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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 ?

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I really like this question... hopefully someone will take a hint and write number (5) and (2) sometime soon! – Dylan Wilson Jan 24 '11 at 10:30
Steve Lack wrote something approximating (2): – Tom Leinster Jan 24 '11 at 11:31
Regarding the Weil conjectures, have you read the appendix to Hartshorne that discusses these? If so, you could also try Nick Katz's exposition on Deligne's work in the Hilbert's Problems book (in the Proceedings of Symposia in Pure Math series) from the 1970s. Also, Deligne's article Weil I is less technical than you might guess, and there is also the textbook by Freitag and Kiehl. – Emerton Jan 24 '11 at 12:44
Qiaochu: Demazure and Gabriel wrote a book using the functor of points approach over 3 decades ago. Some people love this book, while others... – Donu Arapura Jan 24 '11 at 17:55
Maybe there is a place for the dual question: "Books you would like to write (if somebody would just read them)" so people can mention their book ideas and get some feedback. – Gil Kalai Feb 1 '11 at 15:03

37 Answers 37

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

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Dear David, This exists in textbook form: as I noted in another comment, there is the book Modular forms and Fermat's Last Theorem (Cornell, Silverman, Stevens eds.). – Emerton Jan 24 '11 at 12:46
There are also the DDT notes, now available on Darmon's website, along with other related material. – Chandan Singh Dalawat Jan 24 '11 at 14:16
Emerton, I have this book but unfortunately haven't had the time to dive into it yet. My impression (horribly mistaken?) was that the last fifteen years have seen some simplifications and improvements to the proof - e.g. appeal to base change to avoid level lowering, appeal to Jacquet-Langlands to study the Hecke algebras in a more hands-on way, the Diamond-Fujiwara version of patching and concomitant avoidance of appeal to multiplicity one, etc. – David Hansen Jan 25 '11 at 16:53
Dear David, Yes, but these improvements are amply documented in the research literature; I don't see the need for another text at the moment, given the existence of Cornell, Silverman, and Stevens. After all, the paper of Diamond in Inventiones is well-written, so if one understands everything in Cornell, Silverman, and Stevens except the mult. one statements, it is no trouble to modify things so as to incorporate the results of Diamond's article. As for replacing the geometric arguments for level lowering by base change, this is very powerful in those contexts where one doesn't have ... – Emerton Jan 27 '11 at 4:33
... the same tight control of the geometry as one has in the context of modular curves, but it's a matter of one's predilections as to whether it counts as a simplification. (This comment just reflects my own training, which finds Ribet's arithmetic geometry arguments quite a bit easier to follow than the proof of base change.) I think that, with the sole exception of Diamond's paper, which really does count as an unambiguous simplification, these other approaches to the argument just reflect modifications of technique in order to ... – Emerton Jan 27 '11 at 4:39

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

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An English translation? – darij grinberg Jan 27 '11 at 14:25
I think Seamus was ironic ;-) – Julien Puydt Feb 2 '11 at 12:30
That was the easy part. But what's the joke about? – darij grinberg Feb 3 '11 at 17:02

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...

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

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Inter-Universal Geometry for dummies.

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"What is inter-universal geometry?." Might be premature for a book-length exposition, but ... – Joseph O'Rourke Oct 9 '15 at 21:48

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)

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There is a forum for this: But probably you know that – darij grinberg Feb 2 '11 at 11:02

    Mathematical theories deconstructed
A compendium of dependencies by (and for) generalizers and formalizers

We know these theorems are proved. Now we want to know, which precisely pieces of foundations do they use. Does any of these inequalities … require ℝ, or an arbitrary ordered field is sufficient? For which an ordered commutative ring is enough? To which algebras/rings/manifolds can we generalize an analytic function … (although there are no power series)? Does the theory … effectively use the set theory, or it feels well with first-order logic? And with which namely? How different definitions of real numbers affect accessibility of theorems in analysis? What remains provable in topology without the law of the excluded middle? Which namely can be wrong in a theory of … for the statement … to become broken? On which exactly theories relies the best known proof of … theorem? And, generally, what is the mathematical truth?

Mathematics is a huge network of interdependencies but (in literary form) theories are ordered from postulates to conclusions, except for this book. Several nuances in definitions, not expressed before, are explored. And many unsolved problems “can we prove a well-know theorem … without assuming (allegedly true) statement …” are also listed.

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The area of math you are describing is called reverse mathematics. There are several texts on the topic, though perhaps not about the results you are most interested in. – Zach Hamaker Dec 31 '15 at 21:25

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