Kirill Mackenzie / University of Sheffield

General Theory of Lie Groupoids and Lie Algebroids

Series: London Mathematical Society Lecture Note Series (No. 213)

Paperback (ISBN-10: 0521499283 | ISBN-13: 9780521499286)

June 2005 | 540 pages | 298 x 222 mm

Lecturers can request inspection copies of this title.
This is a comprehensive modern account of the theory of Lie groupoids and Lie algebroids, and their importance in differential geometry, in particular their relations with Poisson geometry and general connection theory. It covers much work done since the mid 1980s including the first treatment in book form of Poisson groupoids, Lie bialgebroids and double vector bundles, as well as a revised account of the relations between locally trivial Lie groupoids, Atiyah sequences, and connections in principal bundles. As such, this book will be of great interest to all those concerned with the use of Poisson geometry as a semi-classical limit of quantum geometry, as well as to all those working in or wishing to learn the modern theory of Lie groupoids and Lie algebroids.

● Book includes many results which have never appeared in book form before

● Massive expansion of a successful earlier book

● A thorough and detailed account of the subject

Contents

Part I. The General Theory: 1. Lie groupoids: fundamental theory; 2. Lie groupoids: algebraic constructions; 3. Lie algebroids: fundamental theory; 4. Lie algebroids: algebraic constructions; Part II. The Transitive Theory: 5. Infinitesimal connection theory; 6. Path connections and Lie theory; 7. Cohomology and Schouten calculus; 8. The cohomological obstruction; Part III. The Poisson and Symplectic Theories: 9. Double vector bundles; 10. Poisson structures and Lie algebroids; 11. Poisson and symplectic groupoids; 12. Lie bialgebroids; Appendix; Bibliography; Index.

Alan D. Taylor / Union College, New York

Social Choice and the Mathematics of Manipulation

Series: Outlooks

Hardback (ISBN-10: 0521810523 )
Paperback (ISBN-10: 0521008832 )

June 2005

Honesty in voting, it turns out, is not always the best policy. Indeed, in the early 1970s, Allan Gibbard and Mark Satterthwaite, building on the seminal work of Nobel laureate Kenneth Arrow, proved that with three or more alternatives there is no reasonable voting system that is non-manipulable; voters will always have an opportunity to benefit by submitting a disingenuous ballot. The ensuing decades produced a number of theorems of striking mathematical naturality that dealt with the manipulability of voting systems. This book presents many of these results from the last quarter of the twentieth century, especially the contributions of economists and philosophers, from a mathematical point of view, with many new proofs. The presentation is almost completely self-contained, and requires no prerequisites except a willingness to follow rigorous mathematical arguments. Mathematics students, as well as mathematicians, political scientists, economists and philosophers will learn why it is impossible to devise a completely unmanipulable voting system.

● This is the only book that tries to bring together the main results from the last quarter of the twentieth century that deal with the manipulability of voting systems

● This is a unified treatment; a vast majority of the proofs are new

● This is a self-contained treatment; there are no prerequisites except a willingness to follow mathematical arguments

Contents

1. Introduction; 2. The Gibbard-Satterthwaite theorem; 3. Additional results for single-valued elections; 4. The Duggan-Schwartz theorem; 5. Additional results for multi-valued elections; 6. Ballots that rank sets; 7. Elections with outcomes that are lotteries; 8. Elections with variable agendas; References; Index.

Gustave Choquet - Thierry De Pauw - Pierre de la Harpe
- Jean-Pierre Kahane - Herve Pajot - Bruno Sevennec

Autour du centenaire Lebesgue

Panoramas et Syntheses 18 (2004), xii+156 pages

Resume
:
Ce volume a ete ecrit a l'occasion du centenaire de la publication en 1901 de la fameuse note de Lebesgue introduisant son integrale. Il fait suite a une journee de celebration organisee a l'Ecole normale superieure de Lyon. On y trouvera differents eclairages sur l'heritage de Lebesgue. Le temoignage de Gustave Choquet redonne vie aux mathematiques et mathematiciens de l'epoque de Lebesgue. Les textes de Pierre de la Harpe et Bruno Sevennec sur les mesures finiment additives analysent leurs paradoxes et leurs liens avec la notion de moyennabilite ou l'equirepartition. La contribution de Herve Pajot rend compte des progres considerables qui ont ete faits recemment dans la comprehension de la notion de rectifiabilite, en liaison avec la capacite analytique ou l'operateur de Cauchy; celle de Thierry De Pauw part de l'integrale de Henstock et Kurzweil pour s'interesser aux generalisations possibles de la formule de la divergence. Enfin, la preface de Jean-Pierre Kahane fait un lien entre tous ces eclairages, en meme temps qu'elle lui permet d'evoquer l'influence mathematique de l'integrale de Lebesgue tout au long du vingtieme siecle.

