One night, seven teenagers wake completely alone and in the dark. Their parents are gone, and it seems everyone on earth has disappeared as well. Worse, the electricity is off. Surrounded by darkness, their differences no longer matter; they are the sole survivors of an alien attack and the last hope for humanity. But in the black of night, can they outrun the hulking creatures with the glowing red eyes?

Priya is the exotic girl who says what she thinks. Will was a star athlete before the attack. They find Trevor and Aiyana, the twins; Alex and Ricardo, the tough guys; and finally, Lindsey, the innocent. Together, the seven of them are the last humans on earthbut are they really humans at all? They have no idea that the government has known of the incoming alien attack for decades; to prepare they created special children with very special abilities.

The chosen seven are part alien and part human, and they exist to fight off the otherworldly forces, threatening to conquer the planet. But with such differing personalities, will the seven be able to put their pride be-hind and work together? They must move beyond the grief of lost friends and family and find the strength to go on; if they dont, the world will die, taken over by monstrous creatures that can smell your fear.

Herbert Spencer (1820-1903) is remembered today only as an alleged 'Social Darwinist' who applied the theory of the survival of the fittest to society. Yet he was among the most influential and widely-read philosophers of the nineteenth century. There were few Victorian thinkers and scientists who did not know his work, and who did not formulate their own positions partly in reaction to his.

Michael Taylor's book provides the only detailed and reliable modern survey of the whole corpus of Spencer's thought. Taylor introduces a Spencer very different to his posthumous reputation: not primarily a political philosopher, but the architect of a comprehensive philosophical system that aimed to demonstrate the inevitability of human perfection through universal natural laws. He also locates the Synthetic Philosophy firmly in its place and time by showing how it developed out of the concerns of a group of like-minded British writers and thinkers during the 1850s.

This book will be of interest to historians of philosophy and of science, to social scientists, to scholars and students of nineteenth century literature, and to anyone who wishes to understand one of most important figures in Victorian intellectual life.

A teenage king in 223 BC, Antiochus III inherited an empire in shambles, ravaged by civil strife and eroded by territorial secessions. He proved himself a true heir of Alexander: he defeated rebel armies and embarked on a campaign of conquest and reunification. Although repulsed by Ptolemy IV at the Battle of Raphia, his eastern campaigns reaffirmed Seleucid hegemony as far as modern Afghanistan and Pakistan. Returning westward, he defeated Ptolemy V at Panion (200 BC) and succeeded in adding Koile Syria to the Seleucid realm. ??At the height of his powers, he challenged growing Roman power, unimpressed by their recent successes against Carthage and Macedon. His expeditionary force was crushed at Thermopylae and evacuated. Refusing to bow before Roman demands, Antiochus energetically mobilized against Roman invasion, but was again decisively defeated at the epic battle of Magnesia. Despite the loss of territory and prestige enshrined in the subsequent Peace of Apamea, Antiochus III left the Seleucid Empire in far better condition than he found it. Although sometimes presented as a failure against the unstoppable might of Rome, Antiochus III must rank as one of the most energetic and effective rulers of the Ancient world.??As well as narrating the eventful career of Antiochus III, Michael Taylor examines Seleucid military organization and royal administration.

This second in the series of three volumes builds upon the basic theory of linear PDE given in volume 1, and pursues more advanced topics. Analytical tools introduced here include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure. The book also develops basic differential geometrical concepts, centred about curvature. Topics covered include spectral theory of elliptic differential operators, the theory of scattering of waves by obstacles, index theory for Dirac operators, and Brownian motion and diffusion.

The world is in turmoil. Reeling after a devastating alien invasion that decimates the worlds population, humanity survives in small communities, fighting for their lives. Countless millions died during the initial invasion, and those still alive are seen as nothing more than a food source for the hungry Quar. Governments quickly fall across the globe, and a primitive, brutal, survival mentality now consumes the world.

But amid the chaos, one small pocket of technology held out longer than the rest. Within a secret lab, Earths salvation rested on the small shoulders of a group of genetically modified children known only as the Theta DNA kids. No longer fully human, they carry the genetic traits of humanity withinand they now face the very destiny they had been engineered for. They were humanitys last, best hope for a futureany future.

Once out of the lab, the Theta DNA kids be-gin a dangerous, nomadic adventure. Along the way, they encounter other small groups of other Thetan DNA mutants and struggle to form alliances. Can these young heroes put their own adolescent insecurities aside long enough to truly work together for the common good? To do so, they must learn to work together, to leadand to fight for the very future of humanity.

Their enemies are close, but even the teens dont fully understand how close. Through the horrors of invasion, grief, loss, and brutal truth, these young soldiers fight for every-thing, with everything theyve been genetically enhanced to give.

The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of L Sobolev spaces, H lder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is aimed at graduate students in mathematics, and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis and complex analysis

The first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations.The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.

A powerful and provocative critique of the foundations of Rational Choice theory and the economic way of thinking about the world, written by a former leading practitioner. The target is a dehumanizing ideology that cannot properly recognize that normal people have attachments and commitments to other people and to practices, projects, principles, and places, which provide them with desire-independent reasons for action, and that they are reflective creatures who think about what they are and what they should be, with ideals that can shape and structure the way they see their choices. The author's views are brought to bear on the economic way of thinking about the natural environment and on how and when the norm of fair reciprocity motivates us to do our part in cooperative endeavors. Throughout, the argument is adorned by thought-provoking examples that keep what is at stake clearly before the reader's mind.

