Moduli Spaces of Riemannian Metrics

Author: Wilderich Tuschmann,David J. Wraith
Publisher: Springer
ISBN: 3034809484
Category: Mathematics
Page: 123
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This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.

Geometry Of Spherical Space Form Groups, The (Second Edition)

Author: Gilkey Peter B
Publisher: World Scientific
ISBN: 9813220805
Category: Mathematics
Page: 508
View: 6639

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This volume focuses on discussing the interplay between the analysis, as exemplified by the eta invariant and other spectral invariants, the number theory, as exemplified by the relevant Dedekind sums and Rademacher reciprocity, the algebraic topology, as exemplified by the equivariant bordism groups, K-theory groups, and connective K-theory groups, and the geometry of spherical space forms, as exemplified by the Smith homomorphism. These are used to study the existence of metrics of positive scalar curvature on spin manifolds of dimension at least 5 whose fundamental group is a spherical space form group. This volume is a completely rewritten revision of the first edition. The underlying organization is modified to provide a better organized and more coherent treatment of the material involved. In addition, approximately 100 pages have been added to study the existence of metrics of positive scalar curvature on spin manifolds of dimension at least 5 whose fundamental group is a spherical space form group. We have chosen to focus on the geometric aspect of the theory rather than more abstract algebraic constructions (like the assembly map) and to restrict our attention to spherical space forms rather than more general and more complicated geometrical examples to avoid losing contact with the fundamental geometry which is involved. Contents: Partial Differential OperatorsK Theory and CohomologyEquivariant BordismPositive Scalar CurvatureAuxiliary Materials Readership: Graduate students and researchers interested in global analysis, geometry, and topology. Keywords: Dedekind Sums and Rademacher Reciprocity;K-Theory;Eta Invariant;Spherical Space Form;Lens Space;Quaternion Spherical Space Form;Iterated Jet Bundle;Equivariant Bordism;Smith Homomorphism;Connective K-Theory;Manifolds with Positive Scalar Curvature;Spin Bordism;Unitary Bordism;Spin-C Bordism;Pin-C BordismReview: Key Features: The is a complete revision of the first edition and includes substantial amounts of new material applying the basic material of the book to the examination of metrics of positive scalar curvature on spin manifolds of dimension at least 5 whose fundamental group is a spherical space form groupTo ensure that the book is accessible to wide an audience as possible, there is a review of vector bundle theory, of Clifford module theory, of the Atiyah–Singer index theorem, and of the index theorem with boundaryThere are also tables, which have been simplified and the organization improved from the first edition, giving various K-theory and equivariant bordism groups

A Panoramic View of Riemannian Geometry

Author: Marcel Berger
Publisher: Springer Science & Business Media
ISBN: 3642182453
Category: Mathematics
Page: 824
View: 1818

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This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS

K-Theory for Group C*-Algebras and Semigroup C*-Algebras

Author: Joachim Cuntz,Siegfried Echterhoff,Xin Li,Guoliang Yu
Publisher: Birkhäuser
ISBN: 3319599151
Category: Mathematics
Page: 322
View: 4568

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This book gives an account of the necessary background for group algebras and crossed products for actions of a group or a semigroup on a space and reports on some very recently developed techniques with applications to particular examples. Much of the material is available here for the first time in book form. The topics discussed are among the most classical and intensely studied C*-algebras. They are important for applications in fields as diverse as the theory of unitary group representations, index theory, the topology of manifolds or ergodic theory of group actions. Part of the most basic structural information for such a C*-algebra is contained in its K-theory. The determination of the K-groups of C*-algebras constructed from group or semigroup actions is a particularly challenging problem. Paul Baum and Alain Connes proposed a formula for the K-theory of the reduced crossed product for a group action that would permit, in principle, its computation. By work of many hands, the formula has by now been verified for very large classes of groups and this work has led to the development of a host of new techniques. An important ingredient is Kasparov's bivariant K-theory. More recently, also the C*-algebras generated by the regular representation of a semigroup as well as the crossed products for actions of semigroups by endomorphisms have been studied in more detail. Intriguing examples of actions of such semigroups come from ergodic theory as well as from algebraic number theory. The computation of the K-theory of the corresponding crossed products needs new techniques. In cases of interest the K-theory of the algebras reflects ergodic theoretic or number theoretic properties of the action.

