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1. The Concept of a Riemann Surface
2. Symmetry
3. Space-time-matter
4. The Theory Of Groups And Quantum
5. The Continuum: A Critical Examination
6. Hermann Weyl: 1885-1985: Centenary
7. Groups and Analysis: The Legacy
8. Mind and Nature: Selected Writings
9. Hermann Weyl's Raum - Zeit - Materie
10. Mathematical Heritage of Hermann
11. Exact sciences and their philosophical
12. Das Kontinuum and Anders Monographien
13. Infinite Loop Spaces: Hermann
14. The Classical Groups: Their Invariants
15. Theory of Groups & Quantum
16. Theory of Groups & Quantum
17. Philosophy of Mathematics and
18. Raum, Zeit, Materie: Vorlesungen
19. Temps, espace, matière: leçons
20. Riemanns geometrische Ideen, ihre

1. The Concept of a Riemann Surface
by Hermann Weyl
Paperback: 204 Pages (2010-02-02)
list price: US$9.99 -- used & new: US$6.30
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Asin: 160796239X
Average Customer Review: 4.5 out of 5 stars
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Weyl combined function theory and geometry in this high-level landmark work, forming a new branch of mathematics and the basis of the modern approach to analysis, geometry, and topology. ... Read more

Customer Reviews (3)

3-0 out of 5 stars I don't want to say it's a bad book
Maybe it's excellent.

But I read its first paragraph and immediately saw that it's way beyond my understanding.

Can someone recommend a book which would prepare me for reading this one. My math knowledge includes all high school stuff, 2 years of calculus and some linear algebra.

5-0 out of 5 stars A classic but notrevolutionary
I agree with another reader that this book is a classic.
But the book is not so revolutionary as what the reader said that if you have not read that book, shot yourself.
If that is true, thousands of patients will be jammed into a hospital. Also, the unavailability of this book makes it unknown ot many people. Among various books from Weyl, this is the book out of print. Other books has been re-issued as Dover or continually to be issued by Princeton, only this book remains in oblivion. Also, this book consists only 200+ pages, but the book store sold it at least 240 USD and I think it is very expensive.
May be they think they can speculate that book, but unfortunately, books, unlike watches or jewelleries etc, are the things most unlikely to be speculated.

5-0 out of 5 stars if you haven't hear of this then shoot yourself
this is Weyl's famous exposition of the theory of Riemann surfaces in which the first satisfactory definition of an abstract Riemann Surface is discussed at length ... Read more

2. Symmetry
by Hermann Weyl
Paperback: 176 Pages (1983-01-01)
list price: US$19.95 -- used & new: US$12.02
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Asin: 0691023743
Average Customer Review: 5.0 out of 5 stars
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Defines symmetry through a discussion of its many uses in awide variety of fields both academic and natural. ... Read more

Customer Reviews (8)

5-0 out of 5 stars Enjoyable Book
One of the most enjoyable books I have ever read. Not very long. Requires some knowledge of basic geometry. Good intro book.

5-0 out of 5 stars Overall summetry
Very well written introductory book to symmetry-group theory notions
concise and thought provoking lacking some modern notions and strong mathematical deductions
Can be read from high scool students
an easy pace book ideal for a first contact with the subject

5-0 out of 5 stars Deep and Insightful Book
I thoroughly enjoyed reading this book.It offers very deep insights on the use and implications of symmetry, not only in art, but in physics, chemistry, and other sciences.It also provides a comprehensive mathematical treatment at an accessible level.The reason I did not give it a full 5 stars is that sometimes the mathematical steps were not fully spelled out, or the explanation was vague, so a few passages left me wondering what he really means.More examples, figures, and details would have helped.Still, it was an exciting read - for the first time I really understood the theory of relativity and what it means in terms of symmetry.The author also impresses with his breadth of knowledge, being equally comfortable with the latest mathematical methods as he is with historical development of the field, all the way back to the Egyptians and Greeks.The book is fairly slim and I read it fast, yet learned a lot.

5-0 out of 5 stars a difficult but rewarding introduction to mathematical symmetry and some of it's applications
Symmetry is about the mathematical underpinnings of symmetry as it appears in nature and art. The book is divided into 4 sections, the first Bilateral Symmetry covers reflection. This lecture goes into biology and art. The next lecture is about rotational symmetry. I was able to follow the math presented in this lecture but had trouble in the 3rd lecture titled Ornamental Symmetry. Ornamental symmetry is mostly about tilings of the plane. There is a lot of math presented in this lecture. I had to fall back on my rudimentary knowledge or abstract algebra and linear algebra to understand it. My point is that without this knowledge this lecture and the next one The General Idea of Mathematical Symmetry would have been impossible for me to follow. However, I still recommend this book to people who don't have any of the above background. Symmetry covers the concepts behind symmetry well, and it's applications to nature and art can be followed by anyone.

