Books you would like to read (if somebody would just write them...) - MathOverflow

Books you would like to read (if somebody would just write them...)

2011-01-24T10:23:39Z

I think that the title is self-explanatory but I'm thinking about mathematical subjects that have not received a full treatment in book form or if they have, they could benefit from a different approach. (I do hope this is not inappropriate for MO).

Let me start with some books I would like to read (again with self-explanatory titles)

1) The Weil conjectures for dummies

2) 2-categories for the working mathematician

3) Representations of groups: Linear and permutation representations made side by side

4) The Burnside ring

5) A functor of points approach to algebraic geometry

6) Profinite groups: An approach through examples

Any other suggestions ?

Answer by Dylan Wilson for Books you would like to read (if somebody would just write them...)

2011-01-24T10:33:32Z

Algebraic Geometry from a Homotopical Viewpoint: For the topologist who really wants to like geometry but doesn't know where to start.

Answer by Lorenzo for Books you would like to read (if somebody would just write them...)

2011-01-24T10:41:15Z

I would like to read an SGA-like book on Étale cohomology to replace as a reference SGA 4½. I also have an idea about who could write such a text: Luc Illusie. I'd really love that.

Answer by Will Merry for Books you would like to read (if somebody would just write them...)

2011-01-24T11:00:38Z

I would have killed for this a couple of years ago: a big book on Floer homology, written to be understandable for graduate students. Includes all the analytical details.

Answer by Dirk Basson for Books you would like to read (if somebody would just write them...)

2011-01-24T11:36:09Z

Galois representations.

I know about Serre's Abelian $\ell$-adic representations and elliptic curves, but I am sure that a more general theory has been established since then. There are a few people who have notes on Galois representations on their web pages, but no book that I know of.

Answer by darij grinberg for Books you would like to read (if somebody would just write them...)

2011-01-24T11:56:39Z

You forestalled some of what I would have posted...

Quillen's K-theory without topology
Steenrod algebras through combinatorics and representation theory (as opposed to, through topology)
Ext and Tor defined constructively, with Haskell code
Weyl's "Classical Groups" with the proofs of 1938 but the notations of 2010
The definitive guide to Hochschild homology
Henri Lombardi's "Algèbre Commutative" in English
A documentation to Agda

Answer by David Hansen for Books you would like to read (if somebody would just write them...)

2011-01-24T12:17:34Z

"The proof of the Shimura-Taniyama conjecture, for people who aren't professional algebraists but are willing to try pretty hard."

Answer by John D. Cook for Books you would like to read (if somebody would just write them...)

2011-01-24T12:39:16Z

Three views of differential geometry

I have in mind the most rigorous modern view, the most intuitive undergraduate calculus view, and the physicist's tensor calculus view. These perspectives can be so different that it's hard to keep in mind that they're all ultimately concerned with the same thing.

Take one concept at a time examine it from a rigorous, intuitive, and computational viewpoint. For example, take a gradient and define it as a differential form, as a vector perpendicular to a surface, and as a tensor. Or here's how a differential geometer, a calculus student, and a physicist all view integrating over a surface. Here's how they each view Stokes' theorem etc.

Answer by Johannes Ebert for Books you would like to read (if somebody would just write them...)

2011-01-24T14:07:40Z

"(Counter)examples in Algebraic Topology"

There are many good textbooks in homology and elementary homotopy theory, but the supply of instructive examples they offer is usually appallingly small (spheres and projective spaces are the standard examples, but often there is little beyond). One reason is that to discuss interesting examples, one needs a lot of machinery, whose development consumes time and space. The books by Hatcher or Bredon offer a lot of examples; and I also like Neil Stricklands bestiary:

http://neil-strickland.staff.shef.ac.uk/courses/bestiary/bestiary.pdf,

and together with the unwritten chapter "things left to do", it is pretty close to what I would love to see as a book.

Answer by Chandan Singh Dalawat for Books you would like to read (if somebody would just write them...)

