EMS Tracts in Mathematics Vol. 8
ISBN 978-3-03719-070-8
July 2009, 392 pages, hardcover, 17 x 24 cm.
Quantum statistical mechanics plays a major role in many fields such as, for instance, thermodynamics, plasma physics, solid-state physics, and the study of stellar structure. While the theory of quantum harmonic oscillators is relatively simple, the case of anharmonic oscillators, a mathematical model of a localized quantum particle, is more complex and challenging. Moreover, infinite systems of interacting quantum anharmonic oscillators possess interesting ordering properties with respect to quantum stabilization.
This book presents a rigorous approach to the statistical mechanics of such systems, in particular with respect to their actions on a crystal lattice.
The text is addressed to both mathematicians and physicists, especially those who are concerned with the rigorous mathematical background of their results and the kind of problems that arise in quantum statistical mechanics. The reader will find here a concise collection of facts, concepts, and tools relevant for the application of path integrals and other methods based on measure and integration theory to problems of quantum physics, in particular the latest results in the mathematical theory of quantum anharmonic crystals. The methods developed in the book are also applicable to other problems involving infinitely many variables, for example, in biology and economics.
Memoires de la SMF 113 (2008), viii+197 pages
Resume :
Dans ce travail, nous etudions les groupes de Chow-Witt. Ces groupes ont ete introduits par J. Barge et F. Morel dans le but de comprendre dans quelle situation un -module projectif de rang egal a la dimension de est isomorphe a un module projectif plus simple .Dans un premier temps, nous montrons que ces groupes satisfont a peu de choses pres les proprietes fonctorielles des groupes de Chow classiques. Nous definissons ensuite pour tout -module localement libre de rang (constant) sur un schema regulier de dimension une classe d'Euler qui est un raffinement de la classe de Chern maximale classique . Cette classe d'Euler satisfait elle aussi de bonnes proprietes fonctorielles. Nous obtenons en particulier que si est un projectif de rang sur un anneau regulier de dimension superieure ou egale a tel que alors .Nous calculons dans un second temps les groupes de Chow-Witt maximaux d'un anneau regulier de dimension et d'une -algebre reguliere de dimension quelconque. Il decoule immediatement de ces calculs que si est un -module projectif de rang egal a la dimension de l'anneau on a si et seulement si .Finalement nous examinons les liens entre les groupes de Chow-Witt et les groupes des classes d'Euler introduits par S. Bhatwadekar et R. Sridharan.
Mots-clefs : groupes de Chow-Witt, classe d'Euler, fibres vectoriels
Abstract:
In this work we study the Chow-Witt groups. These groups were defined by J. Barge et F. Morel in order to understand when a projective module of top rank over a ring has a free factor of rank one, i.e., is isomorphic to .We show first that these groups satisfy the same functorial properties as the classical Chow groups. Then we define for each locally free -module of (constant) rank over a regular scheme an Euler class which is a refinement of the usual top Chern class . The Euler classes satisfy also good fonctorial properties. In particular, we get if is a projective module of rank over a regular ring of dimension such that .Next we compute the top Chow-Witt group of a regular ring of dimension and the top Chow-Witt group of a regular -algebra of finite dimension. For such , we get that if is a projective module of rank equal to the dimension of the ring then if and only if .Finally, we examine the links between the Chow-Witt groups and the Euler class groups defined by S. Bhatwadekar and R. Sridharan.
Keywords: Chow-Witt groups, Euler class, vector bundles
Memoires de la SMF 114 ( 2008), viii+150 pages
Resume :
Actions d'un groupoide quantique mesure
Frank Lesieur a introduit dans sa these (maintenant publiee dans une version revisee et completee dans les Memoires de la SMF (2007)) une notion de groupoide quantique mesure, dans le cadre des algebres de von Neumann, et une simplification des axiomes de Lesieur est placee en appendice de cet article. Nous developpons ici les notions d'action d'un groupoide quantique mesure, de produit-croise et un theoreme de bidualite est demontre, en s'inspirant largement de ce qui a ete fait par Stefaan Vaes pour les groupes quantiques localement compacts. Ainsi, nous prouvons que l'inclusion de l'algebre initiale dans son produit croise est de profondeur 2, ce qui fournit une reciproque a un resultat demontre par Jean-Michel Vallin et l'auteur. De plus, a toute action d'un groupoide quantique mesure, on associe un autre groupoide quantique mesure; ainsi, en particulier, on construit un groupoide quantique mesure associe canoniquement a toute action d'un groupe quantique localement compact; quand cette action est exterieure, ce groupoide quantique mesure est le groupe quantique initial.
Mots-clefs : groupoides quantiques mesures, actions, produits croises, bidualite, inclusion de profondeur 2
Abstract:
Franck Lesieur had introduced in his thesis (now published in an expended and revised version in the Memoires de la SMF (2007)) a notion of measured quantum groupoid, in the setting of von Neumann algebras and a simplification of Lesieur's axioms is presented in an appendix of this article. We here develop the notions of actions, crossed-product, and obtain a biduality theorem, following what had been done by Stefaan Vaes for locally compact quantum groups. Moreover, we prove that the inclusion of the initial algebra into its crossed-product is depth 2, which gives a converse of a result proved by Jean-Michel Vallin and the author. More precisely, to any action of a measured quantum groupoid, we associate another measured quantum groupoid. In particular, starting from an action of a locally compact quantum group, we obtain a measured quantum groupoid canonically associated to this action; when the action is outer, this measured quantum groupoid is the initial locally compact quantum group.
Keywords: measured quantum groupoids, actions, crossed-product, biduality theorem, depth 2 inclusions
Memoires de la SMF 115 ( 2008), vi+103 pages
Resume :
Theorie des espaces de Bergman dans la boule unite de
Ces dernieres annees il y a eu un grand nombre de travaux sur les espaces de Bergman ponderes sur la boule unite de , ou et . Nous etendons cette etude, de maniere tres naturelle, au cas ou est un nombre reel quelconque et . Ce traitement unifie couvre tous les espaces de Bergman classiques, les espaces de Besov, de Lipschitz, l'espace de Bloch, l'espace de Hardy, et celui appele espace d'Arveson. Certains de nos resultats autour de la representation entiere, de l'interpolation complexe, des multiplicateurs de coefficients et des mesures de Carleson, sont nouveaux, y compris pour les espaces de Bergman ordinaires (non-ponderes) sur le disque unite.
Mots-clefs : Boule unite, espace de Bergman, espace de Lipschitz, espace de Bloch, espace d'Arveson, espace de Besov, mesure de Carleson, derivee fractionnelle, representation integrale, decomposition atomique, interpolation complexe, coefficient multiplicateur
Abstract:
There has been a great deal of work done in recent years on weighted Bergman spaces on the unit ball of , where and . We extend this study in a very natural way to the case where is any real number and . This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space , and the so-called Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk.
Keywords:
Unit ball, Bergman space, Lipschitz space, Bloch space, Arveson space, Besov space, Carleson measure, fractional derivative, integral representation, atomic decomposition, complex interpolation, coefficient multiplier
Surveys in Differential Geometry,Volume 13
Hardcover. 318 pages.
ISBN-13: 978-1-57146-138-4
To be published: August 2009