Faith is one of the most critical aspects of the Christian life, because it is the means by which we receive all that God has secured for us; yet the nature of faith is one of the most misunderstood concepts of Christianity. Some conceptualize faith as some mental state that one must reach before they are able to “tap into” God’s “resources.” Others view faith as some sort of incorporeal substance that is “out there” for which we must reach. All of these concepts of faith are deficient, if not heretical, and yet all of them are found in the church. Others view faith as a mere fantasy or hope with no basis in reality, and thus religious claims are excluded from the arena of knowledge. “For a number of years my heart has been trilled over the doctrine of faith as it is received and taught among evangelical Christians everywhere. Great emphasis is laid upon faith in orthodox circles, and that is good, but still I am troubled. Specifically, my fear is that the modern conception of faith is not the biblical one; that when the teachers of our day use the word, they do no mean what Bible meant when they used it”. I also share the same concerns with A.W.Tozer, when he said; “the cause of my uneasiness are these”: 1. The lack of spiritual fruits in the lives of so many who claim to have faith. 2. The rarity of a radical change in the conduct and general outlook of persons professing their new faith in Christ as their personal Savior. 3. The failure of our teachers to define or even describe the thing to which the word faith is supposed to refer. 4. The heartbreaking failure of multitudes of seekers is they are even so earnest to make anything out of the doctrine or to receive any satisfying experience through it. 5. The real danger that a doctrine that is parroted so widely and received so uncritically by so many is false as understood by them. 6. I have seen faith put forward as an escape from reality, a refuge from the necessity of hard thinking, a hiding place for weak character, natural optimism, emotional thrills and nervous tics. Therefore, in the pages of this book, I will endeavor to outline for your benefit the dynamics of faith. What is faith? How does faith works? How to grow in faith? The power of faith; the Excellency of faith; the demand of faith; the perseverance of faith; the challenges of faith; the object of faith, the effect of faith; the response of faith; the discipline of faith; the necessity of faith and many more are discussed.

Policymakers are often reluctant to grant independence to the agencies that regulate and supervise the financial sector because of the fear that these agencies, with their wide-ranging responsibilities and powers, could become a law unto themselves. This pamphlet describes mechanisms for making regulatory agencies accountable not only to the government but also to the industry they supervise and the public at large, with examples from a range of countries.

In nearly every major financial crisis of the past decade-from East Asia to Russia, Turkey, and Latin America-political interference in financial sector regulation helped make a bad situation worse. Political pressures not only weakened financial regulation, but also hindered regulators and supervisors from taking action against troubled banks. This paper investigates why, to fulfill their mandate to preserve financial sector stability, financial sector regulators and supervisors need to be independent-from the financial services industry as well as from the government-as well as accountable.

As IBM® Scale Out Network Attached Storage (SONAS) is adopted, it is important to provide information about planning, installation, and daily administration. This IBM Redbooks® publication also describes leading tuning practices information gained by those who implement and support SONAS.

These preferred practices are based on hands-on experience from the field. Monitoring of the SONAS system is included. This IBM Redbooks publication provides information about IBM SONAS features and function at the 1.5.1 level.

This book is the companion to the IBM SONAS Implementation Guide, SG24-7962 IBM Redbooks publication. It is intended for readers who have implemented SONAS and are responsible for daily administration and monitoring.

IBM® Scale Out Network Attached Storage (SONAS) is a scale out network-attached storage offering that is designed to manage vast repositories of information in enterprise environments that require large capacities, high levels of performance, and high availability.

SONAS provides a range of reliable, scalable storage solutions for various storage requirements. These capabilities are achieved by using network access protocols such as Network File System (NFS), Common Internet File System (CIFS), Hypertext Transfer Protocol Secure (HTTPS), File Transfer Protocol (FTP), and Secure Copy Protocol (SCP). Using built-in RAID technologies, all data is well-protected with options to add more protection through mirroring, replication, snapshots, and backup. These storage systems are also characterized by simple management interfaces that make installation, administration, and troubleshooting uncomplicated and straightforward.

This IBM Redbooks® publication is the companion to IBM SONAS Best Practices, SG24-8051. It is intended for storage administrators who have ordered their SONAS solution and are ready to install, customize, and use it. It provides backup and availability scenarios information about configuration and troubleshooting. This book applies to IBM SONAS Version 1.5.5. It is useful for earlier releases of IBM SONAS as well.

In nearly every major financial crisis of the past decade-from East Asia to Russia, Turkey, and Latin America-political interference in financial sector regulation helped make a bad situation worse. Political pressures not only weakened financial regulation, but also hindered regulators and supervisors from taking action against troubled banks. This paper investigates why, to fulfill their mandate to preserve financial sector stability, financial sector regulators and supervisors need to be independent-from the financial services industry as well as from the government-as well as accountable.

Since the mid-1990s, economic observers have kept a watchful eye on the financial sector because of its potential to spark economic crises. Banks in particular have come under close scrutiny. Building Strong Banks through Surveillance and Resolution offers guidance on setting up regulatory and supervisory regimes that can help to prevent crises, and on dealing with turmoil, should a crisis erupt. The book contains a collection of essays-drawn from practical experience-on a wide range of issues germane to bolstering the banking and financial sector, including developing adequate standards for loan classification, and provisioning and promoting deep and liquid money markets. The government’s lender-of-last-resort function, resolution and recapitalization of failed banks, organizing an effective regulatory framework, and moral hazard are all covered in this volume, whose strong practical flavor grows out of the authors’close involvement with these issues.