Mots clefs :

Capacite analytique, courbure de Menger, ensembles, ensembles a perimetre fini, equirepartition, geometrie, groupe compact, groupe moyennable, histoire, integrales, integrale de Cauchy, integration, mesures, mesures finiment additives, mesure de Haar, mesures de Hausdorff, moyennes invariantes, paradoxes, probabilites, rectifiabilite, rectifiabilite uniforme, theoreme de la divergence, trou spectral

Abstract:

Around the Lebesgue centenary
This volume was written on the occasion of the centennial of Lebesgue's Publication in 1901 of his famous Note introducing his integral. It results from a day of celebration at the Ecole normale superieure de Lyon. It provides various viewpoints on Lebesgue's heritage. Gustave Choquet gives a vivid testimony about mathematics and mathematicians of Lebesgue's era. Contributions by Pierre de la Harpe and by Bruno Sevennec on finitely additive measures analyse their paradoxes and their relationship with amenability or equirepartition. Herve Pajot relates the recent and considerable progress made in understanding the notion of rectifiability, in relation with the analytic capacity or with the Cauchy operator. Thierry de Pauw, starting from Henstock and Kurzweil's integral, studies possibilities of generalizing the divergence formula. A preface by Jean-Pierre Kahane synthesizes these viewpoints and highlights Lebesgue's influence in the course of the twentieth century.

Key words:

Amenable groups, analytic capacity, Cauchy integral, compact group, divergence theorem, equidistribution, finitely additive measures, geometry, Haar measure, Hausdorff measures, history, integration, integrals, invariant means, measures, Menger curvature, paradoxes, probabilities, rectifiability, sets, sets of finite perimeter, spectral gap, uniform rectifiability

A D Barbour (University of Zurich, Switzerland)
& Louis H Y Chen (National University of Singapore, Singapore)

STEIN'S METHOD AND APPLICATIONS

Stein's startling technique for deriving probability approximations first appeared about 30 years ago. Since then, much has been done to refine and develop the method, but it is still a highly active field of research, with many outstanding problems, both theoretical and in applications. This volume, the proceedings of a workshop held in honour of Charles Stein in Singapore, August 2003, contains contributions from many of the mathematicians at the forefront of this effort. It provides a cross-section of the work currently being undertaken, with many pointers to future directions. The papers in the collection include applications to the study of random binary search trees, Brownian motion on manifolds, Monte-Carlo integration, Edgeworth expansions, regenerative phenomena, the geometry of random point sets, and random matrices.

Contents:

Zero Biasing in One and Higher Dimensions, and Applications (L Goldstein & G Reinert)
Poisson Limit Theorems for the Appearances of Attributes (O Chryssaphinou et al.)
Normal Approximation in Geometric Probability (M D Penrose & J E Yukich)
Stein's Method, Edgeworth's Expansions and a Formula of Barbour (V Rotar)
Stein's Method for Compound Poisson Approximation via Immigration?Death Processes (A Xia)
The Central Limit Theorem for the Independence Number for Minimal Spanning Trees in the Unit Square (S Lee & Z Su)
Stein's Method, Markov Renewal Point Processes, and Strong Memoryless Time (T Erhardsson)
Multivariate Poisson?Binomial Approximation Using Stein's Method (A D Barbour)
An Explicit Berry?Esseen Bound for Student's t-Statistic via Stein's Method (Q-M Shao)
An Application of Stein's Method to Maxima in Hypercubes (Z D Bai et al.)
Exact Expectations of Minimal Spanning Trees for Graphs with Random Edge Weights (J A Fill & J M Steele)
Limit Theorems for Spectra of Random Matrices with Martingale Structure (F Gotze & A N Tikhomirov)
Characterization of Brownian Motion on Manifolds Through Integration by Parts (E P Hsu)
On the Asymptotic Distribution of Some Randomized Quadrature Rules (W-L Loh)
The Permutation Distribution of Matrix Correlation Statistics (A D Barbour & L H Y Chen)
Applications of Stein's Method in the Analysis of Random Binary Search Trees (L Devroye)

Readership: Researchers and graduate students in probability.

320pp Pub. date: May 2005
ISBN 981-256-281-8

Edited by F. Mezzadri / University of Bristol
N. C. Snaith / University of Bristol

Recent Perspectives in Random Matrix Theory and Number Theory

Series: London Mathematical Society Lecture Note Series

Paperback (ISBN-10: 0521620589 )

June 2005

In recent years the application of random matrix techniques to analytic number theory has been responsible for major advances in this area of mathematics. As a consequence it has created a new and rapidly developing area of research. The aim of this book is to provide the necessary grounding both in relevant aspects of number theory and techniques of random matrix theory, as well as to inform the reader of what progress has been made when these two apparently disparate subjects meet. This volume of proceedings is addressed to graduate students and other researchers in both pure mathematics and theoretical physics. The contributing authors, who are among the world leading experts in this area, have taken care to write self-contained lectures on subjects chosen to produce a coherent volume.

● Self-contained lectures by world-leading experts in the field

● The volume is integrated, indexed and cross-referenced

● This title covers the most important and recent advances in the subject

Contents

1. Introduction; 2. Prime number theory and the Riemann zeta-function; 3. Notes on pair correlation of zeros and prime numbers; 4. Notes on eigenvalue distributions for the classical compact groups; 5. Compound nucleus resonances, random matrices and quantum chaos; 6. Families of L-functions and 1-level densities; 7. Basic analytic number theory; 8. Applications of mean value theorems to the theory of the Riemann zeta function; 9. L-functions and the characteristic polynomials of random matrices; 10. Mock gaussian behaviour; 11. Some specimens of L-functions; 12. Computational methods and experiments in analytic number theory.