Julie Deserti - Dominique Cerveau

Feuilletages et actions de groupes sur les espaces projectifs

Memoires de la SMF 103 (2005), vi+124 pages

Resume :

Un feuilletage holomorphe sur une variete compacte complexe M est un -feuilletage s'il existe une action d'un groupe complexe G telle que les feuilles generiques de soient les orbites de G. On s'interesse essentiellement au cas de la codimension un sur les espaces projectifs dans l'esprit de la theorie des invariants qui ici peuvent etre transcendants. On s'attache a presenter des exemples et des resultats de classification en petite dimension.

Mots clefs : Feuilletage holomorphe, action de groupes, theorie des invariants, calcul formel

Abstract:

Foliations and group actions on projective spaces
A holomorphic foliation on a compact complex manifold M is said to be an -foliation if there exists an action of a complex Lie group G such that the generic leaf of coincides with the generic orbit of G. We study -foliations of codimension one, in particular in projective space, in the spirit of classical invariant theory, but here the invariants are sometimes transcendental ones. We give a list of examples and general properties. Some classification results are obtained in low dimensions.

Key words: Holomorphic foliation, group action, invariant theory, symbolic computation

Class. math. : 37F75, 32M05, 32M17, 32M25, 32S65

ISBN : 2-85629-182-1

Georges Maltsiniotis

La theorie de l'homotopie de Grothendieck

Asterisque 301 (2005), vi+140 pages

Resume :

Le but de ce livre est d'exposer la tres belle theorie de l'homotopie developpee par Grothendieck dans ≪ A la poursuite des champs ≫ . Il s'agit de caracteriser les categories de prefaisceaux qui permettent de modeliser les types d'homotopie, generalisant ainsi la theorie des ensembles simpliciaux. Les criteres degages par Grothendieck montrent que de telles categories, appelees des modelisateurs elementaires , abondent. On expose une construction categorique des extensions de Kan homotopiques a gauche, generalisant une construction des colimites homotopiques par Thomason. On etudie deux classes remarquables de foncteurs, les foncteurs propres et les foncteurs lisses , notions duales l'une de l'autre. Ces foncteurs sont caracterises par des proprietes cohomologiques, inspirees des theoremes de changement de base propre ou lisse, en geometrie algebrique.

Mots clefs :

Aspherique, categorie test, colimite homotopique, ensemble simplicial, equivalence faible, extension de Kan homotopique, foncteur lisse, foncteur propre, homotopie, localisation, modelisateur, prefaisceau

Abstract:

Grothendieck's homotopy theory
The aim of this book is to explain the very beautiful homotopy theory developed by Grothendieck in ≪ Pursuing Stacks ≫ . The question is to characterize categories of presheaves that modelize homotopy types, thus generalizing the theory of simplicial sets. The criteria discovered by Grothendieck show that there are pretty many such categories, called elementary modelizers . We describe a categorical construction of left homotopy Kan extensions, generalizing a construction of homotopy colimits by Thomason. We study two remarkable classes of functors, proper and smooth functors, these two notions being mutually dual. These functors are characterized by cohomological properties inspired by the proper or smooth base change theorem in algebraic geometry.

Key words:

Aspheric, homotopy, homotopy colimit, homotopy Kan extension, localization, modelizer, presheaf, proper functor, simplicial set, smooth functor, test category, weak equivalence

Class. math. : 14F20, 14F35, 18B25, 18F20, 18G10, 18G30, 18G50, 18G55, 55P10, 55P15, 55P60, 55Q05, 55U10, 55U35, 55U40

ISBN : 2-85629-181-3

Luc Robbiano - Claude Zuily

Strichartz estimates for Schrodinger equations with variable coefficients

Memoires de la SMF 101/102 (2005), vi+208 pages

Resume :

Inegalites de Strichartz pour l'equation de Schrodinger a coefficients variables
On demontre les inegalites de Strichartz (locales en temps) pour l'ensemble des indices donnes par l'invariance d'echelle (sauf le point final) pour des perturbations asymptotiquement plates et non captantes du laplacien usuel de , . Le point principal de la preuve, a savoir l'estimation de dispersion, est obtenu en construisant une parametrixe. L'outil principal de cette construction est la theorie de la transformation de FBI construite par Sjostrand.

Mots clefs : Inegalites de Strichartz, equations de Schrodinger, inegalites de dispersion, transformation de FBI, theorie de Sjostrand

Abstract:

We prove the (local in time) Strichartz estimates (for the full range of parameters given by the scaling unless the end point) for asymptotically flat and non trapping perturbations of the flat Laplacian in , . The main point of the proof, namely the dispersion estimate, is obtained in constructing a parametrix. The main tool for this construction is the use of the FBI transform.

Key words: Strichartz estimates, Schrodinger equations, dispersive estimates, FBI transform, Sjostrand's theory

Class. math. : 35A17, 35A22, 35Q40, 35Q55

ISBN : 2-85629-180-5

Roger Nelsen
Claudi Alsina

Math Made Visual
Creating Images for Understanding Mathematics

Hardback (ISBN-13: 9780883857465 | ISBN-10: 0883857464)
available from June 2006

The object of this book is to show how visualization techniques may be employed to produce pictures that have both mathematical and pedagogical interest. Mathematical drawings related to proofs have been produced since antiquity in China, Arabia, Greece and India but only in the last thirty years there has been a growing interest in so-called ‘proofs without words.’ In this book the authors show that behind most of the pictures ‘proving’ mathematical relations are some well-understood methods. The first part of the book consists of twenty short chapters, each one describing a method to visualize some mathematical idea (a proof, a concept, an operation,...) and several applications to concrete cases. Following this the book examines general pedagogical considerations concerning the development of visual thinking, practical approaches for making visualizations in the classroom and a discussion of the role that hands-on material play in this process.

Contents

Introduction; Part I. Visualizing mathematics by creating pictures: 1. Representing numbers by graphical elements; 2. Representing numbers by lengths of segments; 3. Representing numbers by areas of plane figures; 4. Representing numbers by volumes of bodies; 5. Identifying key elements; 6. Employing isometry; 7. Employing similarity; 8. Area preserving transformations; 9. Escaping from the plane; 10. Overlaying tiles; 11. Playing with several copies; 12. Sequential frames; 13. Geometric dissections; 14. Moving frames; 15. Iterative procedures; 16. Introducing colors; 17. Visualization by inclusion; 18. Ingenuity in 3D; 19. Using 3D models; 20. Combining techniques; Part II. Visualization in the classroom; Part III. Hints and solutions to the challenges; References; Index.