5-0 out of 5 stars An excellent introduction to the concept of symmetry
Symmetry is a fundamental characteristic of most living creatures, some natural features such as crystals, the basis of some mathematical models and a beautiful form of art. Most animals possess a form of bilateral symmetry, with only minor differences our right and left sides are mirror images of each other. Weyl gives examples of all of these types of symmetry, images with text explaining the details regarding the symmetry of the object.
At the end, he gives the mathematical explanations of the symmetries, how they can be combined into the construct known as a group. The symmetries can then be sequentially combined to perform multiple actions and generate other actions. This dual examination provides a great deal of insight into the idea of symmetry. Biologically, it is clear that there must be powerful evolutionary advantages to symmetry, as it is universal in the animal kingdom. Humans also have a natural affinity for symmetric objects, as symmetry is a universal theme in the art work of cultures with little or no contact between them.
Weyl has written an excellent introduction to the concept of symmetry. It is an idea that is easy to understand and the different motions of a symmetric object are a very good way to begin the study of group theory. Artists can also obtain some benefit from the additional knowledge of symmetry that they will get from this book.
... Read more

3. Space-time-matter
by Hermann Weyl, Henry Herman Leopold Adolf Brose
 Paperback: 356 Pages (2010-09-13)
list price: US$32.75 -- used & new: US$23.79
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Asin: 1171882882
Average Customer Review: 4.5 out of 5 stars
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"A classic of physics ... the first systematic presentation of Einstein’s theory of relativity"—British Journal for Philosophy and Science. Long one of the standard texts in the field, this excellent introduction probes deeply into Euclidean space, Riemann’s space, Einstein’s general relativity, gravitational waves and energy, laws of conservation.
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Customer Reviews (7)

5-0 out of 5 stars Space-Time-Matter
The book teaches a way to study relativity not as an intellectual jump from Newton to Einstein, but by a method of "analytical continuation" after the concept of "measurement" is cleaned up.
One needs to be a little careful about some of the words such as Force.The use is correct but over time, the phrases are changed and we would say the same things differently.It can be confusing at times.I needed to re-read a few times to see things clearly enough to go on.
I noticed one good omen.I think the book has the author's signature in it (?).If I am right, I never said it and I do not have it.I know nothing!!Even though old, the book became very useful to me in learning the subject.Thanks

Weyl's book is most famous for introducing gauge theory, which was later reborn in the form of phase transformations in quantum theory. Weyl did not live guite long enough to hear of the latter being applied by Yang and Mills, though he socially interacted with Yang in his last year at Princeton.

Einstein and Pauli both criticized Weyl's original unified theory based on general relativity using a length gauge, both as implying false empirical consequences (Einstein), since it implied tiny changes of length dependent on path and as untestable (Pauli). (Obviously it could not both be empirically false and non-empirical.) Yet Eddington and later Einstein himself revived similar theories. Eddington claimed that the length differences were to tiny as to be undetectable, but also that his own gauge theory could be thought of not as literal space/time structure but as a geometrization of an abstract background theory for specific space/time structures.

Thomas Ryckman's The Reign of Relativity: Philosophy in Physics 1915-1925 (Oxford Studies in the Philosophy of Science)has an excellent eighty page discussion of these ideas of Weyl in relativity, as well as chapters on those of Eddington in the 1920s.

Another novelty is Weyl's suggestion that General Relativity could be tied to observation via the conformal structure as representing light cones and the projective structure as particles in free fall. This alternative to the rods and clocks approach, on the basis of which Weyl was criticized, has been developed by Ehlers (who edited the new German edition of this work) Pirani and Schild.

Weyl also introduces what he calls "tensor densities" which Shouten called "Weyl tensors" and Synge and Schild call oriented tensors, often called twisted tensors. These are analogous to and include "axial vectors."

Weyl's introduction of the "affine connection" after criticism of Levi-Civita's notion of parallelism led the way to further notions of connections and generalization of the notion of connection as such by Elie Cartan and others.

These are but a few of the intellectual gems in this work.

The philosophical parts are, unfortunately, almost uniformly mistranslated. The phenomenological introduction is re-translated in Kockelmans and Kisiel, eds. Phenomenology and the Natural Sciences (SPEP). (Courant suggested Weyl as successor to Husserl in the philosophy chair at Goettingen!) This together with the misprints in formulas, makes it desirable that the whole book be retranslated.

