Intuitively, a foliation corresponds to a decomposition of a manifold into a union of connected, disjoint submanifolds of the same dimension, called leaves, which pile up locally like pages of a book. The theory of foliations, as it is known, began with the work of C. Ehresmann and G. Reeb, in the 1940's; however, as Reeb has himself observed, already in the last century P. Painleve saw the necessity of creating a geometric theory (of foliations) in order to better understand the problems in the study of solutions of holomorphic differential equations in the complex field. The development of the theory of foliations was however provoked by the following question about the topology of manifolds proposed by H. Hopf in the 3 1930's: "Does there exist on the Euclidean sphere S a completely integrable vector field, that is, a field X such that X* curl X * 0?" By Frobenius' theorem, this question is equivalent to the following: "Does there exist on the 3 sphere S a two-dimensional foliation?" This question was answered affirmatively by Reeb in his thesis, where he 3 presents an example of a foliation of S with the following characteristics: There exists one compact leaf homeomorphic to the two-dimensional torus, while the other leaves are homeomorphic to two-dimensional planes which accu mulate asymptotically on the compact leaf. Further, the foliation is C"".

This book provides quick access to the theory of Lie groups and isometric actions on smooth manifolds, using a concise geometric approach. After a gentle introduction to the subject, some of its recent applications to active research areas are explored, keeping a constant connection with the basic material. The topics discussed include polar actions, singular Riemannian foliations, cohomogeneity one actions, and positively curved manifolds with many symmetries. This book stems from the experience gathered by the authors in several lectures along the years and was designed to be as self-contained as possible. It is intended for advanced undergraduates, graduate students and young researchers in geometry and can be used for a one-semester course or independent study.

The theory of vertex operator algebras and their representations has been showing its power in the solution of concrete mathematical problems and in the understanding of conceptual but subtle mathematical and physical struc- tures of conformal field theories. Much of the recent progress has deep connec- tions with complex analysis and conformal geometry. Future developments, especially constructions and studies of higher-genus theories, will need a solid geometric theory of vertex operator algebras. Back in 1986, Manin already observed in [Man) that the quantum theory of (super )strings existed (in some sense) in two entirely different mathematical fields. Under canonical quantization this theory appeared to a mathematician as the representation theories of the Heisenberg, Vir as oro and affine Kac- Moody algebras and their superextensions. Quantization with the help of the Polyakov path integral led on the other hand to the analytic theory of algebraic (super ) curves and their moduli spaces, to invariants of the type of the analytic curvature, and so on. He pointed out further that establishing direct mathematical connections between these two forms of a single theory was a "big and important problem. " On the one hand, the theory of vertex operator algebras and their repre- sentations unifies (and considerably extends) the representation theories of the Heisenberg, Virasoro and Kac-Moody algebras and their superextensions.

This book, addressing both researchers and graduate students, reviews equivariant localization techniques for the evaluation of Feynman path integrals. The author gives the relevant mathematical background in some detail, showing at the same time how localization ideas are related to classical integrability. The text explores the symmetries inherent in localizable models for assessing the applicability of localization formulae. Various applications from physics and mathematics are presented.

This treatment of differential geometry and the mathematics required for general relativity makes the subject of this book accessible for the first time to anyone familiar with elementary calculus in one variable and with a knowledge of some vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow. The imaginative text is a major contribution to expounding the subject of differential geometry as applied to studies in relativity, and will prove of interest to a large number of mathematicians and physicists. Review from L'Enseignement Mathématique

Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.

This monograph provides an introduction to the theory of topologies defined on the closed subsets of a metric space, and on the closed convex subsets of a normed linear space as well. A unifying theme is the relationship between topology and set convergence on the one hand, and set functionals on the other. The text includes for the first time anywhere an exposition of three topologies that over the past ten years have become fundamental tools in optimization, one-sided analysis, convex analysis, and the theory of multifunctions: the Wijsman topology, the Attouch--Wets topology, and the slice topology. Particular attention is given to topologies on lower semicontinuous functions, especially lower semicontinuous convex functions, as associated with their epigraphs. The interplay between convex duality and topology is carefully considered and a chapter on set-valued functions is included. The book contains over 350 exercises and is suitable as a graduate text. This book is of interest to those working in general topology, set-valued analysis, geometric functional analysis, optimization, convex analysis and mathematical economics.

