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On the poisson relation for lens spaces / by Donato R. Cianci

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Author	Cianci, Donato R.
Title	On the poisson relation for lens spaces / by Donato R. Cianci
Date	2016
Abstract	Motivated by quantum mechanics and geometric optics, it is a long-standing problem whether the length spectrum of a compact Riemannian manifold can be recovered from its Laplace spectrum. One route to proving that the length spectrum depends on the Laplace spectrum is by computing the singular support of the trace of the corresponding wave group. Indeed, it is well-known that the singular support of the trace of the wave group is a subset of the periods of the geodesic flow: a relation known as the Poisson relation. Proving that the Poisson relation is an equality for a given manifold yields an effective means of recovering the length spectrum of a space from its Laplace spectrum. We investigate whether the Poisson relation is an equality on lens spaces. We show that all lens spaces satisfy the clean intersection hypothesis of Duistermaat and Guillemin. Moreover, we show that the Poisson relation is an equality for homogeneous lens spaces. Finally, we explicitly compute the highest order wave invariants to show that on 3-dimensional lens spaces with fundamental group of prime order the Poisson relation is an equality. Besides spheres and real projective spaces, these are the only other examples of spherical space forms on which it is known that the Poisson relation is an equality.
Description	1 online resource (vi, 70 leaves)
Format	application/pdf
College	Dartmouth College -- Department of Mathematics.
Issuing Body	Dartmouth Dissertations are made available through the Dartmouth Digital Library.
Degree	Thesis (Ph.D.)--Dartmouth College.
Copyright	Made available by the author under a Creative Commons Attribution-NonCommercial License.; Made available by the Trustees of Dartmouth College under a Creative Commons Attribution-NonCommercial License. https://creativecommons.org/licenses/by-nc/4.0/
DOI	https://doi.org/10.1349/ddlp.1942

Title	cianci2016 1
Full Text	ON THE POISSON RELATION FOR LENS SPACES A Thesis Submitted to the Faculty in partial ful llment of the requirements for the degree of Doctor of Philosophy in Mathematics by Donato R. Cianci DARTMOUTH COLLEGE Hanover, New Hampshire April 21, 2016 Examining Committee: Craig J. Sutton, Chair Carolyn S. Gordon Dmitry Jakobson David L. Webb F. Jon Kull, Ph.D. Dean of Graduate Studies

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On the poisson relation for lens spaces / by Donato R. Cianci

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