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Über dieses Buch

This book is an introduction to several active research topics in Foliation Theory and its connections with other areas. It contains expository lectures showing the diversity of ideas and methods converging in the study of foliations. The lectures by Aziz El Kacimi Alaoui provide an introduction to Foliation Theory with emphasis on examples and transverse structures. Steven Hurder's lectures apply ideas from smooth dynamical systems to develop useful concepts in the study of foliations: limit sets and cycles for leaves, leafwise geodesic flow, transverse exponents, Pesin Theory and hyperbolic, parabolic and elliptic types of foliations. The lectures by Masayuki Asaoka compute the leafwise cohomology of foliations given by actions of Lie groups, and apply it to describe deformation of those actions. In his lectures, Ken Richardson studies the properties of transverse Dirac operators for Riemannian foliations and compact Lie group actions, and explains a recently proved index formula. Besides students and researchers of Foliation Theory, this book will be interesting for mathematicians interested in the applications to foliations of subjects like Topology of Manifolds, Differential Geometry, Dynamics, Cohomology or Global Analysis.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Deformation of Locally Free Actions and Leafwise Cohomology

Abstract
These are the notes of the author’s lectures at the Advanced Course on Foliations in the research program Foliations, which was held at the Centre de Recerca Matemàtica in May 2010. In these notes, we discuss the relationship between deformations of actions of Lie groups and the leafwise cohomology of the orbit foliation.
Masayuki Asaoka

Chapter 2. Fundaments of Foliation Theory

Abstract
It is well known that there is no general method to solve differential equations even in the case of the simplest manifold, namely the real line ℝ. Failing that, mathematicians rather try to study the geometrical and topological properties of integral manifolds and their asymptotic behavior. This is exactly the purpose of foliation theory: the qualitative study of differential equations. It was initiated by the works of H. Poincaré and I. Bendixson, and developed later by C.
Aziz El Kacimi Alaoui

Chapter 3. Lectures on Foliation Dynamics

Abstract
The study of foliation dynamics seeks to understand the asymptotic properties of leaves of foliated manifolds, their statistical properties such as orbit growth rates and geometric entropy, and to classify geometric and topological “structures” which are associated to the dynamics, such as the minimal sets of the foliation.
Steven Hurder

Chapter 4. Transversal Dirac Operators on Distributions, Foliations, and G-Manifolds

Abstract
In these lectures, we investigate generalizations of the ordinary Dirac operator to manifolds with additional structure. In particular, if the manifold comes equipped with a distribution and an associated Clifford algebra action on a bundle over the manifold, one may define a transversal Dirac operator associated to this structure. We investigate the geometric and analytic properties of these operators, and we apply the analysis to the settings of Riemannian foliations and of manifolds endowed with Lie group actions.
Ken Richardson
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