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Introduction to Foliations and Lie Groupoids

Author: I. Moerdijk

Publisher: Cambridge University Press

Published: 2003-09-18

Total Pages: 187

ISBN-13: 1139438980

This book gives a quick introduction to the theory of foliations, Lie groupoids and Lie algebroids. An important feature is the emphasis on the interplay between these concepts: Lie groupoids form an indispensable tool to study the transverse structure of foliations as well as their noncommutative geometry, while the theory of foliations has immediate applications to the Lie theory of groupoids and their infinitesimal algebroids. The book starts with a detailed presentation of the main classical theorems in the theory of foliations then proceeds to Molino's theory, Lie groupoids, constructing the holonomy groupoid of a foliation and finally Lie algebroids. Among other things, the authors discuss to what extent Lie's theory for Lie groups and Lie algebras holds in the more general context of groupoids and algebroids. Based on the authors' extensive teaching experience, this book contains numerous examples and exercises making it ideal for graduate students and their instructors.

Introduction to Foliations and Lie Groupoids

Author: Ieke Moerdijk

Publisher:

Published: 2003

Total Pages: 173

ISBN-13: 9780511071539

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This book gives a quick introduction to the theory of foliations and Lie groupoids. It is based on the authors' extensive teaching experience and contains numerous examples and exercises making it ideal either for independent study or as the basis of a graduate course.

Riemannian Foliations

Author: Molino

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 348

ISBN-13: 1468486705

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Foliation theory has its origins in the global analysis of solutions of ordinary differential equations: on an n-dimensional manifold M, an [autonomous] differential equation is defined by a vector field X ; if this vector field has no singularities, then its trajectories form a par tition of M into curves, i.e. a foliation of codimension n - 1. More generally, a foliation F of codimension q on M corresponds to a partition of M into immersed submanifolds [the leaves] of dimension ,--------,- - . - -- p = n - q. The first global image that comes to mind is 1--------;- - - - - - that of a stack of "plaques". 1---------;- - - - - - Viewed laterally [transver 1--------1- - - -- sally], the leaves of such a 1--------1 - - - - -. stacking are the points of a 1--------1--- ----. quotient manifold W of di L..... -' _ mension q. -----~) W M Actually, this image corresponds to an elementary type of folia tion, that one says is "simple". For an arbitrary foliation, it is only l- u L ally [on a "simpIe" open set U] that the foliation appears as a stack of plaques and admits a local quotient manifold. Globally, a leaf L may - - return and cut a simple open set U in several plaques, sometimes even an infinite number of plaques.

General Theory of Lie Groupoids and Lie Algebroids

Author: Kirill C. H. Mackenzie

Publisher: Cambridge University Press

Published: 2005-06-09

Total Pages: 540

ISBN-13: 0521499283

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This 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 andgeneral 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 such, this book will be of great interest to all those working in or wishing to learn the modern theory of Lie groupoids and Lie algebroids.

Foliations and the Geometry of 3-Manifolds

Author: Danny Calegari

Publisher: Oxford University Press on Demand

Published: 2007-05-17

Total Pages: 378

ISBN-13: 0198570082

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This unique reference, aimed at research topologists, gives an exposition of the 'pseudo-Anosov' theory of foliations of 3-manifolds. This theory generalizes Thurston's theory of surface automorphisms and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. Significant themes returned to throughout the text include the importance of geometry, especially the hyperbolic geometry of surfaces, the importance of monotonicity, especially in1-dimensional and co-dimensional dynamics, and combinatorial approximation, using finite combinatorical objects such as train-tracks, branched surfaces and hierarchies to carry more complicated continuous objects.

Introduction to Foliations and Lie Groupoids

Author: I. Moerdijk

Publisher: Cambridge University Press

Published: 2003-09-18

Total Pages: 184

ISBN-13: 9780521831970

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Based on a graduate course taught at Utrecht University, this book provides a short introduction to the theory of Foliations and Lie Groupoids to students who have already taken a first course in differential geometry. Ieke Moerdijk and Janez Mrcun include detailed references to enable students to find the requisite background material in the research literature. The text features many exercises and worked examples.

Poisson Structures and Their Normal Forms

Author: Jean-Paul Dufour

Publisher: Springer Science & Business Media

Published: 2006-01-17

Total Pages: 332

ISBN-13: 3764373350

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The aim of this book is twofold. On the one hand, it gives a quick, self-contained introduction to Poisson geometry and related subjects. On the other hand, it presents a comprehensive treatment of the normal form problem in Poisson geometry. Even when it comes to classical results, the book gives new insights. It contains results obtained over the past 10 years which are not available in other books.

Lectures on Poisson Geometry

Author: Marius Crainic

Publisher: American Mathematical Soc.

Published: 2021-10-14

Total Pages: 479

ISBN-13: 1470466678

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This excellent book will be very useful for students and researchers wishing to learn the basics of Poisson geometry, as well as for those who know something about the subject but wish to update and deepen their knowledge. The authors' philosophy that Poisson geometry is an amalgam of foliation theory, symplectic geometry, and Lie theory enables them to organize the book in a very coherent way. —Alan Weinstein, University of California at Berkeley This well-written book is an excellent starting point for students and researchers who want to learn about the basics of Poisson geometry. The topics covered are fundamental to the theory and avoid any drift into specialized questions; they are illustrated through a large collection of instructive and interesting exercises. The book is ideal as a graduate textbook on the subject, but also for self-study. —Eckhard Meinrenken, University of Toronto

Noncommutative Geometry

Author: Alain Connes

Publisher: Springer Science & Business Media

Published: 2003-12-08

Total Pages: 372

ISBN-13: 9783540203575

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Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.

Poisson Structures

Author: Camille Laurent-Gengoux

Publisher: Springer Science & Business Media

Published: 2012-08-27

Total Pages: 470

ISBN-13: 3642310907

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Poisson structures appear in a large variety of contexts, ranging from string theory, classical/quantum mechanics and differential geometry to abstract algebra, algebraic geometry and representation theory. In each one of these contexts, it turns out that the Poisson structure is not a theoretical artifact, but a key element which, unsolicited, comes along with the problem that is investigated, and its delicate properties are decisive for the solution to the problem in nearly all cases. Poisson Structures is the first book that offers a comprehensive introduction to the theory, as well as an overview of the different aspects of Poisson structures. The first part covers solid foundations, the central part consists of a detailed exposition of the different known types of Poisson structures and of the (usually mathematical) contexts in which they appear, and the final part is devoted to the two main applications of Poisson structures (integrable systems and deformation quantization). The clear structure of the book makes it adequate for readers who come across Poisson structures in their research or for graduate students or advanced researchers who are interested in an introduction to the many facets and applications of Poisson structures.