Basic Noncommutative Geometry

Author: Masoud Khalkhali
Publisher: European Mathematical Society
ISBN: 9783037190616
Category: Mathematics
Page: 223
View: 2653

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"Basic Noncommutative Geometry provides an introduction to noncommutative geometry and some of its applications. The book can be used either as a textbook for a graduate course on the subject or for self-study. It will be useful for graduate students and researchers in mathematics and theoretical physics and all those who are interested in gaining an understanding of the subject. One feature of this book is the wealth of examples and exercises that help the reader to navigate through the subject. While background material is provided in the text and in several appendices, some familiarity with basic notions of functional analysis, algebraic topology, differential geometry and homological algebra at a first year graduate level is helpful. Developed by Alain Connes since the late 1970s, noncommutative geometry has found many applications to long-standing conjectures in topology and geometry and has recently made headways in theoretical physics and number theory. The book starts with a detailed description of some of the most pertinent algebra-geometry correspondences by casting geometric notions in algebraic terms, then proceeds in the second chapter to the idea of a noncommutative space and how it is constructed. The last two chapters deal with homological tools: cyclic cohomology and Connes-Chern characters in K-theory and K-homology, culminating in one commutative diagram expressing the equality of topological and analytic index in a noncommutative setting. Applications to integrality of noncommutative topological invariants are given as well."--Publisher's description.

The Moduli Space of Curves

Author: Robert H. Dijkgraaf,Carel Faber,Gerard B.M. van der Geer
Publisher: Springer Science & Business Media
ISBN: 1461242649
Category: Mathematics
Page: 563
View: 9175

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The moduli space Mg of curves of fixed genus g – that is, the algebraic variety that parametrizes all curves of genus g – is one of the most intriguing objects of study in algebraic geometry these days. Its appeal results not only from its beautiful mathematical structure but also from recent developments in theoretical physics, in particular in conformal field theory.

100-0 - Man Utd-Liverpool/Liverpool-Man Utd

Author: Tim Glynne-Jones,Will Brooks
Publisher: Bantam Press
ISBN: 9780593074596
Category: Sports rivalries
Page: 208
View: 1738

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Two sets of comparisons between Manchester United and Liverpool United, each with its own title page, printed tête-bêche, one set favouring Manchester, the other Liverpool.

Harmonic Analysis on Symmetric Spaces—Higher Rank Spaces, Positive Definite Matrix Space and Generalizations

Author: Audrey Terras
Publisher: Springer
ISBN: 1493934082
Category: Mathematics
Page: 487
View: 1189

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This text is an introduction to harmonic analysis on symmetric spaces, focusing on advanced topics such as higher rank spaces, positive definite matrix space and generalizations. It is intended for beginning graduate students in mathematics or researchers in physics or engineering. As with the introductory book entitled "Harmonic Analysis on Symmetric Spaces - Euclidean Space, the Sphere, and the Poincaré Upper Half Plane, the style is informal with an emphasis on motivation, concrete examples, history, and applications. The symmetric spaces considered here are quotients X=G/K, where G is a non-compact real Lie group, such as the general linear group GL(n,P) of all n x n non-singular real matrices, and K=O(n), the maximal compact subgroup of orthogonal matrices. Other examples are Siegel's upper half "plane" and the quaternionic upper half "plane". In the case of the general linear group, one can identify X with the space Pn of n x n positive definite symmetric matrices. Many corrections and updates have been incorporated in this new edition. Updates include discussions of random matrix theory and quantum chaos, as well as recent research on modular forms and their corresponding L-functions in higher rank. Many applications have been added, such as the solution of the heat equation on Pn, the central limit theorem of Donald St. P. Richards for Pn, results on densest lattice packing of spheres in Euclidean space, and GL(n)-analogs of the Weyl law for eigenvalues of the Laplacian in plane domains. Topics featured throughout the text include inversion formulas for Fourier transforms, central limit theorems, fundamental domains in X for discrete groups Γ (such as the modular group GL(n,Z) of n x n matrices with integer entries and determinant ±1), connections with the problem of finding densest lattice packings of spheres in Euclidean space, automorphic forms, Hecke operators, L-functions, and the Selberg trace formula and its applications in spectral theory as well as number theory.