Intended as a systematic text on topological vector spaces, this text assumes familiarity with the elements of general topology and linear algebra. Similarly, the elementary facts on Hilbert and Banach spaces are not discussed in detail here, since the book is mainly addressed to those readers who wish to go beyond the introductory level. Each of the chapters is preceded by an introduction and followed by exercises, which in turn are devoted to further results and supplements, in particular, to examples and counter-examples, and hints have been given where appropriate. This second edition has been thoroughly revised and includes a new chapter on C DEGREES* and W DEGR

This book presents a comprehensive, encyclopedic approach to the subject of foliations, one of the major concepts of modern geometry and topology. It addresses graduate students and researchers and serves as a reference book for experts in the field.

The main theme of this book is the theory of heights as they appear in various guises. This includes a large body of results on Mahlers measure of the height of a polynomial. The authors'approach is very down to earth as they cover the rationals, assuming no prior knowledge of elliptic curves. The chapters include examples and particular computations, with all special calculation included so as to be self-contained. The authors devote space to discussing Mahlers measure and to giving some convincing and original examples to explain this phenomenon. XXXXXXX NEUER TEXT The main theme of this book is the theory of heights as it appears in various guises. To this End.txt.Int.:, it examines the results of Mahlers measure of the height of a polynomial, which have never before appeared in book form. The authors take a down-to-earth approach that includes convincing and original examples. The book uncovers new and interesting connections between number theory and dynamics and will be interesting to researchers in both number theory and nonlinear dynamics."

This book discusses major theories and applications of fuzzy soft multisets and their generalization which help researchers get all the related information at one place. The primary objective of this book is to help bridge the gap to provide a textbook on the theories in fuzzy soft multisets and their applications in real life. It is targeted to researchers and students working in the field of fuzzy set theory, multiset theory, soft set theory and their applications. Uncertainty, vagueness and the representation of imperfect knowledge have been a problem in many fields of research, including artificial intelligence, network and communication, signal processing, machine learning, computer science, information technology, as well as medical science, economics, environments and engineering. There are many mathematical tools for dealing with uncertainties. They include fuzzy set theory, multiset theory, soft set theory and soft multiset theory.

The theory presented in this book is developed constructively, is based on a few axioms encapsulating the notion of objects (points and sets) being apart, and encompasses both point-set topology and the theory of uniform spaces. While the classical-logic-based theory of proximity spaces provides some guidance for the theory of apartness, the notion of nearness/proximity does not embody enough algorithmic information for a deep constructive development. The use of constructive (intuitionistic) logic in this book requires much more technical ingenuity than one finds in classical proximity theory - algorithmic information does not come cheaply - but it often reveals distinctions that are rendered invisible by classical logic.

In the first chapter the authors outline informal constructive logic and set theory, and, briefly, the basic notions and notations for metric and topological spaces. In the second they introduce axioms for a point-set apartness and then explore some of the consequences of those axioms. In particular, they examine a natural topology associated with an apartness space, and relations between various types of continuity of mappings. In the third chapter the authors extend the notion of point-set (pre-)apartness axiomatically to one of (pre-)apartness between subsets of an inhabited set. They then provide axioms for a quasiuniform space, perhaps the most important type of set-set apartness space. Quasiuniform spaces play a major role in the remainder of the chapter, which covers such topics as the connection between uniform and strong continuity (arguably the most technically difficult part of the book), apartness and convergence in function spaces, types of completeness, and neat compactness. Each chapter has a Notes section, in which are found comments on the definitions, results, and proofs, as well as occasional pointers to future work. The book ends with a Postlude that refers to other constructive approaches to topology, with emphasis on the relation between apartness spaces and formal topology.