2011-01-24T14:11:49Z

The construction of galoisian representations associated to primitive cuspidal eigenforms. I hope the user BCnrd gets the hint.

Answer by John D. Cook for Books you would like to read (if somebody would just write them...)

2011-01-24T14:26:25Z

Categories for the Working Mathematician

I know Saunders Mac Lane already wrote a book by that name, but in my opinion his book doesn't live up to its title. His book would perhaps be better named "Category theory for the working algebraist." I'd like to see a book with more examples, especially examples outside of algebra and algebraic topology.

Answer by gowers for Books you would like to read (if somebody would just write them...)

2011-01-24T14:55:07Z

I don't know for certain that this doesn't exist, so I'm in a no-lose situation: if this is a rubbish answer then it means that a book that I want to exist does exist. Many mathematicians of a pure bent have taken it upon themselves to get a good understanding of theoretical physics. And many have actually managed this. But it seems to me that they usually go native in the process, with the result that I cease to be able to understand what they are saying. It could be that this is just an irreducibly necessary feature of physics, but I doubt it. Out there in book space I believe there exists a book that explains theoretical physics in a way that physicists would dislike intensely but mathematicians would find much easier to read. It may well be that if you want to do serious work in mathematical physics then you have to understand the subject as physicists do. However, this book would be aimed at pure mathematicians who were not necessarily intending to do serious work in mathematical physics but just wanted to understand what was going on from a distance.

I used to have a similar view about explanations of forcing, but I think Timothy Chow's wonderful Forcing for Dummies has filled that gap now.

Answer by Yann Palu for Books you would like to read (if somebody would just write them...)

2011-01-24T15:14:22Z

"Quiver varieties with a wealth of examples" ?

Answer by Dmitri Pavlov for Books you would like to read (if somebody would just write them...)

2011-01-24T17:21:55Z

Remark: Several items below refer to the formalism of locales. Although consistent usage of the language of locales allows one to get rid of the axiom of choice in almost all cases, my main reasons for it are purely pragmatic: The formalism of locales allows one to obtain equivariant and family versions of many theorems without any additional effort, as opposed to the formalism of topological spaces (think of Hahn-Banach theorem, for example).

A general topology textbook written in the language of locales, with no mention of topological spaces.
Textbooks on commutative algebra and algebraic topology written in the language of locales. In particular, such textbooks can usually avoid mentioning maximal ideals, the axiom of choice, or Zorn's lemma.
A measure theory textbook written in the language of locales and commutative von Neumann algebras, with no mention of the set-theoretical approach. The textbook should also have a conceptual exposition of L^p-spaces.
A linear algebra textbook that does not mention coordinates, bases, or matrices.
A textbook on smooth manifolds that never mentions coordinates, charts, or atlases. Such a textbook should have a conceptual exposition of integration and use supermanifolds consistently whenever it makes sense, e.g., for differential forms.
Textbooks on algebraic topology and homological algebra written in the language of (∞,1)-categories.
Higher categories for the working mathematician. This book should contain a lot of examples of higher categories that are actually used in mathematics outside of category theory. (For example, the bicategory of algebras, bimodules, and intertwiners, the tricategory of conformal nets, defects, sectors, and morphisms of sectors etc.)
A textbook on topological vector spaces (in particular, on locally convex, Banach, and nuclear spaces) written from the categorical viewpoint. For example, such a textbook would define a nuclear morphism as a morphism that can be factorized in a certain way (see a recent paper by Stephan Stolz and Peter Teichner). The textbook should consistently use the language of locales. For example, this allows one to prove Hahn-Banach, Gelfand-Neumark, or Banach-Alaoglu theorems without using the axiom of choice.

Answer by John D. Cook for Books you would like to read (if somebody would just write them...)

2011-01-24T19:25:27Z

Whittaker and Watson with a Facelift

There are a number of classic books, such as Whittaker and Watson's Modern Analysis, that I'd like to see typeset in TeX and updated slightly. Sometimes notation or terminology have changed and a little footnote would help greatly.