4-0 out of 5 stars Dated, but a Masterpiece
In 1918, Hermann Weyl developed a theory of the combined gravitational-electromagnetic field that was based on an early form of today's gauge formalism. This book neatly summarizes Weyl's motivations for what can be considered the first serious attempt at unified field theory. This attempt failed, but the gauge idea did not, and in 1929 Weyl transformed it into the gauge-invariant concept of quantum mechanics. Today, gauge invariance is arguably the most profound concept in modern quantum theory, and our understanding of the strong and weak nuclear interactions would not have been possible without it.

Weyl was first and foremost a mathematician, but he also proved to be a visionary theoretical physicist who was greatly admired by the likes of Einstein, Pauli, Dirac and Heisenberg. He was also a great human being who was involved with humanity. In spite of its great age, Space-Time-Matter has earned a place of distinction in the physics literature, if only because of Weyl's gauge idea. The Dover edition costs next to nothing; get it andenlighten yourself.

4-0 out of 5 stars A Classic of Relativity Theory
Not long after Einstein published his general theory of relativity one of the greatest mathematician of his time trumped it. Space-Time-Matter has been published by Dover press for a very long time. My copy was put out in the 5o's. I bought it used in the late 60's. I have never regretted buying it. It is difficult reading even when you know what he is talking about: when I got it , it read like Greek. It isn't an easy read, but he predicts a tenth planet in it that was never found! And lays the foundation of what later became gauge theory. He introduces group theory at a time when quantum groups were just beginning. His tensor discussion is very basic and he doesn't even introduce the Weyl tensor! But he taught me the basic metrical equations and the applications of non Euclidean geometry to relativity. Together with Weinberg's flawed Cosmology this book has been my teacher. I wish I could say he did a good job, but since it took me years to wade through it without falling asleep, I can't say he did!

5-0 out of 5 stars God is the geometer (maybe)
This book is esoteric initiation into spacetime physics. Written with intellectual passion, full of powerful insights & alluring legions of equations- you will enjoy even by immersion in its spirit. Read it slowly and in awe: witness great ideas grow & collapse. Not a textbook, proven wrong in not few points, this is a mutable & profound vision of reality by one of the last universal mathematicians. If you want profit- look elsewhere. ... Read more

4. The Theory Of Groups And Quantum Mechanics
by Hermann Weyl
Hardcover: 448 Pages (2008-06-13)
list price: US$51.95 -- used & new: US$35.08
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Asin: 1436687861
Average Customer Review: 4.5 out of 5 stars
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Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

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5-0 out of 5 stars one the the most important work in quantum mechanics
It's a very important book, written by the father of group theory application in physics (with Wigner and Pauli), and one of the best mathematician of 20th century, Hermann Weyl. Everyone who wants study a deeper view of quantum mechanics, in his intrinsic mathematical formulation, should read this work. After a firt brief introduction to quantum theory, he passes to explain the theory of rapresentation of groups, and its physical application, like the rotation group, or Lorentz group, and finally the theory of simmetry. It's a fundamental book for a good understandig of the importance of simmetries in modern physics. Without any doubt one the the most important work in quantum mechanics.

4-0 out of 5 stars Still a good book
Written in the early years of the quantum theory, the author of this book foresaw the importance of considering symmetry in physics, the use of which now pervades most of theoretical high energy physics. Indeed, with the advent of gauge theories, and their experimental validation, it is readily apparent that symmetry principles are here to stay, and are just not accidental curiosities. A reader of the book can still gain a lot from the perusal of this book, in spite of its date of publication and its somewhat antiquated notation. Older books also have the advantage of discussing the material more in-depth, and do not hesitate to use hand-waving geometrical pictures when appropriate. This approach results in greater insight into the subject, and when coupled with eventual mathematical rigor gives it a solid foundation. One example where the discussion is superior to modern texts is in the author's discussion of group characters and their application to irreducible representations and spectra in atomic systems.

The reader will no doubt probably want to couple the reading of this book with a more modern text so as to alleviate the notational oddities in this book. The author's presentation is clear enough though to make an appropriate translation to modern notation. The reader will then be well prepared to tackle more advanced material in mathematical and theoretical physics that make use of the group-theoretic constructions that take place in this book.

5-0 out of 5 stars A wonderful book
This is my favorite introduction to quantum mechanics. It is a difficult book, because it is succinct, though clear, and reflects Weyl's powerful intellect and original approach at every step. Each page is a challenge, but worth the effort.