Largely an exposition of the authors' own research, this is the first book dealing with the apartness approach to constructive topology, and is a valuable addition to the literature on constructive mathematics and on topology in computer science. It is aimed at graduate students and advanced researchers in theoretical computer science, mathematics, and logic who are interested in constructive/algorithmic aspects of topology.

Largely an exposition of the authors' own research, this is the first book dealing with the apartness approach to constructive topology, and is a valuable addition to the literature on constructive mathematics and on topology in computer science. It is aimed at graduate students and advanced researchers in theoretical computer science, mathematics, and logic who are interested in constructive/algorithmic aspects of topology.

This book is intended as a textbook for a first-year graduate course on algebraic topology, with as strong flavoring in smooth manifold theory. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory. It covers most of the topics all topologists will want students to see, including surfaces, Lie groups and fibre bundle theory. With a thoroughly modern point of view, it is the first truly new textbook in topology since Spanier, almost 25 years ago. Although the book is comprehensive, there is no attempt made to present the material in excessive generality, except where generality improves the efficiency and clarity of the presentation.

The aim of this book is to throw light on various facets of geometry through development of four geometrical themes.

The first theme is about the ellipse, the shape of the shadow east by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola.

The third theme is about certain properties of geometrical figures related to the problem of finding the largest area that can be enclosed by a curve of given length. This problem is called the isoperimetric problem. In itself, this topic contains motivation for major parts of the curriculum in mathematics at college level and sets the stage for more advanced mathematical subjects such as functions of several variables and the calculus of variations. Here, three types of conic section are discussed briefly.

The emergence of non-Euclidean geometries in the beginning of the nineteenth century represents one of the dramatic episodes in the history of mathematics. In the last theme the non-Euclidean geometry in the Poincare disc model of the hyperbolic plane is developed.

In this monograph the authors present detailed and pedagogic proofs of persistence theorems for normally hyperbolic invariant manifolds and their stable and unstable manifolds for classes of perturbations of the NLS equation, as well as for the existence and persistence of fibrations of these invariant manifolds. Their techniques are based on an infinite dimensional generalisation of the graph transform and can be viewed as an infinite dimensional generalisation of Fenichels results. As such, they may be applied to a broad class of infinite dimensional dynamical systems.

This volume describes the status of fractal imaging research and looks to future directions. It will be useful to researchers in the areas of fractal image compression, analysis, and synthesis, iterated function systems, and fractals in education. In particular it includes a vision for the future of these areas. It aims to provide an efficient means by which researchers can look back over the last decade at what has been achieved, and look forward towards second-generation fractal imaging. The articles in themselves are not meant to be detailed reviews or expositions, but to serve as signposts to the state of the art in their areas. What is important is what they mention and what tools and ideas are seen now to be relevant to the future. The contributors, a number of whom have been involved since the start, are active in fractal imaging, and provide a well-informed viewpoint on both the status and the future. Most were invited participants at a meeting on Fractals in Multimedia held at the IMA in January 2001. Some goals of the mini-symposium, shared with this volume, were to demonstrate that the fractal viewpoint leads to a broad collection of useful mathematical tools, common themes, new ways of looking at and thinking about existing algorithms and applications in multimedia, and to consider future developments. This book should be useful to commercial and university researchers in the rapidly evolving field of digital imaging, specifically, chief information officers, professors, software engineers, and graduate students in the mathematical sciences. While much of the content is quite technical, it contains pointers to the state-of-the-art and the future in fractal imaging.

Banach spaces provide a framework for linear and nonlinear functional analysis, operator theory, abstract analysis, probability, optimization and other branches of mathematics. This book introduces the reader to linear functional analysis and to related parts of infinite-dimensional Banach space theory. Key Features: - Develops classical theory, including weak topologies, locally convex space, Schauder bases and compact operator theory - Covers Radon-Nikodym property, finite-dimensional spaces and local theory on tensor products - Contains sections on uniform homeomorphisms and non-linear theory, Rosenthal's L1 theorem, fixed points, and more - Includes information about further topics and directions of research and some open problems at the end of each chapter - Provides numerous exercises for practice The text is suitable for graduate courses or for independent study. Prerequisites include basic courses in calculus and linear. Researchers in functional analysis will also benefit for this book as it can serve as a reference book.