Fractals in Engineering

From Theory to Industrial Applications
Author: Jacques Levy Vehel,Evelyne Lutton,Claude Tricot
Publisher: Springer Science & Business Media
ISBN: 1447109953
Category: Computers
Page: 402
View: 957

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Fractal analysis research is expanding into a variety of engineering domains. The strong potential of this work is now beginning to be seen in important applications in real industrial situations. Recent research progress has already led to new developments in domains such as signal processing and chemical engineering, and the major advances in fractal theory that underlie such developments are detailed here. New domains of applications are also presented, among them environmental science and rough surface analysis. Sections include multifractal analysis, iterated function systems, random processes, network traffic analysis, fractals and waves, image compression, and applications in physics. Fractals in Engineering emphasizes the connection between fractal analysis research and applications to industry. It is an important volume that illustrates the scientific and industrial value of this exciting field.

An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces

Author: Martin Schlichenmaier
Publisher: Springer Science & Business Media
ISBN: 3540711759
Category: Science
Page: 217
View: 3836

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This book gives an introduction to modern geometry. Starting from an elementary level, the author develops deep geometrical concepts that play an important role in contemporary theoretical physics, presenting various techniques and viewpoints along the way. This second edition contains two additional, more advanced geometric techniques: the modern language and modern view of Algebraic Geometry and Mirror Symmetry.

The Convenient Setting of Global Analysis

Author: Andreas Kriegl,Peter W. Michor
Publisher: American Mathematical Soc.
ISBN: 0821807803
Category: Mathematics
Page: 618
View: 629

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This book lays the foundations of differential calculus in infinite dimensions and discusses those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory. The approach is simple: a mapping is called smooth if it maps smooth curves to smooth curves. Up to Frechet spaces, this notion of smoothness coincides with all known reasonable concepts. In the same spirit, calculus of holomorphic mappings (including Hartogs' theorem and holomorphic uniform boundedness theorems) and calculus of real analytic mappings are developed. Existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions, the relation between tangent vectors and derivations, and differential forms are discussed thoroughly. Special emphasis is given to the notion of regular infinite dimensional Lie groups. Many applications of this theory are included: manifolds of smooth mappings, groups of diffeomorphisms, geodesics on spaces of Riemannian metrics, direct limit manifolds, perturbation theory of operators, and differentiability questions of infinite dimensional representations.

Discrete Differential Geometry

Integrable Structure
Author: Alexander I. Bobenko,Yuri B. Suris
Publisher: American Mathematical Soc.
ISBN: 0821847007
Category: Mathematics
Page: 404
View: 5730

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An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of Integrable systems. One of the main goals of this book Is to reveal this integrable structure of discrete differential geometry. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question "How do we discretize differential geometry?" arising in their specific field.

Geometric Problems on Maxima and Minima

Author: Titu Andreescu,Oleg Mushkarov,Luchezar Stoyanov
Publisher: Springer Science & Business Media
ISBN: 0817644733
Category: Mathematics
Page: 264
View: 1639

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Presents hundreds of extreme value problems, examples, and solutions primarily through Euclidean geometry Unified approach to the subject, with emphasis on geometric, algebraic, analytic, and combinatorial reasoning Applications to physics, engineering, and economics Ideal for use at the junior and senior undergraduate level, with wide appeal to students, teachers, professional mathematicians, and puzzle enthusiasts

Brownian Motion, Martingales, and Stochastic Calculus

Author: Jean-François Le Gall
Publisher: Springer
ISBN: 3319310895
Category: Mathematics
Page: 273
View: 8545

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This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments. Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.

The Geometry of Four-manifolds

Author: S. K. Donaldson,P. B. Kronheimer
Publisher: Oxford University Press
ISBN: 9780198502692
Category: Fiction
Page: 440
View: 5125

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This book provides the first lucid and accessible account to the modern study of the geometry of four-manifolds. It has become required reading for postgraduates and research workers whose research touches on this topic. Pre-requisites are a firm grounding in differential topology, and geometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that the fundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodular forms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central theme of this book is that the appropriate geometrical tools for investigating these questions come from mathematical physics: the Yang-Mills theory and anti-self dual connections over four-manifolds. One of the many consquences of this theory is that 'exotic' smooth manifolds exist which are homeomorphic but not diffeomorphic to (4, and that large classes of forms cannot be realized as intersection forms whereas distinct manifolds may share the same form. These result have hadfar-reaching consequences in algebraic geometry, topology, and mathematical physics, and will continue to be a mainspring of mathematical research for years to come.