5-0 out of 5 stars One of the two great classics on group theory in physics
The other one is Wigner's "Group Theory and Quantum Mechanics". As it is true of the other great books by Weyl, this is not an easy book, but it is, by all means, accessible. Don't try to read it in front of the TV set. Get pencil and paper, put yourself in a calm and contemplative mood and patiently read the words of the master. Hermann Weyl, one of the great minds of the 20th century, wrote this book with utmost care to make it self-contained. Sometimes you have to be deep in order to be brief, so the book requires some thought. But the main ideas are all there, and the connection of group theory with quantum mechanics has here its best treatment, in my humble opinion. But in less humble too: this was the only book concerning physics which Enrico Fermi read as a grown up. Once, Max Born had to write a synthetic exposition of Quantum Mechanics. After he finished it, he saw, for the first time, this book, and Weyl's synthesis of QM. He felt depressed by the superiority of Weyl's text. The book was originally written in German, but the translation is excellent, due to the great American cosmologist H. P. Robertson, of Robertson-Walker fame.

4-0 out of 5 stars Classic from the early days of quantum mechanics
Although published by Dover in 1984, this book dates back to about 1930, when Weyl was the big proponent of group theory in quantum mechanics. Because of this date, much of what modern books on group theory wouldinclude, is absent from the book. It mainly discusses the permutationgroup. The book is, however, of historic interest, as Weyl (mathematician)tried to convince the physicists to exploit group theory - which even gaverise to some irritation ("group pest"). ... Read more

5. The Continuum: A Critical Examination of the Foundation of Analysis
by Hermann Weyl
Paperback: 130 Pages (1994-03-14)
list price: US$8.95 -- used & new: US$6.09
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Asin: 0486679829
Average Customer Review: 4.5 out of 5 stars
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Concise classic consists of two chapters dealing with the conceptual problem posed by the continuum—the set of all real numbers. Chapter One deals with the logic and mathematics of set and function, while Chapter Two focuses on the concept of number and the continuum. Advanced-level mathematical landmark will interest anyone working in foundational analysis. Bibliography. Orig. pub. 1918.
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Customer Reviews (3)

3-0 out of 5 stars Not Weyl's finest writing
Space, Time, Matterand Symmetry are two true classics by this arthor.
This book translates badly and the subject has been covered better and more clearly by others like:The Principles of Mathematics.
The KAM theorem made Weyl's arguments obsolete
as it gave a better technology for the rational number continuum
approximations used in thesekinds of discussions. He never quite made his circulus vitiosusidea work.

5-0 out of 5 stars Correction
I feel obliged to place a 5-star review as compensation for the incompetent review by "GangstaLawya". He is a true amateur, the exact opposite of Hermann Weyl. "GangstaLawya" obviously couldn't understand the book and blames Weyl for this, trying to prevent other people from understanding it, how childish.
"GangstaLawya" claims in his "review" that Weyl didn't contribute to mathematics. Look for the "Weyl character formula" on the internet to see that he is wrong and thus incompetent.

5-0 out of 5 stars a fascinating detour
I first learned of this book from Eves and Newsom back in the early 1960's. It sounded fascinating but I couldn't read German. Now we're lucky to have it in Englsh translation with an introduction that relates Weyl's notation and terminology to the current one. (Or, if you're really out of date like me, you can use it in reverse to catch up on the modern field of foundations studies).
Precise statement is the essence of the study of the foundations of mathematics and what follows won't rise to that level but I hope it won't be seriously misleading either. In real life definitions are often circular; dictionaries define words in terms of otherwords, etc. Ordinarily this is not a problem but vicious circles can happen. In 1872 Dedekind published a definition of real numbers in terms of sets of rational numbers.
This fulfilled a long term dream of defining the reals without reference to geometric concepts. Encouraged by this Frege began his project of deriving all of mathematics from basic logical notions. He was largely successful but Russell found a contradiction within his system. It wasn't clear what caused this problem and Poincare suggested that it arose because Frege had allowed a certain kind of circular definition called 'impredicative'. While it was true that the contradiction could be eliminated by avoiding impredicative definitions, this solution was very drastic: it also barred Dedekind's defintion of the real. Most mathenaticians, including Whitehead and Russell, shrank from this step and proposed more moderate ways of fixing the foundations of mathematics. Working in the aftermath of World War I, Weyl was attracted to the more radical idea of trying to develop mathematics without using any impredicative definitions. He managed to derive some, but far from all, of analysis and the result was this book. Subsequently, Weyl was attracted to an even more radical critique of mathematical foundations proposed by Brouwer (you can read about this in Mancosu's great anthology "From Brouwer to Hilbert". At the same time Weyl remained passionately attached to mainstream mathematics. As far as I know, he never resolved his own conflicts about this. Naturally, anything by Weyl is brilliant and worth reading and this book is no exception. ... Read more

