Harmonic Analysis and Fractal Geometry

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Emilio Corso
Speaker Affiliation: 

February 8, 2023

ESB 4127 (at PIMS)

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What happens to a progressively dilating body when folding the
space in which it lives? For a start, we shall examine the problem in a
Euclidean context, surveying results of Randol and Strichartz through a
classical Fourier-analytic approach. Afterwards, following a recent joint
work with Ravotti, we take up the same question in the hyperbolic plane,
focusing however on its upgrade to the unit tangent bundle. This allows
abstract harmonic analysis to enter the picture, in the form of the theory
of unitary representations of the special linear group. In conjunction with
an ingenious strategy pioneered by Ratner in the eighties, the method leads
to a full quantitative understanding of the uniform distribution properties
of expanding dilates of homogeneous curves on compact hyperbolic surfaces,
thereby refining the qualitative analogue established by Margulis in his
doctoral thesis.

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