Also by Watson, I'd like to see his 1922 book "A Treatise on the Theory of Bessel Functions" with updated typography and notation. A scan of the book is available here. Apparently the book has entered the public domain and so there would be no legal barrier to producing an updated version.

Answer by Yemon Choi for Books you would like to read (if somebody would just write them...)

2011-01-24T19:36:45Z

Somewhat frivolous/exasperated suggestion:

The Homology of Banach and Topological Algebras, Vol. II: Collected folklore and missing bookwork.

I only suggest this because I have been needing to cite this book, on and off, for much of the last five years, and the fact it's not been written hasn't really helped.

Answer by Thomas Riepe for Books you would like to read (if somebody would just write them...)

2011-01-24T22:25:29Z

"Faltings explained" : Several of his articles are very hard to read and existing surveys on his concepts don't really fill the gap. I would like to read a book about his work, his themes, background ideas and techniques which is a readable walk through all that, something like Connes' "NCG"-book + Connes/Marcolli's "noncommutative garden".
"Morava explained" : The same as above on Morava's work, containing a (for the arithmetic geometry inclined reader) readable description of the homotopy theory background. With comments from Manin, Kontsevich and Connes, and a (sci-fi ?) chapter on how homotopy theory and number theory may mutually interfuse (e.g. through "brave new rings").
Mumford suggested in a letter to Grothendieck to publish a suitable edited selection of letters by Grothendieck to his friends, because the letters he received from him were "by far the most important things which explained your ideas and insights ... vivid and unencumbered by the customary style of formal french publications ... express(ing) succintly the essential ideas and motivations and often giv(ing) quite complete ideas about how to overcome the main technical problems ... a clear alternative (to the existing texts) for students who wish to gain access rapidly to the core of your ideas". (Found in the very beautifull 2nd collection)

Answer by Ariyan Javanpeykar for Books you would like to read (if somebody would just write them...)

2011-01-25T08:15:46Z

There are precisely two books on Arakelov geometry. One by Lang and one by Soule. I would love to see a book written on the subject which focuses mainly on the two dimensional (and one-dimensional) case. Sections 8.3 and 9.1 of Liu's book do this greatly for example (but considers only intersection multiplicities at the finite points). It should include all the theorems done so far. Something like

Chapter 0. Prerequisites

Chapter 1. Arithmetic curves (Riemann-Roch, slopes method, etc. One should include a paragraph or appendix on algebraic curves stating all the theorems that can and have been generalized.)

(N.B. An arithmetic curve is the spec of a ring of integers.)

Chapter 2. Arithmetic surfaces (This would contain all the "arithmetic" analogues of the theorems mentioned in the Appendix. For example, there has been a lot of work on Riemann-Roch theorems, trace formulas, Dirichlet's higher-dimensional unit theorem, Bogomolov inequalities, etc. Also, there are four intersection theories (which are compatible) I know of at the moment. The one developed by Arakelov-Faltings, then Gillet-Soule, then Bost and then Kuhn. The book should include a detailed description of them.

Appendix A. Algebraic surfaces. (A survey of all the classical theorems for algebraic surfaces that have an analogue in Arakelov geometry. This includes Faltings' generalizations of the Riemann-Roch theorem, Noether theorem, etc. but also the theorems generalized to Arakelov theory by Gasbarri, Tang, Rossler, Kuhn, Moriwaki, Bost, etc.)

Appendix B. Riemann surfaces (Just the necessary. Differential forms and Green functions basically.)

Answer by Gunnar Magnusson for Books you would like to read (if somebody would just write them...)

2011-01-25T09:16:14Z

"Examples in complex geometry."

The algebraic and differential geometry and Hodge theory side of complex geometry is well established in many books, but I've had real trouble finding examples that are worked out in detail (which would be perfect as exercises, perhaps if given with hints) that show how the theory works in practise and provide counterexamples to some implications. For example, an ample line bundle does not have to admit any global sections, but I've never seen an example of such a bundle given in a textbook.

