac.uk
CONTACT:Matthew Colbrook
DESCRIPTION:On gas giant planets\, the speed of sound is isotropic and ten
 ds to zero at the surface. Geometrically\, this corresponds to a Riemannia
 n manifold whose metric tensor has a conformal blow-up near the boundary. 
 The blow-up is tamer than in asymptotically hyperbolic geometry: The bound
 ary is at a finite distance.\n\nWe consider the acoustic wave and the acou
 sto-gravitational equations of gas giants. The geometry of acoustic waves 
 is modeled by the above mentioned singular Riemannian\, "gas giant" metric
 . We give an overview of the basic properties of the geometry\, including 
 properties of geodesics near the boundary\, the Hausdorff dimension of the
  boundary\, and the discreteness of the (acoustic) spectrum of the Laplace
 –Beltrami operator. We present the spectral analysis of this operator an
 d derive the Weyl law. The involved exponents depend on the Hausdorff dime
 nsion\, which\, in the supercritical case (the relevant case for Jupiter a
 nd Saturn)\, is larger than the topological dimension. The Weyl asymptotic
 s determines the blow-up in the supercritical case.\n\nWe then consider va
 rious inverse problems for simple gas giant planets\, proving that the met
 ric is uniquely determined by its boundary distance data and that the geod
 esic ray transform is injective. We study the determination of a conformal
  factor of the metric from boundary distance data using methods involving 
 singular microlocal analysis of the normal operator corresponding to the g
 eodesic ray transform. Moreover\, we present the boundary observability of
  acoustic waves given full and local observations and show how the resulti
 ng observability inequality can be turned into a scanning protocol where t
 he observation set moves in time. We conclude with some remarks on inertia
 -gravity modes on gas planets forming the essential spectrum.\n\nJoint res
 earch with Y. Colin de Verdìère\, C. Dietze\, J. Ilmavirta\, A. Kykkäne
 n\, R. Mazzeo and E. Trélat.\n
LOCATION:Centre for Mathematical Sciences\, MR14
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