​The theory of distributions constitutes an essential tool in the study of partial differential equations. This textbook would offer, in a concise, largely self-contained form, a rapid introduction to the theory of distributions and its applications to partial differential equations, including computing fundamental solutions for the most basic differential operators: the Laplace, heat, wave, Lam\'e and Schrodinger operators.​

10 Things You Must Know About Auto Navigation Systems Today's auto navigation systems aren't just for the car, truck, or SUV. More and more manufacturers are discovering that offering drivers the option of taking their navigation systems off road is an incredible boon to sales. With this realization they are delivering on many levels previously unheard of. There are even GPS and navigation systems that are designed in order to assist in finding a great catch. Perfect for the fisherman in your life. Here's a preview of what you will learn: - 4 Great Reasons to Enjoy Geocaching - 5 Ways Auto Navigation Systems and GPS Technology Improve Lives - 5 Ways to Utilize Auto Navigation Systems and GPS Technology While Traveling - and More GRAB YOUR COPY TODAY!

The topics in this research monograph are at the interface of several areas of mathematics such as harmonic analysis, functional analysis, analysis on spaces of homogeneous type, topology, and quasi-metric geometry. The presentation is self-contained with complete, detailed proofs, and a large number of examples and counterexamples are provided.

Unique features of Metrization Theory for Groupoids: With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis include:

* treatment of metrization from a wide, interdisciplinary perspective, with accompanying applications ranging across diverse fields;

* coverage of topics applicable to a variety of scientific areas within pure mathematics;

* useful techniques and extensive reference material;

* includes sharp results in the field of metrization.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

* coverage of topics applicable to a variety of scientific areas within pure mathematics;

* useful techniques and extensive reference material;

* includes sharp results in the field of metrization.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

* useful techniques and extensive reference material;

* includes sharp results in the field of metrization.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

* includes sharp results in the field of metrization.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

Biomechanics is the application of mechanical principles to living organisms, and it is one of the most exciting and fastest growing research areas. In forensic science, it is biomechanics that explains trauma to the body at a crime scene or the fracture of fibers and textiles, and helps interpret blood spatter. Forensic Biomechanics is a comprehensive overview of the role of biomechanics in forensics. Well-illustrated with real-life case studies, and using a multidisciplinary approach, this unique book is an invaluable reference for practicing forensic scientists, lawyers, and researchers.

Partial differential equations is a many-faceted subject. Created to describe the mechanical behavior of objects such as vibrating strings and blowing winds, it has developed into a body of material that interacts with many branches of math ematics, such as differential geometry, complex analysis, and harmonic analysis, as weIl as a ubiquitous factor in the description and elucidatiön of problems in mathematical physics. This work is intended to provide a course of study of some of the major aspects ofPDE.1t is addressed to readers with a background in the basic introductory grad uate mathematics courses in American universities: elementary real and complex analysis, differential geometry, and measure theory. Chapter 1 provides background material on the theory of ordinary differential equations (ODE). This includes both very basic material-on topics such as the existence and uniqueness of solutions to ODE and explicit solutions to equations with constant coefficients and relations to linear algebra-and more sophisticated resuIts-on ftows generated by vector fields, connections with differential geom etry, the calculus of differential forms, stationary action principles in mechanics, and their relation to Hamiltonian systems. We discuss equations of relativistic motion as weIl as equations of classical Newtonian mechanics. There are also applications to topological resuIts, such as degree theory, the Brouwer fixed-point theorem, and the Jordan-Brouwer separation theorem. In this chapter we also treat scalar first-order PDE, via Hamilton-Jacobi theory.

Multivariable analysis is an important subject for mathematicians, both pure and applied. Apart from mathematicians, we expect that physicists, mechanical engi neers, electrical engineers, systems engineers, mathematical biologists, mathemati cal economists, and statisticians engaged in multivariate analysis will find this book extremely useful. The material presented in this work is fundamental for studies in differential geometry and for analysis in N dimensions and on manifolds. It is also of interest to anyone working in the areas of general relativity, dynamical systems, fluid mechanics, electromagnetic phenomena, plasma dynamics, control theory, and optimization, to name only several. An earlier work entitled An Introduction to Analysis: from Number to Integral by Jan and Piotr Mikusinski was devoted to analyzing functions of a single variable. As indicated by the title, this present book concentrates on multivariable analysis and is completely self-contained. Our motivation and approach to this useful subject are discussed below. A careful study of analysis is difficult enough for the average student; that of multi variable analysis is an even greater challenge. Somehow the intuitions that served so well in dimension I grow weak, even useless, as one moves into the alien territory of dimension N. Worse yet, the very useful machinery of differential forms on manifolds presents particular difficulties; as one reviewer noted, it seems as though the more precisely one presents this machinery, the harder it is to understand.

The general aim of the present monograph is to study boundary-value problems for second-order elliptic operators in Lipschitz sub domains of Riemannian manifolds. In the first part (ss1-4), we develop a theory for Cauchy type operators on Lipschitz submanifolds of co dimension one (focused on boundedness properties and jump relations) and solve the $L^p$-Dirichlet problem, with $p$ close to $2$, for general second-order strongly elliptic systems. The solution is represented in the form of layer potentials and optimal non tangential maximal function estimates are established.This analysis is carried out under smoothness assumptions (for the coefficients of the operator, metric tensor and the underlying domain) which are in the nature of best possible. In the second part of the monograph, ss5-13, we further specialize this discussion to the case of Hodge Laplacian $\Delta:=-d\delta-\delta d$. This time, the goal is to identify all (pairs of) natural boundary conditions of Neumann type. Owing to the structural richness of the higher degree case we are considering, the theory developed here encompasses in a unitary fashion many basic PDE's of mathematical physics. Its scope extends to also cover Maxwell's equations, dealt with separately in s14. The main tools are those of PDE's and harmonic analysis, occasionally supplemented with some basic facts from algebraic topology and differential geometry.

For the past 25 years the theory of pseudodifferential operators has played an important role in many exciting and deep investigations into linear PDE. Over the past decade, this tool has also begun to yield interesting results in nonlinear PDE. This book is devoted to a summary and reconsideration of some used of pseudodifferential operator techniques in nonlinear PDE.