Answer by Mikhail Bondarko for Books you would like to read (if somebody would just write them...)

2011-01-26T20:25:21Z

My answer is quite simple and stupid. I don't know French; so I would like to read EGA, SGA, and BBD in English (or in Russian:)). I also suspect that these books could be updated in the process of translation.:)

Answer by Chandan Singh Dalawat for Books you would like to read (if somebody would just write them...)

2011-01-27T04:17:52Z

I'm surprised that nobody has expressed the desire to read Bourbaki's Théorie des nombres.

Answer by Chandan Singh Dalawat for Books you would like to read (if somebody would just write them...)

2011-01-27T04:19:52Z

As I have been telling many people involved in mathematical publishing, the one book I would like to read is The Serre-Tate correspondence.

Answer by Seamus for Books you would like to read (if somebody would just write them...)

2011-01-27T13:35:36Z

An English translation of Curtis and Reiner, Methods of representation theory with applications to finite groups and orders would be nice.

Answer by Jim Humphreys for Books you would like to read (if somebody would just write them...)

2011-01-30T18:29:48Z

The Springer Correspondence

Tonny Springer developed a subtle correspondence between Weyl group representations (say over $\mathbb{C}$ ) and nilpotent orbits of the related semisimple Lie algebra, showing in particular how to realize the finite group representations in the top cohomology of fibers in his special desingularization of the nilpotent variety. By now the ideas involved have permeated much of the work in Lie theory due to Lusztig and many other people. But there is no systematic treatise on the subject and its connections with other areas of Lie theory, algebraic geometry, combinatorics. In my 1995 book Conjugacy Classes in Semisimple Algebraic Groups I included toward the end a very short survey of Springer theory, following a treatment of the unipotent and nilpotent varieties. But I realized at the time that I didn't understand the subject deeply enough to write a comprehensive account. (I still don't.)

My first exposure to Springer's ideas unfortunately didn't take hold right away. I recall making a short visit to Utrecht around 1975, where I had lunch with Springer at an Indonesian restaurant and he jotted down the new ideas he was excited about. No napkin or other scrap of paper survives, but anyway I understood only later how amazing his insights were. They deserve a thorough treatment in book form.

Answer by Qfwfq for Books you would like to read (if somebody would just write them...)

2011-01-31T13:34:33Z

Counterexamples in scheme theory

Answer by Holzinger Raphael for Books you would like to read (if somebody would just write them...)

2011-01-31T13:56:45Z

C.P. Snow once used such persuasion as he had to get G.H.Hardy to write another book, which Hardy promised him to do. It was to be called 'A Day at the Oval' and was to consist of himself watching cricket for a whole day, spreading himself in disquisitions on the game, human nature, his reminiscences, life in general. Unfortunately Hardy's final years of his life were not of delight and the book, though destined to be an eccentric minor classic was never written.

I would love to see such a book, written with incomparable style and mathematical touch.

Answer by Daniel Moskovich for Books you would like to read (if somebody would just write them...)

2011-01-31T14:59:33Z

Spaces of Diffeomorphisms
For 60+ years this has been a foundation of differential topology, featuring prominently in work of Smale, Cerf, Hatcher, Thurston, and many others; but I don't know any adequate reference. Indeed, it seems only a handful of brilliant people know this stuff, and everyone else uses their work as if it were a collection of black boxes.
My dream book would include, among other things, a modern introduction to Cerf theory from the perspective of Igusa's theory of framed functions, leading up to a readable and self-contained proof of Kirby's Theorem. It would also contain exposition and simplification of theorems of Hatcher, Cerf, Kirby, and Seibenmann.
This is a cheerful prod to a certain prospective author of such a book, that when it is written it will surely become an instant classic; I, for one, will pre-order.

