Download or read book The Spectrum of Hyperbolic Surfaces written by Nicolas Bergeron and published by Springer. This book was released on 2016-02-19 with total page 375 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Le spectre des surfaces hyperboliques written by Nicolas Bergeron and published by . This book was released on 2011 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book G om trie des surfaces hyperboliques written by Emmanuel Philippe and published by . This book was released on 2008 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dans ce mémoire, on décrit le début du spectre des longueurs de tous les groupes de triangles associés à un triangle hyperbolique (r,p,q) avec r,p,q entiers ordonnés dans l'ordre croissant. On montre alors que la donnée du spectre des longueurs caractérise, sauf si r=3 , la classe d'isométrie d'un tel groupe parmi tous les groupes de triangles
Download or read book Progress in Inverse Spectral Geometry written by Stig I. Andersson and published by Birkhäuser. This book was released on 2012-12-06 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most polynomial growth on every half-space Re (z) ::::: c. Moreover, Op(t) depends holomorphically on t for Re t> O. General references for much of the material on the derivation of spectral functions, asymptotic expansions and analytic properties of spectral functions are [A-P-S] and [Sh], especially Chapter 2. To study the spectral functions and their relation to the geometry and topology of X, one could, for example, take the natural associated parabolic problem as a starting point. That is, consider the 'heat equation': (%t + p) u(x, t) = 0 { u(x, O) = Uo(x), tP which is solved by means of the (heat) semi group V(t) = e- ; namely, u(·, t) = V(t)uoU· Assuming that V(t) is of trace class (which is guaranteed, for instance, if P has a positive principal symbol), it has a Schwartz kernel K E COO(X x X x Rt, E* ®E), locally given by 00 K(x, y; t) = L>-IAk(~k ® 'Pk)(X, y), k=O for a complete set of orthonormal eigensections 'Pk E COO(E). Taking the trace, we then obtain: 00 tA Op(t) = trace(V(t)) = 2::>- k. k=O Now, using, e. g., the Dunford calculus formula (where C is a suitable curve around a(P)) as a starting point and the standard for malism of pseudodifferential operators, one easily derives asymptotic expansions for the spectral functions, in this case for Op.
Download or read book Annals of Mathematics written by and published by . This book was released on 1980 with total page 650 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book SURFACES ALGEBRIQUES HYPERBOLIQUES PROPRIETES DE NEGATIVITE DE LACOURBURE written by JAWHER.. EL GOUL and published by . This book was released on 1997 with total page 76 pages. Available in PDF, EPUB and Kindle. Book excerpt: LE THEME CENTRAL DE CETTE THESE EST L'ETUDE DE CERTAINES PROPRIETES DE NEGATIVITE DE COURBURE DES SURFACES ALGEBRIQUES HYPERBOLIQUES AU SENS DE KOBAYASHI. L'ETUDE QUE NOUS AVONS FAITE EST DIVISEE EN DEUX PARTIES : DANS UN PREMIER TEMPS, NOUS INTRODUISONS LA NOTION DE CONNEXION PROJECTIVE PARTIELLE A COEFFICIENTS MEROMORPHES, ET MONTRONS COMMENT DE TELLES CONNEXIONS PEUVENT SERVIR A CONSTRUIRE DES OPERATEURS WRONSKIENS GLOBAUX AGISSANT SUR LES JETS DE COURBES HOLOMORPHES (UN TYPE TRES PARTICULIER DE JETS DE DIFFERENTIELLES INVARIANTES). GRACE A UN THEOREME D'ANNULATION DU WRONSKIEN REPOSANT SUR DES HYPOTHESES DE NEGATIVITE DE LA COURBURE DE RICCI ET GENERALISANT DES RESULTATS ANTERIEURS DE GREEN-GRIFFITHS, SIU ET NADEL, NOUS DONNONS DES EXEMPLES EXPLICITES DE FAMILLES ALGEBRIQUES DE SURFACES HYPERBOLIQUES DANS L'ESPACE PROJECTIF, DE DEGRE QUELCONQUE SUPERIEUR OU EGAL A ONZE. CECI ILLUSTRE UNE (TOUTE PETITE) PARTIE D'UNE CONJECTURE CELEBRE DE S. KOBAYASHI, QUI PREVOIT QU'UNE HYPERSURFACE GENERIQUE DE L'ESPACE PROJECTIF DE DEGRE