One goal has been to build a bridge between two approaches which have been used in a number of papers written in the last decade, one being the theory of paradifferential operators, pioneered by Bony and Meyer, the other the study of pseudodifferential operators whose symbols have limited regularity. The latter approach is a natural successor to classical devices of deriving estimates for linear PDE whose coefficients have limited regularity in order to obtain results in nonlinear PDE.

After developing the requisite tools, we proceed to demonstrate their effectiveness on a range of basic topics in nonlinear PDE. For example, for hyperbolic systems, known sufficient conditions for persistence of solutions are both sharpened and extended in scope. In the treatment of parabolic equations and elliptic boundary problems, it is shown that the results obtained here interface particularly easily with the DeGiorgi-Nash-Moser theory, when that theory applies.

To make the work reasonable self-contained, there are appendices treating background topics in harmonic analysis and the DeGiorgi-Nash-Moser theory, as well as an introductory chapter on pseudodifferential operators as developed for linear PDE.

The book should be of interest to graduate students, instructors, and researchers interested in partial differential equations, nonlinear analysis in classical mathematical physics and differential geometry, and in harmonic analysis.

Systematically constructing an optimal theory, this monograph develops and explores several approaches to Hardy spaces in the setting of Alhlfors-regular quasi-metric spaces. The text is divided into two main parts, with the first part providing atomic, molecular, and grand maximal function characterizations of Hardy spaces and formulates sharp versions of basic analytical tools for quasi-metric spaces, such as a Lebesgue differentiation theorem with minimal demands on the underlying measure, a maximally smooth approximation to the identity and a Calderon-Zygmund decomposition for distributions. These results are of independent interest. The second part establishes very general criteria guaranteeing that a linear operator acts continuously from a Hardy space into a topological vector space, emphasizing the role of the action of the operator on atoms. Applications include the solvability of the Dirichlet problem for elliptic systems in the upper-half space with boundary data from Hardy spaces. The tools established in the first part are then used to develop a sharp theory of Besov and Triebel-Lizorkin spaces in Ahlfors-regular quasi-metric spaces. The monograph is largely self-contained and is intended for mathematicians, graduate students and professionals with a mathematical background who are interested in the interplay between analysis and geometry.

Native American sports team mascots represent a contemporary problem for modern Native American people. The ideas embedded in the mascot representations, however, are as old as the ideas constructed about the Indian since contact between the peoples of Western and the Eastern hemispheres. Such ideas conceived about Native Americans go hand-in-hand with the machinations of colonialism and conquest of these people. This research looks at how such ideas inform the construction of identity of white males from historic experiences with Native Americans. Notions of “playing Indian” and of “going Native” are precipitated from these historic contexts such that in the contemporary sense of considering Native Americans, popular culture ideas dress Native Americans in feathers and buckskin in order to satisfy stereotypic expectations of Indian-ness.

Many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator in a given domain. When the differential operator under discussion is of second order a variety of tools are available for dealing with such problems, including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. When the differential operator is of higher-order (as is the case, e.g., with anisotropic plate bending when one deals with a fourth order operator) only a few options could be successfully implemented. In the 1970s Alberto Calderón, one of the founders of the modern theory of Singular Integral Operators, advocated the use of layer potentials for the treatment of higher-order elliptic boundary value problems. The present monograph represents the first systematic treatment based on this approach.

This research monograph lays, for the first time, the mathematical foundation aimed at solving boundary value problems for higher-order elliptic operators in non-smooth domains using the layer potential method and addresses a comprehensive range of topics, dealing with elliptic boundary value problems in non-smooth domains including layer potentials, jump relations, non-tangential maximal function estimates, multi-traces and extensions, boundary value problems with data in Whitney–Lebesque spaces, Whitney–Besov spaces, Whitney–Sobolev- based Lebesgue spaces, Whitney–Triebel–Lizorkin spaces,Whitney–Sobolev-based Hardy spaces, Whitney–BMO and Whitney–VMO spaces.

This thesis reports on the development of the first quantum enhanced microscope and on its applications in biological microscopy. The first quantum particle-tracking microscope, described in detail here, represents a pioneering advance in quantum microscopy, which is shown to be a powerful and relevant technique for future applications in science and medicine.

The microscope is used to perform the first quantum-enhanced biological measurements -- a central and long-standing goal in the field of quantum measurement. Sub diffraction-limited quantum imaging is achieved, also for the first time, with a scanning probe imaging configuration allowing 10-nanometer resolution.

Partial differential equations is a many-faceted subject. Created to describe the mechanical behavior of objects such as vibrating strings and blowing winds, it has developed into a body of material that interacts with many branches of math ematics, such as differential geometry, complex analysis, and harmonic analysis, as weil as a ubiquitous factor in the description and elucidation of problems in mathematical physics. This work is intended to provide a course of study of some of the major aspects of PDE. It is addressed to readers with a background in the basic introductory grad uate mathematics courses in American universities: elementary real and complex analysis, differential geometry, and measure theory. Chapter 1 provides background material on the theory of ordinary differential equations (ODE). This includes both very basic material-on topics such as the existence and uniqueness of solutions to ODE and explicit solutions to equations with constant coefficients and relations to linear algebra-and more sophisticated results-on flows generated by vector fields, connections with differential geom etry, the calculus of differential forms, stationary action principles in mechanics, and their relation to Hamiltonian systems. We discuss equations of relativistic motion as weIl as equations of c1assical Newtonian mechanics. There are also applications to topological results, such as degree theory, the Brouwer fixed-point theorem, and the Jordan-Brouwer separation theorem. In this chapter we also treat scalar first-order PDE, via Hamilton-Jacobi theory.