Answer by Mark Schwarzmann for Books you would like to read (if somebody would just write them...)

2011-02-02T02:20:17Z

Algebraic groups by example

There are currently several books on Lie theory which take a very concrete approach, containing many examples (e.g. Rossmann, Hall, Stillwell). Basically they can be read by a student with some knowledge in calculus, linear algebra and perhaps some mathematical maturity. However, I have yet to find a book on the theory of (linear) algebraic groups which doesn't delve into topics from commutative algebra and algebraic geometry before even defining what an algebraic group is, and even then, most texts take a very abstract approach - most proofs seem like general nonsense to me, but maybe that's just because I'm not an algebraist in heart. In any case, I would very like to see a book on the subject which takes a very concrete approach through examples and constructive proofs.

Answer by Andres Caicedo for Books you would like to read (if somebody would just write them...)

2011-02-02T03:53:49Z

AD${}^+$ by Hugh Woodin.

Answer by Marko Amnell for Books you would like to read (if somebody would just write them...)

2011-02-02T08:57:37Z

I would like to read a comprehensive, step-by-step introduction to the Langlands Programme written for non-experts. An Introduction to the Langlands Program (edited by Joseph Bernstein and Stephen Gelbart) is good, but it is a collection of articles, not a textbook or monograph. Stephen Gelbart's "An Elementary Introduction to the Langlands Program" (Bulletin of the AMS, Vol. 10, No. 2, 1984, pp. 177-219) has the right approach, but while quite long, is not a book-length treatment. David Nadler's excellent new article "The Geometric Nature of the Fundamental Lemma" is another example of the sort of expository approach I would like to see in a full-length book about the Langlands Programme.

Answer by Anonymous for Books you would like to read (if somebody would just write them...)

2011-02-02T10:26:59Z

I, as an undergraduate student in physics, would really like a comprehensive solutions book for Roger Penrose's The Road to Reality: a complete guide to the laws of the universe (Vintage, 2004)

Answer by Andreas Holmstrom for Books you would like to read (if somebody would just write them...)

2011-02-05T02:59:39Z

Introduction to algebraic cycles.

With lots of examples...

Answer by Colin Tan for Books you would like to read (if somebody would just write them...)

2011-03-06T07:13:08Z

In pursuit of Hilbert's Problems

I think Hilbert's 23 problems form an organizatory framework for mathematics, that is much more organic than say the AMS classification. I believe that a book that traces the mathematics that grew from these problems can help to organize the burgeoning state mathematics is currently in.

I'm aware there is a book called "The Honours Class" that gives a history of Hilbert's problems up to their solution. However, this book is more biography than mathematics. Also, I'm interested in what happens after the problem is solved. A case study is the 17th problem, which lead to much of real algebra and real algebraic geometry today.

Answer by Zeeshan Mahmud for Books you would like to read (if somebody would just write them...)

2011-10-25T20:02:26Z

For a popular account an autobiographical Six Million Dollar Man: How I solved all six of the millennium problems in 1 year by anonymous author would definitely top my shelf.

On a bit more serious note, I am looking forward to...

Continuum Hypothesis Part I and II with a chapter headed The Art of Forcing
Five Pillars of Mahtmeatical Logic (an encyclopedia in the same vein as the Russian EOM with 8000 entries from Logic only)
On formalizing predicative notion: From zero to Γ0 in 2 seconds...
Alan Turing's unpublished papers
Ω: Absolute Infinity (perhaps this being sequel to Heller and Woodin edited Infinity)

Answer by Todd Eisworth for Books you would like to read (if somebody would just write them...)

2011-10-25T20:06:22Z

"Cardinal Arithmetic: The New Corrected Edition (including index)" by Saharon Shelah...

Answer by nabil for Books you would like to read (if somebody would just write them...)

2012-12-13T15:42:48Z

Algebraic topology demystified Differential topology demystified Algebraic geometry demystified Differential geometry demystified D-branes demystified