By Nicolas Bergeron

ISBN-10: 3319276646

ISBN-13: 9783319276649

ISBN-10: 3319276662

ISBN-13: 9783319276663

This textual content is an advent to the spectral idea of the Laplacian on compact or finite zone hyperbolic surfaces. For a few of these surfaces, referred to as “arithmetic hyperbolic surfaces”, the eigenfunctions are of mathematics nature, and one may possibly use analytic instruments in addition to robust equipment in quantity concept to check them.

After an advent to the hyperbolic geometry of surfaces, with a distinct emphasis on these of mathematics kind, after which an advent to spectral analytic equipment at the Laplace operator on those surfaces, the writer develops the analogy among geometry (closed geodesics) and mathematics (prime numbers) in proving the Selberg hint formulation. besides very important quantity theoretic functions, the writer indicates purposes of those instruments to the spectral records of the Laplacian and the quantum special ergodicity estate. The latter refers back to the mathematics quantum certain ergodicity theorem, lately proved by way of Elon Lindenstrauss.

The fruit of numerous graduate point classes at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces permits the reader to study an array of classical effects after which to be led in the direction of very lively parts in sleek mathematics.

Show description

Read Online or Download The Spectrum of Hyperbolic Surfaces PDF

Best geometry books

Download PDF by Werner Ballmann: Lectures on Kähler Manifolds (Esi Lectures in Mathematics

Those notes are in line with lectures the writer gave on the collage of Bonn and the Erwin Schrödinger Institute in Vienna. the purpose is to provide an intensive creation to the idea of Kähler manifolds with distinct emphasis at the differential geometric facet of Kähler geometry. The exposition begins with a brief dialogue of complicated manifolds and holomorphic vector bundles and an in depth account of the elemental differential geometric homes of Kähler manifolds.

Lectures on Discrete Geometry by Jiří Matoušek (auth.), Jiří Matoušek (eds.) PDF

Discrete geometry investigates combinatorial homes of configurations of geometric gadgets. To a operating mathematician or computing device scientist, it deals refined effects and methods of significant range and it's a origin for fields reminiscent of computational geometry or combinatorial optimization.

Get Discrete Geometry for Computer Imagery: 10th International PDF

This publication constitutes the refereed court cases of the tenth foreign convention on electronic Geometry for laptop Imagery, DGCI 2002, held in Bordeaux, France, in April 2002. The 22 revised complete papers and thirteen posters provided including three invited papers have been conscientiously reviewed and chosen from sixty seven submissions.

Additional resources for The Spectrum of Hyperbolic Surfaces

Sample text

Q; Q/ 38 2 Arithmetic Hyperbolic Surfaces which induces an isomorphism on the real groups1 D1a;b =f˙1g ! q/: Proof It suffices to construct the first isomorphism. Q/ by interior automorphisms. P/ D P. S˛ /jP . Q/ ! Q/ D Q . Before continuing we make a few preliminary calculations. x 2 P/: Already we see that if ˇ is the rotation of angle with axis ˇ, we have ˇ D ˇ D sˇ . q; Q/. ˇi / ¤ 0, the integer r being even if u is in SO and odd otherwise. x/ D . 2; 1/ the connected component of the identity of the special orthogonal group of the quadratic form q on P.

One can then define, by induction on n, the notion of an admissible family of vectors in L. This is a family . f1 ; : : : ; fn / of vectors in L such that • f1 is a vector in L f0g of minimal norm, • the images fP2 ; : : : ; fPn of f2 ; : : : ; fn in the lattice LP WD L=Zf1 of the Euclidean space P and Rn =Rf1 form an admissible family of L, • the vectors fi are of minimal norm among the vectors of L whose image in LP is fPi . ) It is a standard fact that the Z-module L admits an admissible family .

Assume for example that ax21 is non-zero. x21 bx23 /. x21 bx23 /; x0 x1 C bx2 x3 / is an integer solution to ax21 bx22 C abx23 D 0. x1 ; 0; x3 / is a non-zero integer solution (since x1 ¤ 0) to ax21 bx22 C abx23 D 0. 7, the quadratic forms q D ax21 bx22 C abx23 and q0 D x21 x22 C x33 are similar over Q. 3; Q/. q0 ; Z/. 2; R/ then implies that the groups a;b and 1;1 are commensurable in G. 2; Z/. 2; Z/ if and only if ax2 by C abz2 is similar to x2 y2 C z2 . On the other hand 2 ab. abz/2 ; so that the quadratic forms ax2 by2 C abz2 and x2 ay2 bz2 are themselves similar.

Download PDF sample

The Spectrum of Hyperbolic Surfaces by Nicolas Bergeron


by Christopher
4.2

Rated 4.90 of 5 – based on 32 votes