Series:De Gruyter Textbook
Paperback
Publication Date: August 2015
ISBN: 978-3-11-037949-5
Presents Riemannian geometry and Lie group analysis in partial differential equations and modeling
Designed for developing analytical skills in classical and new methods
Practical, concise and allowing easy access to the topic
This book is based on the experience of teaching the subject by the author in Russia, France, South Africa and Sweden. The author provides students and teachers with an easy to follow textbook spanning a variety of topics on tensors, Riemannian geometry and geometric approach to partial differential equations. Application of approximate transformation groups to the equations of general relativity in the de Sitter space simplifies the subject significantly.
Series:De Gruyter Expositions in Mathematics 61
Hardcover
Publication Date: December 2015
ISBN: 978-3-11-028145-3
This is the fourth volume of a comprehensive and elementary treatment of finite p-group theory. As in the previous volumes, minimal nonabelian p-groups play an important role. Topics covered in this volume include:
subgroup structure of metacyclic p-groups
Ishikawa’s theorem on p-groups with two sizes of conjugate classes
p-central p-groups
theorem of Kegel on nilpotence of H p-groups
partitions of p-groups
characterizations of Dedekindian groups
norm of p-groups
p-groups with 2-uniserial subgroups of small order
The book also contains hundreds of original exercises and solutions and a comprehensive list of more than 500 open problems. This work is suitable for researchers and graduate students with a modest background in algebra.
Series:De Gruyter Expositions in Mathematics 62
Hardcover
ISBN 978-3-11-029534-4
This is the fifth volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume include theory of linear algebras and Lie algebras.
The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.
Series:Radon Series on Computational and Applied Mathematics 19
Hardcover
To be published: October 2016
ISBN: 978-3-11-037013-3
Tensor-structured numerical methods in scientific computing provide a powerful tool for efficient computations in higher dimensions
MATLAB exercises included in the book
Applications in Engeneering and Quantum Many Body Physics
This book presents an introduction to the modern tensor-structured numerical methods in scientific computing. In recent years these methods have been shown to provide a powerful tool for efficient computations in higher dimensions, thus overcoming the so-called “curse of dimensionality,” a problem that encompasses various phenomena that arise when analyzing and organizing data in high-dimensional spaces.
Asterisque 372 (2015), xxi+177 pages
Rigidite des varietes graphees de grande dimension
Ce texte est consacre a la definition et a l'etude systematique des varietes graphees de grande dimension. Celles-ci sont des varietes lisses, ayant une decomposition en un nombre fini de morceaux geometriques. Chaque morceau est diffeomorphe au produit d'un tore et d'une variete hyperbolique de volume fini dont tous les bouts sont des tores. Les morceaux sont recolles par des applications affines des tores qui en sont les bords. Nous exigeons que le facteur hyperbolique dans chaque morceau soit de dimension 3. Notre but principal est d'etablir divers resultats de rigidite pour cette classe de varietes graphees. Nous demontrons, en dimension 6, la conjecture de Borel pour les varietes graphees : une variete quelconque est homotopiquement equivalente a une variete graphee si et seulement si elle est homeomorphe a cette meme variete graphee. Nous etablissons la rigidite lisse pour la classe des varietes graphees : deux varietes graphees sont homotopiquement equivalentes si et seulement si elles sont diffeomorphes. Du point de vue de la geometrie a grande echelle, la distorsion des groupes fondamentaux des morceaux dans le groupe fondamental de la variete graphee joue un role essentiel. Nous introduisons la notion de variete graphee irreductible. Elles forment une sous-classe pour laquelle ces sous-groupes sont toujours non-distordus. Ceci nous permet d'analyser la structure des groupes quasi-isometriques au groupe fondamental d'une variete graphee irreductible: un tel groupe a (virtuellement) une action sur un arbre, avec de fortes contraintes sur les stabilisateurs de sommets et d'aretes. Cette analyse comprend, entre autre, une classification des groupes quasi-isometriques au produit d'un groupe abelien libre et d'un reseau non-uniforme dans SO(n,1). Nous presentons plusieurs exemples de varietes graphees qui n'admettent aucune metrique localement CAT(0). Certains de nos resultats s'appliquent aussi bien en presence de morceaux ayant commes facteurs des surfaces hyperboliques. Nous precisons que, en dimension trois, notre notion de variete graphee ne coincide pas avec la notion classique de variete graphee. Nos varietes forment une classe comprenant certaines des varietes graphees classiques (mais pas toutes: nous excluons certaines sous-varietes de Seifert), ainsi que des varietes que ne sont pas des varietes graphees classiques (nous admettons des morceaux purement hyperboliques).