ASSEZ GRAND (PAR RAPPORT A LA DIMENSION) EST HYPERBOLIQUE. DANS UN DEUXIEME TEMPS, NOUS MONTRONS LA PRESQUE AMPLITUDE AU SENS DE MIYAOKA DU FIBRE COTANGENT A TOUTE SURFACE DE TYPE GENERAL FIBREE SUR UNE COURBE DE GENRE SUPERIEUR OU EGAL A DEUX, ET QUI ADMET AU MOINS UNE FIBRE SINGULIERE. COMME APPLICATION, NOUS OBTENONS UNE REPONSE POSITIVE A UNE CONJECTURE DE DEMAILLY (QUI RELIE L'HYPERBOLICITE A UNE PROPRIETE DE NEGATIVITE DE COURBURE AU SENS DES JETS) DANS LE CAS D'UNE SURFACE FIBREE HYPERBOLIQUE STABLE SUR UNE COURBE DE GENRE AU MOINS DEUX. CES RESULTATS SONT OBTENUS GRACE A UNE ETUDE ALGEBRO-GEOMETRIQUE QUI NOUS PERMET DE CONSTRUIRE DES SECTIONS GLOBALES DES FIBRES EN DROITES TAUTOLOGIQUES ASSOCIES AUX ESPACES DES JETS. LES TECHNIQUES QUE NOUS UTILISONS SONT ASSEZ VARIEES, ET COMBINENT DES OUTILS ALGEBRIQUES TELS QUE DES THEOREMES DE RIGIDITE ET D'ADDITIVITE DES DIMENSIONS DE KODAIRA-IITAKA, ET DES OUTILS ANALYTIQUES DE GEOMETRIE DIFFERENTIELLE COMPLEXE : METRIQUES HERMITIENNES SINGULIERES ET TECHNIQUES DE RECOLLEMENT DE METRIQUES.
Download or read book Zeta Functions in Geometry written by Kurokawa N. (Nobushige) and published by . This book was released on 1992 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains accounts of work presented during the research conference, ``Zeta Functions in Geometry,'' held at the Tokyo Institute of Technology in August 1990. The aim of the conference was to provide an opportunity for the discussion of recent results by geometers and number theorists on zeta functions in several different categories. The exchange of ideas produced new insights on various geometric zeta functions, as well as the classical zeta functions. The zeta functions covered here are the Selberg zeta functions, the Ihara zeta functions, spectral zeta functions, and those associated with prehomogeneous vector spaces. Accessible to graduate students with background in geometry and number theory, Zeta Functions in Geometry will prove useful for its presentation of new results and up-to-date surveys.
Download or read book Dynamical Numbers Interplay between Dynamical Systems and Number Theory written by S. F. Koli︠a︡da and published by American Mathematical Soc.. This book was released on 2010 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains papers from the special program and international conference on Dynamical Numbers which were held at the Max-Planck Institute in Bonn, Germany in 2009. These papers reflect the extraordinary range and depth of the interactions between ergodic theory and dynamical systems and number theory. Topics covered in the book include stationary measures, systems of enumeration, geometrical methods, spectral methods, and algebraic dynamical systems.
Download or read book Random Surfaces written by Scott Sheffield and published by . This book was released on 2005 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Geodesic and Horocyclic Trajectories written by Françoise Dal’Bo and published by Springer Science & Business Media. This book was released on 2010-11-12 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geodesic and Horocyclic Trajectories presents an introduction to the topological dynamics of two classical flows associated with surfaces of curvature −1, namely the geodesic and horocycle flows. Written primarily with the idea of highlighting, in a relatively elementary framework, the existence of gateways between some mathematical fields, and the advantages of using them, historical aspects of this field are not addressed and most of the references are reserved until the end of each chapter in the Comments section. Topics within the text cover geometry, and examples, of Fuchsian groups; topological dynamics of the geodesic flow; Schottky groups; the Lorentzian point of view and Trajectories and Diophantine approximations.