6. Hermann Weyl: 1885-1985: Centenary Lectures
Hardcover: 132 Pages (1986-11-01)
list price: US$98.00 -- used & new: US$84.96
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Asin: 3540168435
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7. Groups and Analysis: The Legacy of Hermann Weyl (London Mathematical Society Lecture Note Series)
Paperback: 354 Pages (2008-11-03)
list price: US$70.00 -- used & new: US$27.75
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Asin: 0521717884
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Many areas of mathematics were deeply influenced or even founded by Hermann Weyl, including geometric foundations of manifolds and physics, topological groups, Lie groups and representation theory, harmonic analysis and analytic number theory as well as foundations of mathematics. In this volume, leading experts present his lasting influence on current mathematics, often connecting Weyl's theorems with cutting edge research in dynamical systems, invariant theory, and partial differential equations. In a broad and accessible presentation, survey chapters describe the historical development of each area alongside up-to-the-minute results, focusing on the mathematical roots evident within Weyl's work. ... Read more

8. Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics
by Hermann Weyl
Hardcover: 272 Pages (2009-03-31)
list price: US$37.50 -- used & new: US$24.95
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Asin: 0691135452
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Hermann Weyl (1885-1955) was one of the twentieth century's most important mathematicians, as well as a seminal figure in the development of quantum physics and general relativity. He was also an eloquent writer with a lifelong interest in the philosophical implications of the startling new scientific developments with which he was so involved. Mind and Nature is a collection of Weyl's most important general writings on philosophy, mathematics, and physics, including pieces that have never before been published in any language or translated into English, or that have long been out of print. Complete with Peter Pesic's introduction, notes, and bibliography, these writings reveal an unjustly neglected dimension of a complex and fascinating thinker. In addition, the book includes more than twenty photographs of Weyl and his family and colleagues, many of which are previously unpublished.

Included here are Weyl's exposition of his important synthesis of electromagnetism and gravitation, which Einstein at first hailed as "a first-class stroke of genius"; two little-known letters by Weyl and Einstein from 1922 that give their contrasting views on the philosophical implications of modern physics; and an essay on time that contains Weyl's argument that the past is never completed and the present is not a point. Also included are two book-length series of lectures, The Open World (1932) and Mind and Nature (1934), each a masterly exposition of Weyl's views on a range of topics from modern physics and mathematics. Finally, four retrospective essays from Weyl's last decade give his final thoughts on the interrelations among mathematics, philosophy, and physics, intertwined with reflections on the course of his rich life.

... Read more

9. Hermann Weyl's Raum - Zeit - Materie and a General Introduction to his Scientific Work (Oberwolfach Seminars) (English and German Edition)
Paperback: 412 Pages (2001-03-01)
list price: US$69.95 -- used & new: US$46.48
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Asin: 3764364769
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Historical interest and studies of Weyl's role in the interplay between 20th-century mathematics, physics and philosophy have been increasing since the middle 1980s, triggered by different activities at the occasion of the centenary of his birth in 1985, and are far from being exhausted.

The present book takes Weyl's Raum - Zeit - Materie (Space - Time - Matter) as center of concentration and starting field for a broader look at his work.The contributions in the first part of this volume discuss Weyl's deep involvement in relativity, cosmology and matter theories between the classical unified field theories and quantum physics from the perspective of a creative mind struggling against theories of nature restricted by the view of classical determinism.

In the second part of this volume, a broad and detailed introduction is given to Weyl's work in the mathematical sciences in general and in philosophy. It covers the whole range of Weyl's mathematical and physical interests: real analysis, complex function theory and Riemann surfaces, elementary ergodic theory, foundations of mathematics, differential geometry, general relativity, Lie groups, quantum mechanics, and number theory. ... Read more

10. Mathematical Heritage of Hermann Weyl (Proceedings of Symposia in Pure Mathematics; v. 48)
 Paperback: 344 Pages (1988-08)
list price: US$45.00 -- used & new: US$49.04
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Asin: 0821814826
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Hermann Weyl was one of the most influential mathematicians of the twentieth century. Viewing mathematics as an organic whole rather than a collection of separate subjects, Weyl made profound contributions to a wide range of areas, including analysis, geometry, number theory, Lie groups, and mathematical physics, as well as the philosophy of science and of mathematics. The topics he chose to study, the lines of thought he initiated, and his general perspective on mathematics have proved remarkably fruitful and have formed the basis for some of the best of modern mathematical research.