We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic manifold with toric cusps. The various pieces are attached together via affine maps of the boundary tori. We require all the hyperbolic factors in the pieces to have dimension 3. Our main goal is to study this class of graph manifolds from the viewpoint of rigidity theory. We show that, in dimensions 6, the Borel conjecture holds for our graph manifolds. We also show that smooth rigidity holds within the class: two graph manifolds are homotopy equivalent if and only if they are diffeomorphic. We introduce the notion of irreducible graph manifolds. These form a subclass which has better coarse geometric properties, in that various subgroups can be shown to be quasi-isometrically embedded inside the fundamental group. We establish some structure theory for finitely generated groups which are quasi-isometric to the fundamental group of an irreducible graph manifold: any such group has a graph of groups splitting with strong constraints on the edge and vertex groups. Along the way, we classify groups which are quasi-isometric to the product of a free abelian group and a non-uniform lattice in SO(n,1). We provide various examples of graph manifolds which do not support any locally CAT(0) metric. Several of our results can be extended to allow pieces with hyperbolic surface factors. We emphasize that, in dimension =3, our notion of graph manifold does not coincide with the classical graph manifolds. Rather, it is a class of 3-manifolds that contains some (but not all) classical graph 3-manifolds (we don't allow general Seifert fibered pieces), as well as some non-graph 3-manifolds (we do allow hyperbolic pieces).
Keywords: Quasi-isometry, quasi-action, graph of groups, CAT(0) space, Borel conjecture, smooth rigidity, Baum-Connes conjecture, asymptotic cone, mapping class group, Kazhdan’s property (T), Tits’ alternative, co-Hopf property, C?-simplicity, SQ-universality.
Class. math. : 53C24, 20F65; Secondary:53C23, 20E08, 20F67, 20F69, 19D35.
ISBN : 978-2-85629-809-1
*
Asterisque 373 (2015), viii+281 pages
Etats d’equilibre en courbure negative
Les mesures de Gibbs, utilisees d’abord en thermodynamique et en dynamique symbolique, sont des outils cruciaux pour l’etude de la theorie ergodique des flots geodesiques des varietes de courbure strictement negative. Nous introduisons (via les densite?s de Patterson-Sullivan) un cadre qui permet de s’affranchir d’hypotheses de compacite sur la variete, et nous demontrons de nombreux resultats d’existence, d’unicite et de finitude des mesures de Gibbs. Nous en donnons moultes applications, au principe variationnel, au comptage et a l’equidistribution des points d’orbites et des periodes, a l’unique ergodicite du feuilletage fortement instable, et a la classification des mesures de Gibbs sur certains revetements riemanniens.
With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many existence, uniqueness and finiteness results of Gibbs measures. We give many applications, to the Variational Principle, the counting and equidistribution of orbit points and periods, the unique ergodicity of the strong unstable foliation and the classification of Gibbs densities on some Riemannian covers.
Keywords: Geodesic flow, negative curvature, Gibbs state, periods, orbit counting, Patterson density, pressure, variational principle, strong unstable foliation.
Class. math. : 37D35, 53D25, 37D40, 37A25, 37C35, 53C12
ISBN : 978-2-85629-818-3