Greatest Books in Mathematics?

[ExploreScience]

What are your favorite books on mathematics?

 

Are there any great autobiographies? What about books on the application of mathematics to various fields?

[ajb]

One great book is Foundations of Mechanics by Ralph Abraham and Jerrold E. Marsden. (Marsden died last year)

 

The first edition from 1967 seems years ahead of its time. I refer to this book quite often.

[imatfaal]

Fermat's Last Theorem by Simon Singh is a great popular science read.

[DrRocket]

One great book is Foundations of Mechanics by Ralph Abraham and Jerrold E. Marsden. (Marsden died last year)

 

The first edition from 1967 seems years ahead of its time. I refer to this book quite often.

 

Damn. I had not heard that he died.

 

The latest edition of that book is available from the American Mathematical Society.

 

Any of Walter Rudin's books are superb(in order of difficulty and sophistication):

 

Principles of Mathematical Analysis

 

Real and Complex Analysis

 

Functional Analysis

 

Fourier Analysis on Groups

Also

 

Analysis, Manifolds and Physics (including Part II, a second volume) by Yvonne Choquet-Bruhat and Cecile DeWitt-Morette

[ajb]

Damn. I had not heard that he died.

 

Have a look at his webpage.

[DrRocket]

Have a look at his webpage.

 

 

Thanks. I had not visited that page in a long time.

 

Now that I know more of your background it occurs to me that the book Quantum Mechanics, A Functional Integral Point of View by Glimm and Jaffe might be of interest.

[ajb]

Now that I know more of your background it occurs to me that the book Quantum Mechanics, A Functional Integral Point of View by Glimm and Jaffe might be of interest.

 

This is an attempt at a rigorous definition of the path integral? I am not familiar with the books, but I have come across the authors before.

[sound]

What are your favorite books on mathematics?

 

Are there any great autobiographies? What about books on the application of mathematics to various fields?

the one i had in grade 9 was a good one i think the depth of the book was really had good

[the tree]

A great all round book is The Pleasures Of Counting which covers an awful lot of applied mathematics, not in much depth with enough to give you a feel for what is going on and lots of historical context. I think beyond that the question is really dependent on what area of mathematics interests you the most, since there is a hell of a lot of literature to chose from.

the one i had in grade 9 was a good one i think the depth of the book was really had good

Can you please, please, please take a second to consider what the point of this post was? Ignoring the grammar and whatnot, you haven't actually named a book.

[DrRocket]

This is an attempt at a rigorous definition of the path integral? I am not familiar with the books, but I have come across the authors before.

 

I don't think anyone has managed that. It is however an attempt to do quantum field theory rigorously and looks at the subject from the perspective of a mathematician.

 

I am FAR from expert in the subject. I am basically struggling to figure it out and books by physicists confuse me quickly. In Zee's book by page 13 he is explicitly evaluating integrals that I can prove do not exist (although I can certainly get them as a principle value of a singular integral, but not by Zee's hand-waving methods). So I am (very) slowly reading this book.

 

This may be up your alley, and I think you might like the Glimm, Jaffe book. I am not at all surprised that their names are familiar to you. I would be surprised if they were not. If you do look at the book, I would be very interested in your thoughts on it.

Edited February 21, 2011 by DrRocket

[ajb]

It is not uncommon in QFT to define perturbation expansions formally. In effect you don't worry about an infinite number of terms (may or may not be that many) or their convergence properties.

 

Then you have renormalisation effects, again people tend to be very slack with the convergence issues.

 

So, mathematically a lot of QFT (I think you can get rigorous results in 2 dimensions) is quite sick. However, the framework allows construction of theories that agree with nature to some stupidly high degree of accuracy.

 

I will, when I get chance have a look at the book by Glimm and Jaffe. Reading the description of the book via Amazon, they concentrate on two dimensions and give lots of existence proofs. QFT in low dimensions has lots of nice mathematical properties that make it worth studying as a mathematician. As a physicist, one would like to extend these results to 4d (or maybe 10d ).

 

John C. Baez, Irving E. Segal and Zhengfang Zhou have written a book Introduction to Algebraic and Constructive Quantum Field Theory. It is out of print, but an pdf version is available here. This book is also on my list of books to read at some point.

 

 

Algebraic field theory is a relative of constructive field theory that uses c*-algebras and von Neumann algebras to describe the algebra of (local) observables. The approach is heavy on functional analysis and seems quite removed from the more standard path integral approach. To my knowledge, only a few important results have really been established here. For example, an understanding of superselection rules and the possibility of QFT on non-globally hyperbolic space-times. I don't believe that a proper analysis of any realistic QFT has been achieved in this framework. (Scalar field theory I think has)

 

THE book on algebraic field theory is Haag, Local Quantum Physics: Fields, Particles, Algebras. This I have read, but need to do so again and probably again as it will take a while to really absorb the material. Doing research in this area would require a lot more effort than just reading this book! (The same as reading Hawking's Brief History does not make you a cosmologist!)

Edited February 21, 2011 by ajb

[ajb]

Another book I find very useful is Concise Encyclopedia of Supersymmetry: And Noncommutative Structures in Mathematics and Physics, edited by S. Duplij, Springer (31 Dec 2003).

 

It contains hundreds of short articles by experts in the field. Topics include quantum field theory, supersymmetry, supergeometry, supergravity, M-theory, noncommutative geometry, representation theory, categories and quantum groups.

 

 

 

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[khaled]

I read free books on my iPod, using iBooks ...

 

1. Thomas Hariot the mathematician, the philosopher, and the scholar

2. Amusements in Mathematics

3. Logic