This volume contains the proceedings of the AMS Symposium on the Mathematical Heritage of Hermann Weyl, held in May 1987 at Duke University. In addition to honoring Weyl's great accomplishments in mathematics, the symposium also sought to stimulate the younger generation of mathematicians by highlighting the cohesive nature of modern mathematics as seen from Weyl's ideas. The symposium assembled a brilliant array of speakers and covered a wide range of topics. All of the papers are expository and will appeal to a broad audience of mathematicians, theoretical physicists, and other scientists. ... Read more

11. Exact sciences and their philosophical foundations =: Exakte Wissenschaften und ihre philosophische Grundlegung : Vortrage des Internationalen Hermann-Weyl-Kongresses, Kiel 1985
 Unknown Binding: 552 Pages (1988)

Isbn: 382049328X
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12. Das Kontinuum and Anders Monographien (4 Volumes in 1) (AMS Chelsea Publishing)
by Hermann Weyl
Hardcover: 400 Pages (1972-01)
list price: US$20.00 -- used & new: US$20.00
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Asin: 0821840975
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The present volume is a reprint, under one cover, of four short books written by outstanding German mathematicians. The first book, The Continuum by Hermann Weyl, is the study of the continuum, particularly its role in the foundations of analysis. The second piece is Mathematical Analysis of Space Problems, also by Weyl. The two main topics are infinitesimal geometry and the use of group theory in the analysis of space problems. These are followed by a selection of twelve additional topics, such as Mobius transformations, computing Riemannian curvature, and elements of Lie theory. Next in the collection is Edmund Landau's short book Statement and Proof of Some New Results in Function Theory. In eight short chapters, Landau considers a variety of topics, such as Tauberian theorems, Hadamard's three circle theorem, and Koebe's distortion theorem for schlicht functions. In his preface, Landau writes that he has chosen the topics for their great elegance. The final piece of this volume is Bernard Riemann's ground-breaking lecture On the Hypotheses Which Lie at the Foundations of Geometry. In this one remarkable lecture, Riemann laid out the foundations of what became Riemannian geometry, particularly the notion of curvature for arbitrary manifolds. The story is told that, of the audience, only Gauss fully appreciated the depth of what Riemann was saying. ... Read more

13. Infinite Loop Spaces: Hermann Weyl Lectures, The Institute for Advanced Study. (AM-90) (Annals of Mathematics Studies)
by John Frank Adams
Paperback: 230 Pages (1978-09-01)
list price: US$50.00 -- used & new: US$47.07
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Asin: 0691082065
Average Customer Review: 5.0 out of 5 stars
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Customer Reviews (2)

5-0 out of 5 stars Rigorous, but very understandable
Although published in 1978, this book could be used as an introduction to the theory of operads and other recent work on homotopy theory and vertex operators. Vertex operators are not discussed in this book, but the theory elucidated herein is good background material for their study.

The author does a great job in motivating the subject in chapter 1. Loop spaces are function spaces of maps from the unit interval to a space with a chosen basepoint, with the property that each map sends 0 and 1 to the base point. The mathematician Jean Pierre Serre introduced the path space in order to study loop spaces, resulting in the famous Serre fibering. The nth homotopy group of the loop space can be shown to be equivalent to the (n+1)-th homotopy group of the original space. The homology of loop spaces can be calculated for some types of spaces, such as wedges of spheres. Infinite loop spaces are essentially sequences of spaces such that the nth element of this sequence is equivalent to the loop space of the (n+1)-th element. This sequence is also known as an "Omega-spectrum" and has the infinite loop space as its zeroth term. The name "spectrum" comes from general considerations involving sequences of spaces where the nth term is equivalent to the loop space of the (n+1)-th term; equivalently, where the suspension of the nth term is equal to the (n+1)-th term. The author reviews how a generalized cohomology theory yields an Omega-spectrum, giving two examples involving Eilenberg-Maclane spaces and complex and real K-theory. One can also start with a spectrum and construct a generalized homology and cohomology theory. Spectra and cohomology theory are thus essentially equivalent.

Chapter 2 is an overview of techniques needed to construct a category of spaces with enough structure so that the infinite loop space functor yields an equivalence from the category of spectra to the category of certain spaces. An example of the latter is given by the Stasheff A-infinity space, and its now ubiquitous property of having a product which is strictly associative. This property allows one to prove that a space is equivalent to a loop space if and only if the space is a Stasheff A-infinity space and that the zeroth homotopy of the space is a group. The Stasheff A-infinity spaces are also used to motivate the construction of 'operads'.

The next chapter the author is concerned with the concept of a space being like another one without being equivalent to it. He discusses the use of 'localization' in homotopy theory, an idea that is analogous to the one in algebra. The use of localization in homotopy theory is due to D. Sullivan, and involves use of the notion of a space being 'A-local', where A is a subring of the rationals. Remembering that a Z-module is A-local if it has the structure of an A-module, a space is A-local if its homotopy groups are A-local. Examples of the use of localization in constructing certain spaces are given. The author also discusses the use of the 'plus construction' that allows the alteration of fundamental groups without affecting the cohomology groups. Then after the construction of the Quillen higher algebraic K-theory groups in this regard, the author describes the relation between a topological monoid and the loop space of the classifying space of this monoid. This involves the notion of 'group completion', which is essentially an isomorphism between the homology of the path components of the monoid and the homology of the loop space of the classifying space of the monoid, but in the (infinite) direct limit.