This book gives an intuitive and hands-on introduction to Topological Data Analysis (TDA). Covering a wide range of topics at levels of sophistication varying from elementary (matrix algebra) to esoteric (Grothendieck spectral sequence), it offers a mirror of data science aimed at a general mathematical audience. The required algebraic background is developed in detail. The first third of the book reviews several core areas of mathematics, beginning with basic linear algebra and applications to data fitting and web search algorithms, followed by quick primers on algebra and topology. The middle third introduces algebraic topology, along with applications to sensor networks and voter ranking. The last third covers key contemporary tools in TDA: persistent and multiparameter persistent homology. Also included is a user's guide to derived functors and spectral sequences (useful but somewhat technical tools which have recently found applications in TDA), and an appendix illustrating a number of software packages used in the field. Based on a course given as part of a masters degree in statistics, the book is appropriate for graduate students.

Carl Ludwig Siegel gave a course of lectures on the Geometry of Numbers at New York University during the academic year 1945-46, when there were hardly any books on the subject other than Minkowski's original one. This volume stems from Siegel's requirements of accuracy in detail, both in the text and in the illustrations, but involving no changes in the structure and style of the lectures as originally delivered. This book is an enticing introduction to Minkowski's great work. It also reveals the workings of a remarkable mind, such as Siegel's with its precision and power and aesthetic charm. It is of interest to the aspiring as well as the established mathematician, with its unique blend of arithmetic, algebra, geometry, and analysis, and its easy readability.

This introduction to some basic ideas in algebraic topology is devoted to the foundations and applications of homology theory. After the essentials of singular homology and some important applications are given, successive topics covered include attaching spaces, finite CW complexes, cohomology products, manifolds, Poincare duality, and fixed point theory. This second edition includes a chapter on covering spaces and many new exercises.

Knot theory is a rapidly developing field of research with many applications not only for mathematics. The present volume, written by a well-known specialist, gives a complete survey of knot theory from its very beginnings to today's most recent research results. The topics include Alexander polynomials, Jones type polynomials, and Vassiliev invariants. With its appendix containing many useful tables and an extended list of references with over 3,500 entries it is an indispensable book for everyone concerned with knot theory. The book can serve as an introduction to the field for advanced undergraduate and graduate students. Also researchers working in outside areas such as theoretical physics or molecular biology will benefit from this thorough study which is complemented by many exercises and examples.

Topology-based methods are of increasing importance in the analysis and visualization of dataset from a wide variety of scientific domains such as biology, physics, engineering, and medicine. Current challenges of topology-based techniques include the management of time-dependent data, the representation large and complex datasets, the characterization of noise and uncertainty, the effective integration of numerical methods with robust combinatorial algorithms, etc. (see also below for a list of selected issues). While there is an increasing number of high-quality publications in this field, many fundamental questions remain unsolved. New focused efforts are needed in a variety of techniques ranging from the theoretical foundations of topological models, algorithmic issues related to the representation power of computer-based implementations as well as their computational efficiency, user interfaces for presentation of quantitative topological information, and the development of new techniques for systematic mapping of science problems in topological constructs that can be solved computationally. In this forum the editors have brought together the most prominent and best recognized researchers in the field of topology-based data analysis and visualization for a joint discussion and scientific exchange of the latest results in the field. The 2009 workshop in Snowbird, Utah, follows the two successful workshops in 2005 (Budmerice, Slovakia) and 2007 (Leipzig, Germany).

Riemannian manifolds, particularly those with positive or nonnegative curvature, are constructed from only a handful by means of metric fibrations or deformations thereof. This text documents some of these constructions, many of which have only appeared in journal form. The emphasis is less on the fibration itself and more on how to use it to either construct or understand a metric with curvature of fixed sign on a given space.