This IMA Volume in Mathematics and its Applications SINGULARITIES AND OSCILLATIONS is based on the proceedings of a very successful one-week workshop with the same title, which was an integral part of the 1994-1995 IMA program on "Waves and Scattering. " We would like to thank Joseph Keller, Jeffrey Rauch, and Michael Taylor for their excellent work as organizers of the meeting. We would like to express our further gratitude to Rauch and Taylor, who served as editors of the proceedings. We also take this opportunity to thank the National Science Foun dation (NSF), the Army Research Office (ARO) and the Office of Naval Research (ONR), whose financial support made the workshop possible. Avner Friedman Robert Gulliver v PREFACE Thestudyofsingularitiesand oscillationsofwaves has progressed along several fronts. A key common feature is the presence of a small scale in the solutions. Recent emphasis has been on nonlinear waves. Nonlinear problems are generally less amenable than linear problems to broad unified approaches. As a result there is a justifiable tendency to concentrate on problems of particular geometric or physical interest. This volume con tains a multiplicity of approaches brought to bear on problems varying from the formation ofcaustics and the propagation ofwaves at a boundary to the examination ofviscous boundary layers. There is an examination of the foundations of the theory of high-frequency electromagnetic waves in a dielectric or semiconducting medium. Unifying themes are not entirely absent from nonlinear analysis.

Economic voting is a phenomenon that political scientists and economists can hardly overlook. There is ample evidence for a strong link between economic conditions and government popularity. However, not everything is that simple and this edited collection focuses on 'the comparative puzzle' of economic voting.
Economic Voting emphasises the importance of comparative research design and argues that the psychology of the economic voter model needs to be developed further.

The book discusses the extensions of basic Fourier Analysis techniques to the Clifford algebra framework.
Topics covered: construction of Clifford-valued wavelets, Calderon-Zygmund theory for Clifford valued singular integral operators on Lipschitz hyper-surfaces, Hardy spaces of Clifford monogenic functions on Lipschitz domains. Results are applied to potential theory and elliptic boundary value problems on non-smooth domains. The book is self-contained to a large extent and well-suited for graduate students and researchers in the areas of wavelet theory, Harmonic and Clifford Analysis.
It will also interest the specialists concerned with the applications of the Clifford algebra machinery to Mathematical Physics.

This book presents fourteen new essays by international scholars about the intersections between pragmatism, education, and philosophy with children. Pragmatism from its beginnings has sought a revolution in learning, and is itself a special kind of philosophy of education. What can the applications of pragmatism to pedagogy around the world teach us today?

Written primarily for students who have completed the standard first courses in calculus and linear algebra, Elementary Differential Geometry, Revised 2nd Edition, provides an introduction to the geometry of curves and surfaces.

The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard.

This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this material supplements the content but no computer skill is necessary to take full advantage of this comprehensive text.

Over 36,000 copies sold worldwideAccessible, practical yet rigorous approach to a complex topic--also suitable for self-studyExtensive update of appendices on Mathematica and Maple software packagesThorough streamlining of second edition's numbering systemFuller information on solutions to odd-numbered problemsAdditional exercises and hints guide students in using the latest computer modeling tools

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.

This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.

Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

This outstanding textbook by a distinguished mathematical scholar introduces the differential geometry of curves and surfaces in three-dimensional Euclidean space. The subject is presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the geometric significance and theoretical and practical importance of the different concepts, methods and results involved.
The first chapters of the book focus on the basic concepts and facts of analytic geometry, the theory of space curves, and the foundations of the theory of surfaces, including problems closely related to the first and second fundamental forms. The treatment of the theory of surfaces makes full use of the tensor calculus.
The later chapters address geodesics, mappings of surfaces, special surfaces, and the absolute differential calculus and the displacement of Levi-Cività. Problems at the end of each section (with solutions at the end of the book) will help students meaningfully review the material presented, and familiarize themselves with the manner of reasoning in differential geometry.

The principal aim of analysis of tensors is to investigate those relations which remain valid when we change from one coordinate system to another. This book on Tensors requires only a knowledge of elementary calculus, differential equations and classical mechanics as pre-requisites. It provides the readers with all the information about the tensors along with the derivation of all the tensorial relations/equations in a simple manner. The book also deals in detail with topics of importance to the study of special and general relativity and the geometry of differentiable manifolds with a crystal clear exposition. The concepts dealt within the book are well supported by a number of solved examples. A carefully selected set of unsolved problems is also given at the end of each chapter, and the answers and hints for the solution of these problems are given at the end of the book. The applications of tensors to the fields of differential geometry, relativity, cosmology and electromagnetism is another attraction of the present book.

This book is intended to serve as text for postgraduate students of mathematics, physics and engineering. It is ideally suited for both students and teachers who are engaged in research in General Theory of Relativity and Differential Geometry.

Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student.
Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Dr. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there.
For this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. The result was to further increase the merit of this stimulating, thought-provoking text — ideal for classroom use, but also perfectly suited for self-study. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.

Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature,...) andobjectives,inparticularto understand certain classes of (compact) Riemannian manifolds de?ned by curvature conditions (constant or positive or negative curvature,...). Bywayofcontrast,g- metric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two ?elds complement each other very well; geometric analysis o?ers tools for solving di?cult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting am- tious goals. It is the aim of this book to be a systematic and comprehensive introduction to Riemannian geometry and a representative introduction to the methods of geometric analysis. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. The present work is the ?fth edition of my textbook on Riemannian geometry and geometric analysis. It has developed on the basis of several graduate courses I taught at the Ruhr-University Bochum and the University of Leipzig. The main new features of the present edition are the systematic inclusion of ?ow equations and a mathematical treatment of the nonlinear sigma model of quantum ?eld theory. These new topics also led to a systematic reorganization of the other material. Naturally, I have also included several smaller additions and minor corrections (for which I am grateful to several readers).