Chapter 4 introduces the concept of a transfer map. A very elusive idea at first glance, the transfer map is motivated via the n-sheeted covering map of a space on another. The (singular) simplices of each then get matched up by the covering, and the transfer map between the spaces is then defined so that it is equal to the sum of the singular simplices of the covering space. It is in fact a chain map as shown by the author. The transfer maps are related to homotopy classes of the 'structure' maps of chapter 2, and the author gives a few examples of how they are used.

Chapter 5 is a quick overview of the Adams conjecture, which is essentially an assertion that the image of KO(X) in KF(X) can be characterized explicitly. Detailed proofs are omitted but references are given for the interested reader.

In chapter 6, the author restricts his attention to the K-theory of spectra. The treatment is concerned in large degree with the question of the existence of infinite loop map between infinite loop structures, and finding such a map, checking whether it is unique. This question is answered for particular types of spectra, via the Madsen, Snaith, and Tornehave theorem. Also, the Adams-Priddy theorem is proved, showing that one can construct on a space a unique infinite loop space structure. The reader gets more examples of the use of localization, in that some spaces can become equivalent as infinite loop spaces upon localization. The origin of K-theory in this chapter comes from the replacing of spectra that are not known by ones that are (namely the ones in classical K-theory). The author shows how the Madsen-Snaith-Tornehave theorem works in the context of both complex and real (periodic) K-theory. Detailed proofs are given for all of these results.

5-0 out of 5 stars A charming and readable introduction to infinite loop spaces
Reading this book made me excited about infinite loop spaces, which I had always imagined to be a very dry topic. Adams informal style reads as smoothly as a purely expository work, but gave me enough understanding andinsight to make me feel like I could fill in the details myself if I neededto. Even if this isn't always literally true, it certainly oriented me wellenough to be able to make sense of the literature. I especially liked thediscussion of A_infty spaces. ... Read more

14. The Classical Groups: Their Invariants and Representations
by Hermann Weyl
Paperback: 316 Pages (1997-10-13)
list price: US$67.50 -- used & new: US$39.99
(price subject to change: see help)
Asin: 0691057567
Average Customer Review: 4.5 out of 5 stars
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Product Description
In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. Analysis and topology are used wherever appropriate. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are of great importance in understanding the group-theoretic structure of quantum mechanics. ... Read more

Customer Reviews (3)

3-0 out of 5 stars maybe a liitle more difficult than it has to be?
His presentation of simplectic groups ( although short) was helpful.
That said having had Weyl's Space, Time, Matter for many years
I'm used to his notation, but here he is very intense in his presentation
and somewhat less than clear. As he is the founder of gauge group theory
one expects some mathematics, just not where it seems more difficult than necessary?
Also the, now, standard classifications of Cartan groups aren't mentioned ( although might be
because of the original publishing date of 1939?).
I got this book looking for the Weyl root groups used in making Cartan invariant matrices
which I couldn't find.
So on these points the book is a disappointment while still being a classic text.

5-0 out of 5 stars Hard-core Group Theory
Forget about that pansy abstract axiom approach. This is the WWF of group theory. Weyl will take anyone to the mat with this book. It is packed with detail and demonstrations.He follows the vector space/matrix representation approach common to digital systems, physics and chemistry rather than axiomatic, generators/permutations approach more common in Abstract Algebra courses. This is the lineage that develops matrix transforms as groups starting from "the full group of all non-singular linear transformations and .. the orthogonal groups" (p. vii). The latter chapters cover characters and invariants. Galois and field theory have been vanquished. Chapter 2, "Remembrance of things past" is very entertaining. My favorite quote, "Here there is only one man to mention - Hilbert. His papers (1890/92) mark a turning point in the history of invariants theory. He solves the main problems and thus almost kills the whole subject." It's funny because it's true.This is almost a botanical treatise in which the matrix groups are studied as specimens in the jungle -- "...after all each group stands in its own right and does not deserve to be looked upon merely as a subgroup of...Her All-embracing Majesty GL(n)." (p 136). Historic references throughout provide motivation and entertainment. You couldn't possibly be disappointed with this book.