Download or read book Spectral Geometry written by Pierre H. Berard and published by Springer. This book was released on 2006-11-14 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Old and New Aspects in Spectral Geometry written by M.-E. Craioveanu and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is known that to any Riemannian manifold (M, g ) , with or without boundary, one can associate certain fundamental objects. Among them are the Laplace-Beltrami opera tor and the Hodge-de Rham operators, which are natural [that is, they commute with the isometries of (M,g)], elliptic, self-adjoint second order differential operators acting on the space of real valued smooth functions on M and the spaces of smooth differential forms on M, respectively. If M is closed, the spectrum of each such operator is an infinite divergent sequence of real numbers, each eigenvalue being repeated according to its finite multiplicity. Spectral Geometry is concerned with the spectra of these operators, also the extent to which these spectra determine the geometry of (M, g) and the topology of M. This problem has been translated by several authors (most notably M. Kac). into the col loquial question "Can one hear the shape of a manifold?" because of its analogy with the wave equation. This terminology was inspired from earlier results of H. Weyl. It is known that the above spectra cannot completely determine either the geometry of (M , g) or the topology of M. For instance, there are examples of pairs of closed Riemannian manifolds with the same spectra corresponding to the Laplace-Beltrami operators, but which differ substantially in their geometry and which are even not homotopically equiva lent.
Download or read book S minaire de th orie spectrale et g om trie written by and published by . This book was released on 2000 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Twisted Isospectrality Homological Wideness and Isometry written by Gunther Cornelissen and published by Springer Nature. This book was released on 2023-05-10 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: The question of reconstructing a geometric shape from spectra of operators (such as the Laplace operator) is decades old and an active area of research in mathematics and mathematical physics. This book focusses on the case of compact Riemannian manifolds, and, in particular, the question whether one can find finitely many natural operators that determine whether two such manifolds are isometric (coverings). The methods outlined in the book fit into the tradition of the famous work of Sunada on the construction of isospectral, non-isometric manifolds, and thus do not focus on analytic techniques, but rather on algebraic methods: in particular, the analogy with constructions in number theory, methods from representation theory, and from algebraic topology. The main goal of the book is to present the construction of finitely many “twisted” Laplace operators whose spectrum determines covering equivalence of two Riemannian manifolds. The book has a leisure pace and presents details and examples that are hard to find in the literature, concerning: fiber products of manifolds and orbifolds, the distinction between the spectrum and the spectral zeta function for general operators, strong isospectrality, twisted Laplacians, the action of isometry groups on homology groups, monomial structures on group representations, geometric and group-theoretical realisation of coverings with wreath products as covering groups, and “class field theory” for manifolds. The book contains a wealth of worked examples and open problems. After perusing the book, the reader will have a comfortable working knowledge of the algebraic approach to isospectrality. This is an open access book.
Download or read book Annales de l Institut Fourier written by and published by . This book was released on 1949 with total page 1072 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematical Reviews written by and published by . This book was released on 2005 with total page 700 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Spectral Analysis in Geometry and Number Theory written by Motoko Kotani and published by American Mathematical Soc.. This book was released on 2009 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is an outgrowth of an international conference in honor of Toshikazu Sunada on the occasion of his sixtieth birthday. The conference took place at Nagoya University, Japan, in 2007. Sunada's research covers a wide spectrum of spectral analysis, including interactions among geometry, number theory, dynamical systems, probability theory and mathematical physics. Readers will find papers on trace formulae, isospectral problems, zeta functions, quantum ergodicity, random waves, discrete geometric analysis, value distribution, and semiclassical analysis. This volume also contains an article that presents an overview of Sunada's work in mathematics up to the age of sixty.