An Introduction to Intersection Homology Theory, Second Edition

Author: Frances Kirwan,Jonathan Woolf
Publisher: CRC Press
ISBN: 9781584881841
Category: Mathematics
Page: 248
View: 2722

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Now more that a quarter of a century old, intersection homology theory has proven to be a powerful tool in the study of the topology of singular spaces, with deep links to many other areas of mathematics, including combinatorics, differential equations, group representations, and number theory. Like its predecessor, An Introduction to Intersection Homology Theory, Second Edition introduces the power and beauty of intersection homology, explaining the main ideas and omitting, or merely sketching, the difficult proofs. It treats both the basics of the subject and a wide range of applications, providing lucid overviews of highly technical areas that make the subject accessible and prepare readers for more advanced work in the area. This second edition contains entirely new chapters introducing the theory of Witt spaces, perverse sheaves, and the combinatorial intersection cohomology of fans. Intersection homology is a large and growing subject that touches on many aspects of topology, geometry, and algebra. With its clear explanations of the main ideas, this book builds the confidence needed to tackle more specialist, technical texts and provides a framework within which to place them.

Moonwatch Only

60 Years of OMEGA Speedmaster
Author: Grégoire Rossier,Anthony Marquie
Publisher: Sarl
ISBN: 9782940506170
Page: 566
View: 5793

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-A new edition of this definitive book, marking the 60th anniversary of the Speedmaster -Includes new features and additional historical information "The OMEGA Speedmaster Professional - the Moonwatch - has done things that no other timepiece has done and it has been worn in places that only a few human beings have been." - Captain Eugene Cernan, last man on the moon "It is an indescribable reference work and a true must-have for every Speedmaster collector." - Forbes There are very few timepieces in the world that deserve a definitive and comprehensive book. The OMEGA Speedmaster Professional Moonwatch is one of them. Initially designed for automobile racing teams and engineers, the Omega Speedmaster embarked on a very different trajectory when NASA chose it to accompany astronauts heading for the Moon in 1965. Its involvement in the space adventure has propelled the Moonwatch to the top of the list of celebrated timepieces. After years of research and observation, the authors present a complete panorama of the Moonwatch in a systematic work that is both technical and attractive, making it the unparalleled reference book for this legendary watch. This new edition, marking the 60th anniversary of the Speedmaster, has been enriched with numerous new features and additional historical information. Contents: Foreword by Raynald Aeschlimann, President and CEO of OMEGA; Foreword by Captain Eugene Cernan, Commander of Apollo 17; Why a Speedmaster Moonwatch guide?; Part 1 - Speedmaster History; 1, Major Dates; 2, Speedmaster and NASA 25; Part 2 - Main Components and Accessories; 1, An Original Approach; 2, The Caliber; 3, The Caseband; 4, The Dial; 5, The Bezel; 6, The Hands; 7, The Caseback; 8, The Crown; 9, The Pushers; 10, The Glass; 11, The Bracelet; 12, The Presentation Box; 13, The Documents; Part 3 - The Models; 1, Introduction; 2, Standard Production; 3, Special and Limited Series; 4, Personalized Models and Special Projects; 5, The Alaska Project; Part 4 - 60 Years of Innovation; Part 5 - How to Start Collecting Speedmasters; 1, Budget; 2, Choosing a Model; 3, Sales Channels; Part 6 - Appendices; 1, Model Codes; 2, Tables & Bibliography; 3, Contributions; 4, Identification Aid

Metric Structures for Riemannian and Non-Riemannian Spaces

Author: Mikhail Gromov
Publisher: Springer Science & Business Media
ISBN: 0817645837
Category: Mathematics
Page: 586
View: 2011

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This book is an English translation of the famous "Green Book" by Lafontaine and Pansu (1979). It has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices, by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures, as well as an extensive bibliography and index round out this unique and beautiful book.