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing. This collection will be useful to students and researchers for decades to come.

The contributors are Marco Abate, Marco Arizzi, Alexander Blokh, Thierry Bousch, Xavier Buff, Serge Cantat, Tao Chen, Robert Devaney, Alexandre Dezotti, Tien-Cuong Dinh, Romain Dujardin, Hugo García-Compeán, William Goldman, Rotislav Grigorchuk, John Hubbard, Yunping Jiang, Linda Keen, Jan Kiwi, Genadi Levin, Daniel Meyer, John Milnor, Carlos Moreira, Vincente Muñoz, Viet-Anh Nguyên, Lex Oversteegen, Ricardo Pérez-Marco, Ross Ptacek, Jasmin Raissy, Pascale Roesch, Roberto Santos-Silva, Dierk Schleicher, Nessim Sibony, Daniel Smania, Tan Lei, William Thurston, Vladlen Timorin, Sebastian van Strien, and Alberto Verjovsky.

Based on a course given at Oxford over many years, this book is a short and concise exposition of the central ideas of general relativity. Although the original audience was made up of mathematics students, the focus is on the chain of reasoning that leads to the relativistic theory from the analysis of distance and time measurements in the presence of gravity, rather than on the underlying mathematical structure. The geometric ideas - which are central to the understanding of the nature of gravity - are introduced in parallel with the development of the theory, the emphasis being on laying bare how one is led to pseudo-Riemannian geometry through a natural process of reconciliation of special relativity with the equivalence principle. At centre stage are the "local inertial coordinates" set up by an observer in free fall, in which special relativity is valid over short times and distances.

In more practical terms, the book is a sequel to the author's Special Relativity in the same series, with some overlap in the treatment of tensors. The basic theory is presented using techniques, such as phase-plane analysis, that will already be familiar to mathematics undergraduates, and numerous problems, of varying levels of difficulty, are provided to test understanding. The latter chapters include the theoretical background to contemporary observational tests - in particular the detection of gravitational waves and the verification of the Lens-Thirring precession - and some introductory cosmology, to tempt the reader to further study.

While primarily designed as an introduction for final-year undergraduates and first-year postgraduates in mathematics, the book is also accessible to physicists who would like to see a more mathematical approach to the ideas.

The theory of Riemann surfaces occupies a very special place in mathematics. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. It provides a model for a large number of more recent developments in areas including manifold topology, global analysis, algebraic geometry, Riemannian geometry, and diverse topics in mathematical physics. This graduate text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind. Part I sets up the interplay between complex analysis and topology, with the latter treated informally. Part II works as a rapid first course in Riemann surface theory, including elliptic curves. The core of the book is contained in Part III, where the fundamental analytical results are proved. Following this section, the remainder of the text illustrates various facets of the more advanced theory.

A solid introduction to the methods of differential geometry and tensor calculus, this volume is suitable for advanced undergraduate and graduate students of mathematics, physics, and engineering. Rather than a comprehensive account, it offers an introduction to the essential ideas and methods of differential geometry.
Part 1 begins by employing vector methods to explore the classical theory of curves and surfaces. An introduction to the differential geometry of surfaces in the large provides students with ideas and techniques involved in global research. Part 2 introduces the concept of a tensor, first in algebra, then in calculus. It covers the basic theory of the absolute calculus and the fundamentals of Riemannian geometry. Worked examples and exercises appear throughout the text.

One of the most widely used texts in its field, this volume introduces the differential geometry of curves and surfaces in both local and global aspects. The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. Many examples and exercises enhance the clear, well-written exposition, along with hints and answers to some of the problems.
The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. Suitable for advanced undergraduates and graduate students of mathematics, this text's prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables. For this second edition, the author has corrected, revised, and updated the entire volume.

This textbook is distinguished from other texts on the subject by the depth of the presentation and the discussion of the calculus of moving surfaces, which is an extension of tensor calculus to deforming manifolds.

Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation and dynamic fluid film equations.

The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author’s skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 – when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject.

The last part of the textbook is devoted to the Calculus of Moving Surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems and dynamic fluid film equations developed by the author in recent years. Furthermore, the moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss-Bonnet theorem.

These lecture notes are based on the joint work of the author and Arthur Fischer on Teichmiiller theory undertaken in the years 1980-1986. Since then many of our colleagues have encouraged us to publish our approach to the subject in a concise format, easily accessible to a broad mathematical audience. However, it was the invitation by the faculty of the ETH Ziirich to deliver the ETH N achdiplom-Vorlesungen on this material which provided the opportunity for the author to develop our research papers into a format suitable for mathematicians with a modest background in differential geometry. We also hoped it would provide the basis for a graduate course stressing the application of fundamental ideas in geometry. For this opportunity the author wishes to thank Eduard Zehnder and Jiirgen Moser, acting director and director of the Forschungsinstitut fiir Mathematik at the ETH, Gisbert Wiistholz, responsible for the Nachdiplom Vorlesungen and the entire ETH faculty for their support and warm hospitality. This new approach to Teichmiiller theory presented here was undertaken for two reasons. First, it was clear that the classical approach, using the theory of extremal quasi-conformal mappings (in this approach we completely avoid the use of quasi-conformal maps) was not easily applicable to the theory of minimal surfaces, a field of interest of the author over many years. Second, many other active mathematicians, who at various times needed some Teichmiiller theory, have found the classical approach inaccessible to them.

