Published: 27 April 2021
Publisher: International Press of Boston, Inc.
Hardcover 9781571463975
836 pages
For the past forty years, Robert Bartnik has been one of the leading mathematicians working on mathematical general relativity and geometric analysis. Since his early dissertation work on the existence of maximal hypersurfaces in general asymptotically flat spacetimes, done under the guidance of S.T. Yau at the Institute for Advanced Study at Princeton, Bartnikfs work has had a major impact on a number of different areas in mathematical relativity.
His careful definition of the ADM mass on asymptotically Euclidean geometries, together with his analysis of the Laplace operator on such geometries, has been highly influential in geometric analysis. This work led in turn to his insightful definition of gquasi-local mass,h a topic of intense interest to this day.
Bartnikfs collaboration with his student John McKinnon resulted in their iconic discovery of a globally regular static solution of the Einstein?Yang?Mills equations. His proof that there exist globally hyperbolic spacetime solutions of Einsteinfs equations, which contain no constant mean curvature Cauchy surfaces, was very surprising, and has led to a variety of further results of this nature. The procedure he developed for generating solutions of the Einstein constraint equations using a parabolic PDE system, has already led to important applications and is likely to be very useful in the future.
With the publication of this volume, the editors wish to honor Robert Bartnikfs great contributions to their field. Included in this collection are most of his published papers, together with short essays by friends and colleagues who have been strongly influenced him. The editors dedicate this collection to Robert, and to all those who will greatly benefit from being introduced to his work.
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An inviting, intuitive, and visual exploration of differential geometry and forms
Hardcover ISBN: 9780691203690
Paperback ISBN: 9780691203706
Published: Jul 13, 2021
Copyright: 2021
Pages: 584
Size: 7 x 10 in.
Overview
Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newtonfs geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner.
Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gaussfs famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einsteinfs field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwellfs equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartanfs method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book.
Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.
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Offers a concise introduction to spectral theory designed for newcomers to
functional analysis
Illustrates a variety of applications of spectral theory to differential
operators, including the Dirichlet Laplacian and Schrodinger operators
Incorporates a brief introduction to functional analysis, with a focus on
unbounded operators and separable Hilbert spaces
This textbook offers a concise introduction to spectral theory, designed for newcomers to
functional analysis. Curating the content carefully, the author builds to a proof of the spectral
theorem in the early part of the book. Subsequent chapters illustrate a variety of application
areas, exploring key examples in detail. Readers looking to delve further into specialized topics
will find ample references to classic and recent literature. Beginning with a brief introduction to
functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as
the essential tools needed for the subsequent theory. A thorough discussion of the concepts of
spectrum and resolvent follows, leading to a complete proof of the spectral theorem for
unbounded self-adjoint operators. Applications of spectral theory to differential operators
comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian
operator, Schrodinger operators, operators on graphs, and the spectral theory of Riemannian
manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further
study in any number of different directions. A background in real and complex analysis is
assumed; the author presents the requisite tools from functional analysis within the text. This
introductory treatment would suit a functional analysis course intended as a pathway to linear
PDE theory. Independent later chapters allow for flexibility in selecting applications to suit
specific interests within a one-semester course.
1st ed. 2020, X, 338 p. 31
illus., 30 illus. in color.
Hardcover
ISBN 978-3-030-38001-4
Softcover
ISBN 978-3-030-38004-5
Product category : Graduate/advanced undergraduate textbook
Series : Graduate Texts in Mathematics
Mathematics : Partial Differential Equations
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Reveals the myriad aspects of Big Data collection and analysis
Discusses critical issues and ethical aspects of the use of Big Data
Offers various case studies
This book reveals the myriad aspects of Big Data collection and analysis, by defining and
clarifying the meaning of Big Data and its unique characteristics in a non-technical and
easy-tofollow way. Moreover, it discusses critical issues and problems related to the Big Data
revolution and their implications for both Statistics as a discipline and for our everyday lives.
The author identifies various problems and limitations in the quantitative analysis of Big Data,
with regard to e.g. its volume, velocity and variety, as well as its reliability and veridicity.
Dedicated chapters focus on the epistemological aspects of data-based knowledge and ethical
aspects of the use of Big Data, while also addressing paradigmatic cases such as Cambridge
Analytica and the use of data from social networks to influence election outcomes.
1st ed. 2021, XI, 54 p. 5 illus., 1 illus. in color.
Softcover
ISBN 978-3-030-73029-1
Product category : Brief
Series : SpringerBriefs in Statistics
Economics : Econometrics
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The first and the only monograph to treat topological, group-theoretic, and
geometric problems of ideal hydrodynamics and magneto-hydrodynamics
from a unified point of view. It has become a classical book and the standard
reference by now: Google Scholar includes nearly 1800 citations of this book
It includes over 70 figures, which helps the reader to "visualise", as opposed
to "compute" hydrodynamics
It includes the survey of recent developments for the last 20 years in this
now-flourishing field of topological and geometric hydrodynamics. The
corresponding survey bibliography includes over 250 titles
First published in 1998 this unique monograph treats topological, group-theoretic, and
geometric problems of ideal hydrodynamics and magneto-hydrodynamics from a unified point
of view. It describes the necessary preliminary notions both in hydrodynamics and pure
mathematics with numerous examples and figures. This book, now accepted as one of the
main references in the field, is accessible to graduates as well as pure and applied
mathematicians working in hydrodynamics, Lie groups, dynamical systems, and differential
geometry. The updated second edition also contains a survey of recent developments in this
now-flourishing field of topological and geometric hydrodynamics.
2nd ed. 2021, XX, 455 p. 79 illus.
Hardcover
ISBN 978-3-030-74277-5
Product category : Monograph
Series : Applied Mathematical Sciences
Mathematics : Mathematics (general)