5-0 out of 5 stars Great
Although this is a dated work, lacking some of the more modern language, it is still worth owning and reading.It is, after all, a designated "classic."And the material presented has been incorporatedwithin so many aspects of physics that one simply cannot avoid needing abook such as this.There are better books on the subject, for bothmathematicians and physicists, but this book still proves its worth. ... Read more

15. Theory of Groups & Quantum Mechanics 2ND Edition
by Hermann Weyl
 Hardcover: Pages (1931)

Asin: B003TN0YQA
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16. Theory of Groups & Quantum Mechanics
by Hermann Weyl
Paperback: Pages (1931)
-- used & new: US$10.00
(price subject to change: see help)
Asin: B001THPJ2C
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The Theory of Groups and Quantum Mechanics, Good Condition, 1931. ... Read more

17. Philosophy of Mathematics and Natural Science
by hermann weyl
 Paperback: Pages (1949)

Asin: B0012KJYAY
Average Customer Review: 4.0 out of 5 stars
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Customer Reviews (2)

3-0 out of 5 stars I didn't find it that interesting
Hermann Weyl was certainly an excellent mathematician, and the work he did on gauge theories (although they proved useless for uniting gravitational and electromagnetic forces) paid off much later.

However, I didn't get much insight from this book (I read the original version, from a library). He doesn't give a lot of detail about the mathematical or physical ideas, the book is really about the philosophy of these topics, with no discussion of the scientific or mathematical content at all. It's just not as inspiring as I would have expected.

5-0 out of 5 stars Please put this back in print!
Herman Weyl (1885-1955) has been described by Roger Penrose as the most influential mathematician who worked entirely in the 20th century. Many of his books are in print but not this one; Atheneum did a very small font reprint in the 1960's and even that is not available. Yet this book, which I have to read in a library copy, is a treasure. Weyl wrote it for a German-language encyclopedia in the 1920's and revised it in the late '40's for an English translation. It gives brief and clear accounts of Weyl's unique approaches to general relativity and quantum mechanics. He was way ahead of his time in recognizing the importance of symmetries and in taking a sophisticated geometric approach. He was particularly astute in noting important features that unified the different branches of physics (notably in the case of gauge invariance which he doesn't mention much here). His analysis of the problem of the ether is particularly lucid and stimulating as his discussion of the relationship between kinematics and dynamics. He also gives an account of the foundations of mathematics up to the discoveries of Godel. Weyl regarded physics and mathematics as essentially one, both being parts of the study of nature and thought that the same philosophical principles underlay both. He was an unusually openminded scholar who was quick to abandon an innovative but less successful approach for a better one. I don't think he would have fit in with a popular late 20th century style of doing physics that scorned philosophy, and insisted that the current generation was saying the last word on basic physical theory. Maybe that's why this beautiful, intelligent and very readable book was allowed to fall out of print. Its time to bring it back! ... Read more

18. Raum, Zeit, Materie: Vorlesungen über allgemeine Relativitätstheorie (German Edition)
by Hermann Weyl
 Hardcover: 349 Pages (1993-10-25)
list price: US$55.95
Isbn: 3540569782
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Aus dem Vorwort von Jürgen Ehlers zur 7. Auflage: "Die ... Entwicklung der Physik macht verständlich, warum ein so "altes" Werk wie Raum, Zeit, Materie noch aktuell ist: Die Riemann-Einsteinsche Raumzeitstruktur, die von Weyl so meisterhaft beschrieben und aus ihren mathematischen und physikalischen Wurzeln hervorwachsend dargestellt wird, ist immer noch die physikalisch umfassendste und erfolgreichste Raumzeittheorie, die bisher entwickelt und mit der Erfahrung konfrontiert wurde. (...) Als erstes Lehrbuch der noch neuen Theorie setzt es sich gründlicher als spätere Bücher mit den historischen Wurzeln und den sachlichen Motiven auseinander, die zur Einführung der damals neuen Begriffe wie Zusammenhang und Krümmung in die Physik geführt haben. Zweitens ist es von dem vielleicht letzten Universalisten geschrieben worden, der alle wesentlichen Entwicklungen der Mathematik und Physik seiner Zeit nicht nur überblickte, sondern in wesentlichen Teilen mitgestaltete. Das Studium dieses Werkes vermittelt nicht nur die Grundzüge der beiden Relativitätstheorien, sondern zeigt Zusammenhänge mit anderen Ideen, nicht zuletzt auch der Naturphilosophie auf." ... Read more

19. Temps, espace, matière: leçons sur la théorie de la relativité générale (French Edition)
by Hermann Weyl
Paperback: 312 Pages (2010-05-18)
list price: US$29.75 -- used & new: US$17.57
(price subject to change: see help)
Asin: 1149550724
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Product Description
This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

20. Riemanns geometrische Ideen, ihre Auswirkung und ihre Verknupfung mit der Gruppentheorie (German Edition)
by Hermann Weyl
 Unknown Binding: 46 Pages (1988)

Isbn: 0387194851
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