This monograph presents a concise yet elementary account of exterior differential system theory so that it can be quickly applied to problems. The first part of the monograph, Chapters 1-5, deals with the general theory: the Cartan-Kaehler theorem is proved, the notions of involution and prolongation are carefully laid out, quasi-linear differential systems are examined in detail, and explicit examples of the Spencer cohomology groups and the characteristic variety are given. The second part of the monograph, Chapters 6 and 7, deals with applications to problems in differential geometry: the isometric embedding theorem of Cartan-Janet and its various geometric ramifications are discussed, a proof of the Andreotti-Hill theorem on the O-R embedding problem is given, and embeddings of abstract projective structures are discussed. For researchers and graduate students who would like a good introduction to exterior differential systems. This volume will also be particularly useful to those whose work involves differential geometry and partial differential equations.

There has been fundamental progress in complex differential geometry in the last two decades. For one, The uniformization theory of canonical Kähler metrics has been established in higher dimensions, and many applications have been found, including the use of Calabi-Yau spaces in superstring theory. This monograph gives an introduction to the theory of canonical Kähler metrics on complex manifolds. It also presents some advanced topics not easily found elsewhere.

The emergence of topological quantum ?eld theory has been one of the most important breakthroughs which have occurred in the context of ma- ematical physics in the last century, a century characterizedbyindependent developments of the main ideas in both disciplines, physics and mathematics, which has concluded with two decades of strong interaction between them, where physics, as in previous centuries, has acted as a source of new mat- matics. Topological quantum ?eld theories constitute the core of these p- nomena, although the main drivingforce behind it has been the enormous e?ort made in theoretical particle physics to understand string theory as a theory able to unify the four fundamental interactions observed in nature. These theories set up a new realm where both disciplines pro?t from each other. Although the most striking results have appeared on the mathema- calside,theoreticalphysicshasclearlyalsobene?tted,sincethecorresponding developments have helped better to understand aspects of the fundamentals of ?eld and string theory.

This book gives a presentation of topics in Hamilton's Ricci flow for graduate students and mathematicians interested in working in the subject. The authors have aimed at presenting technical material in a clear and detailed manner. In this volume, geometric aspects of the theory have been emphasized. The book presents the theory of Ricci solitons, Kahler-Ricci flow, compactness theorems, Perelman's entropy monotonicity and no local collapsing, Perelman's reduced distance function and applications to ancient solutions, and a primer of 3-manifold topology. Various technical aspects of Ricci flow have been explained in a clear and detailed manner. The authors have tried to make some advanced material accessible to graduate students and nonexperts. The book gives a rigorous introduction to Perelman's work and explains technical aspects of Ricci flow useful for singularity analysis. Throughout, there are appropriate references so that the reader may further pursue the statements and proofs of the various results.

This invaluable monograph has arisen in part from E Witten's lectures on topological quantum field theory in the spring of 1989 at Princeton University. At that time Witten unified several important mathematical works in terms of quantum field theory, most notably the Donaldson polynomial, the Gromov-Floer homology and the Jones polynomials.In his lectures, among other things, Witten explained his intrinsic three-dimensional construction of Jones polynomials via Chern-Simons gauge theory. He provided both a rigorous proof of the geometric quantization of the Chern-Simons action and a very illuminating view as to how the quantization arises from quantization of the space of connections. He constructed a projective flat connection for the Hilbert space bundle over the space of complex structures, which becomes the Knizhik-Zamolodchikov equations in a special case. His construction leads to many beautiful applications, such as the derivation of the skein relation and the surgery formula for knot invariant, a proof of Verlinde's formula, and the establishment of a connection with conformal field theory.In this book, Sen Hu has added material to provide some of the details left out of Witten's lectures and to update some new developments. In Chapter 4 he presents a construction of knot invariant via representation of mapping class groups based on the work of Moore-Seiberg and Kohno. In Chapter 6 he offers an approach to constructing knot invariant from string theory and topological sigma models proposed by Witten and Vafa. The localization principle is a powerful tool to build mathematical foundations for such cohomological quantum field theories.In addition, some highly relevant material by S S Chern and E Witten has been included as appendices for the convenience of readers: (1) Complex Manifold without Potential Theory by S S Chern, pp148-154. (2) “Geometric quantization of Chern-Simons gauge theory” by S Axelrod, S D Pietra and E Witten. (3) “On holomorphic factorization of WZW and Coset models” by E Witten.

...there are reasons enough to warrant a coherent treatment of the main body of differential topology in the realm of Banach manifolds, which is at the same time correct and complete. This book fills the gap: whenever possible the manifolds treated are Banach manifolds with corners. Corners add to the complications and the authors have carefully fathomed the validity of all main results at corners. Even in finite dimensions some results at corners are more complete and better thought out here than elsewhere in the literature. The proofs are correct and with all details. I see this book as a reliable monograph of a well-defined subject; the possibility to fall back to it adds to the feeling of security when climbing in the more dangerous realms of infinite dimensional differential geometry.

Peter W. Michor

Up until recently, Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead (and still are given in some places) have come gradually to be viewed as anachronisms. However, there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date, that is to say, which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education, and what might be the appropriate level of abstractness of their exposition. The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. The subject-matter and level of abstractness of its exposition were dictated by the view that, in addition to the geometry of curves and surfaces, the following topics are certainly useful in the various areas of application of mathematics (especially in elasticity and relativity, to name but two), and are therefore essential: the theory of tensors (including covariant differentiation of them); Riemannian curvature; geodesics and the calculus of variations (including the conservation laws and Hamiltonian formalism); the particular case of skew-symmetric tensors (i. e.

