Detailed calculations of a number of concrete examplesWritten for both mathematicians who want to see something of the applications of topology and geometry to modern physics Written for physicists who want to see the foundations of their subject treated with mathematical rigor
Gregory Naber is a Professor at Drexel University in the Department of Mathematics
Like any books on a subject as vast as this, this book has to have a point-of-view to guide the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. The reader is asked to join the author on some vague notion of what an electromagnetic field might be, to be willing to accept a few of the more elementary pronouncements of quantum mechanics, and to have a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. In return, the book offers an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1.|Through detailed calculations of a number of concrete examples, this book displays an introduction to topology and geometry that reflects the recent impact of gauge theory. This text should be accessible to first year graduate students of mathematics and physics.
Contents: Preface.- Physical and geometrical motivation 1 Topological spaces.- Homotopy groups.- Principal bundles.- Differentiable manifolds and matrix Lie groups.- Gauge fields and Instantons. Appendix. References. Index.
This is a book on topology and geometry, and like any book on subjects as vast as these, it has a point of view that guided the selection of topics. The author´s point of view is that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The goal is to weave together rudimentary notions from the classical gauge theories of physics and the topological and geometrical concepts that became the mathematical models of these notions. The reader is assumed to have a minimal understanding of what an electromagnetic field is, a willingness to accept a few of the more elementary pronouncements of quantum mechanics, and a solid background in real analysis and linear algebra with some of the vocabulary of modern algebra. To such a reader we offer an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2)-connections on S4 with instanton number -1. This second edition of the book includes a new chapter on singular homology theory and a new appendix outlining Donaldson´s beautiful application of gauge theory to the topology of compact, simply connected , smooth 4-manifolds with definite intersection form.Reviews of the first edition:"It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a deep knowledge of his subject with an excellent informal writing style.” (NZMS Newsletter)"...this book should be very interesting for mathematicians and physicists (as well as other scientists) who are concerned with gauge theories."(ZENTRALBLATT FUER MATHEMATIK) "The book is well written and the examples do a great service to the reader. It will be a helpful companion to anyone teaching or studying gauge theory ...” (Mathematical Reviews)
First Edition Review:
"Naber´s book, together with its predecessor[N4] subtitled Foundations, occupies a less populated niche in the market. This is the sector of teachable texts on differential geometry and its use in physics. Teachability does not refer to a definition-theorem-proof format. Nor does it imply anything about the depth of the treatment. Rather, it has to do with the organization of the topics, the selection of examples, the amount of instructive details provided, the ability to anticipate questions from the reader, and knowing when to stop."
--SIAM REVIEW
This is a book on topology and geometry and, like any books on subjects as vast as these, it has a point-of-view that guided the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. We ask the reader to come to us with some vague notion of what an electromagnetic field might be, a willingness to accept a few of the more elementary pronouncements of quantum mechanics, a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. To such a reader we offer an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1. Iwould go over both volumes thoroughly and make some minor changes in terminology and notation and correct any errors I find.In this new edition, a chapter on Singular Homology will be added as well as minor changes in notation and terminology throughout and some sections have been rewritten or omitted. Reviews of First Edition:"It is unusual to find a book so carefully tailored to the needs of this interdisciplinary area of mathematical physics...Naber combines a knowledge of his subject with an excellent informal writing style."(NZMS Newletter)"...this book should be very interesting for mathematicians and physicists (as well as other scientists) who are concerned with gauge theories."(Zentralblatt Fuer Mathematik)
First Edition Review:
"Naber's book, together with its predecessor[N4] subtitled Foundations, occupies a less populated niche in the market. This is the sector of teachable texts on differential geometry and its use in physics. Teachability does not refer to a definition-theorem-proof format. Nor does it imply anything about the depth of the treatment. Rather, it has to do with the organization of the topics, the selection of examples, the amount of instructive details provided, the ability to anticipate questions from the reader, and knowing when to stop."
--SIAM REVIEW
Über den Autor
Gregory Naber is a Professor at Drexel University in the Department of Mathematics
Inhaltsverzeichnis
Contents: Preface.- Physical and geometrical motivation 1 Topological spaces.- Homotopy groups.- Principal bundles.- Differentiable manifolds and matrix Lie groups.- Gauge fields and Instantons. Appendix. References. Index.
Klappentext
Like any books on a subject as vast as this, this book has to have a point-of-view to guide the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. The reader is asked to join the author on some vague notion of what an electromagnetic field might be, to be willing to accept a few of the more elementary pronouncements of quantum mechanics, and to have a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. In return, the book offers an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1.
Through detailed calculations of a number of concrete examples, this book displays an introduction to topology and geometry that reflects the recent impact of gauge theory. This text should be accessible to first year graduate students of mathematics and physics.
