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The Spectrum of Hyperbolic Surfaces (Paperback, 1st ed. 2016)
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The Spectrum of Hyperbolic Surfaces (Paperback, 1st ed. 2016)
Series: Universitext
Expected to ship within 10 - 15 working days
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This text is an introduction to the spectral theory of the
Laplacian on compact or finite area hyperbolic surfaces. For some
of these surfaces, called "arithmetic hyperbolic surfaces", the
eigenfunctions are of arithmetic nature, and one may use analytic
tools as well as powerful methods in number theory to study them.
After an introduction to the hyperbolic geometry of surfaces, with
a special emphasis on those of arithmetic type, and then an
introduction to spectral analytic methods on the Laplace operator
on these surfaces, the author develops the analogy between geometry
(closed geodesics) and arithmetic (prime numbers) in proving the
Selberg trace formula. Along with important number theoretic
applications, the author exhibits applications of these tools to
the spectral statistics of the Laplacian and the quantum unique
ergodicity property. The latter refers to the arithmetic quantum
unique ergodicity theorem, recently proved by Elon Lindenstrauss.
The fruit of several graduate level courses at Orsay and Jussieu,
The Spectrum of Hyperbolic Surfaces allows the reader to review an
array of classical results and then to be led towards very active
areas in modern mathematics.
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