The main aim of this book is to introduce Lie groups and allied algebraic and geometric concepts to a robotics audience. These topics seem to be quite fashionable at the moment, but most of the robotics books that touch on these topics tend to treat Lie groups as little more than a fancy notation. I hope to show the power and elegance of these methods as they apply to problems in robotics. A subsidiary aim of the book is to reintroduce some old ideas by describing them in modem notation, particularly Study's Quadric-a description of the group of rigid motions in three dimensions as an algebraic variety (well, actually an open subset in an algebraic variety)-as well as some of the less well known aspects of Ball's theory of screws. In the first four chapters, a careful exposition of the theory of Lie groups and their Lie algebras is given. Except for the simplest examples, all examples used to illustrate these ideas are taken from robotics. So, unlike most standard texts on Lie groups, emphasis is placed on a group that is not semi-simple-the group of proper Euclidean motions in three dimensions. In particular, the continuous subgroups of this group are found, and the elements of its Lie algebra are identified with the surfaces of the lower Reuleaux pairs. These surfaces were first identified by Reuleaux in the latter half of the 19th century.

This concise guide to the differential geometry of curves and surfaces can be recommended to ?rst-year graduate students, strong senior students, and students specializing in geometry. The material is given in two parallel streams. The ?rst stream contains the standard theoretical material on differential ge- etry of curves and surfaces. It contains a small number of exercises and simple problems of a local nature. It includes the whole of Chapter 1 except for the pr- lems (Sections 1.5, 1.7, 1.10) and Section 1.11, about the phase length of a curve, and the whole of Chapter 2 except for Section 2.6, about classes of surfaces, T- orems 2.8.1–2.8.4, the problems (Sections 2.7.4, 2.8.3) and the appendix (S- tion 2.9). The second stream contains more dif?cult and additional material and for- lations of some complicated but important theorems, for example, a proof of A.D. Aleksandrov’s comparison theorem about the angles of a triangle on a convex 1 surface, formulations of A.V. Pogorelov’s theorem about rigidity of convex s- faces, and S.N. Bernstein’s theorem about saddle surfaces. In the last case, the formulations are discussed in detail. A distinctive feature of the book is a large collection (80 to 90) ofnonstandard andoriginalproblems that introduce the student into the real world of geometry.

The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds.

In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers.

Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.

Geometry in ancient Greece is said to have originated in the curiosity of mathematicians about the shapes of crystals, with that curiosity culminating in the classification of regular convex polyhedra addressed in the final volume of Euclid’s Elements. Since then, geometry has taken its own path and the study of crystals has not been a central theme in mathematics, with the exception of Kepler’s work on snowflakes. Only in the nineteenth century did mathematics begin to play a role in crystallography as group theory came to be applied to the morphology of crystals.

This monograph follows the Greek tradition in seeking beautiful shapes such as regular convex polyhedra. The primary aim is to convey to the reader how algebraic topology is effectively used to explore the rich world of crystal structures. Graph theory, homology theory, and the theory of covering maps are employed to introduce the notion of the topological crystal which retains, in the abstract, all the information on the connectivity of atoms in the crystal. For that reason the title Topological Crystallography has been chosen.

Topological crystals can be described as “living in the logical world, not in space,” leading to the question of how to place or realize them “canonically” in space. Proposed here is the notion of standard realizations of topological crystals in space, including as typical examples the crystal structures of diamond and lonsdaleite. A mathematical view of

the standard realizations is also provided by relating them to asymptotic behaviors of random walks and harmonic maps. Furthermore, it can be seen that a discrete analogue of algebraic geometry is linked to the standard realizations.

Applications of the discussions in this volume include not only a systematic enumeration of crystal structures, an area of considerable scientific interest for many years, but also the architectural design of lightweight rigid structures. The reader therefore can see the agreement of theory and practice.

For more than five decades Bertram Kostant has been one of the major architects of modern Lie theory. Virtually all his papers are pioneering with deep consequences, many giving rise to whole new fields of activities. His interests span a tremendous range of Lie theory, from differential geometry to representation theory, abstract algebra, and mathematical physics. Some specific topics cover algebraic groups and invariant theory, the geometry of homogeneous spaces, representation theory, geometric quantization and symplectic geometry, Lie algebra cohomology, Hamiltonian mechanics, modular forms, Whittaker theory, Toda lattice, and much more. It is striking to note that Lie theory (and symmetry in general) now occupies an ever increasing larger role in mathematics than it did in the fifties.

During his years as professor at the Massachusetts Institute of Technology from 1962 until retiring from teaching in 1993, he received many honors and prizes: election to the National Academy of Sciences USA, the American Academy of Arts and Sciences, the AMS Steele Prize, Honorary Doctorates from University of Córdoba, Argentina, the University of Salamanca, Spain, Purdue University. Now in the sixth decade of his career, he continues to produce results of astonishing beauty and significance for which he is invited to lecture all over the world.

This is the first volume (1955-1966) of a five-volume set of Bertram Kostant's collected papers. A distinguished feature of this first volume is Kostant's commentaries and summaries of his papers in his own words.

Differential geometry arguably offers the smoothest transition from the standard university mathematics sequence of the first four semesters in calculus, linear algebra, and differential equations to the higher levels of abstraction and proof encountered at the upper division by mathematics majors. Today it is possible to describe differential geometry as "the study of structures on the tangent space," and this text develops this point of view.

This book, unlike other introductory texts in differential geometry, develops the architecture necessary to introduce symplectic and contact geometry alongside its Riemannian cousin. The main goal of this book is to bring the undergraduate student who already has a solid foundation in the standard mathematics curriculum into contact with the beauty of higher mathematics. In particular, the presentation here emphasizes the consequences of a definition and the careful use of examples and constructions in order to explore those consequences.