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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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    $\begingroup$ I really like this question... hopefully someone will take a hint and write number (5) and (2) sometime soon! $\endgroup$
    – Dylan Wilson
    Commented Jan 24, 2011 at 10:30
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    $\begingroup$ 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. $\endgroup$
    – Emerton
    Commented Jan 24, 2011 at 12:44
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    $\begingroup$ Qiaochu: Demazure and Gabriel wrote a book using the functor of points approach over 3 decades ago. Some people love this book, while others... $\endgroup$
    – Donu Arapura
    Commented Jan 24, 2011 at 17:55
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    $\begingroup$ It strikes me that many of the knowledgeable participants who made wonderfully detailed suggestions for a book on a coherent topic from a particular viewpoint, are well-positioned to write the very book they wish to read! $\endgroup$
    – Joseph O'Rourke
    Commented Feb 1, 2011 at 0:07
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    $\begingroup$ 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. $\endgroup$
    – Gil Kalai
    Commented Feb 1, 2011 at 15:03

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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.

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    $\begingroup$ Michael Spivak has recently written a book called "Physics for Mathematicians: Mechanics I". I haven't seen it and it's a bit expensive on Amazon, but it might be just what you want (but as far as I can tell it's "only" about classical mechanics...) $\endgroup$
    – Gonçalo Marques
    Commented Jan 24, 2011 at 15:07
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    $\begingroup$ "Physics for Mathematicians: Mechanics I" is apparently a reworked and expanded version of these notes: math.uga.edu/~shifrin/Spivak_physics.pdf. Now that I know about it, I'm really looking forward to reading it!!! +1 $\endgroup$
    – Vectornaut
    Commented Jan 24, 2011 at 15:36
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    $\begingroup$ +lots. Physics books are usually written in a way that teaches the mathematics through physical intuition... The trouble is that I have no physical intuition. I'd like a book that teaches the physics through mathematical intuition. $\endgroup$
    – Dylan Wilson
    Commented Jan 24, 2011 at 16:32
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    $\begingroup$ Have you read Vladimir Arnold's "Mathematical Methods in Classical Mechanics"? I would say that it fits the bill, but maybe you've read it and it falls short in some way. $\endgroup$
    – arsmath
    Commented Jan 24, 2011 at 16:41
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    $\begingroup$ I really like Folland's book "Quantum field theory, A tourist guide for mathematicians". $\endgroup$
    – Rob Harron
    Commented Feb 2, 2011 at 2:43
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Counterexamples in scheme theory

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    $\begingroup$ Maybe the examples chapter stacks.math.columbia.edu/tag/026Z in the Stacks project can be seen as an approximation of the book you would like to read. $\endgroup$
    – pbelmans
    Commented Jun 19, 2019 at 17:47
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  • "(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:

https://strickland1.org/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.

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    $\begingroup$ Before we have a book "(Counter)examples in Algebraic Topology" can be the title for a good and useful MO question. $\endgroup$
    – Gil Kalai
    Commented Feb 13, 2011 at 14:22
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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.

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    $\begingroup$ Does Chriss and Ginzburg, <i>Representation Theory and Algebraic Geometry</i>, do the job? I generally consider this an excellent book, and it has a chapter on the Springer correspondence, but I haven't read that chapter. $\endgroup$
    – David E Speyer
    Commented Jan 31, 2011 at 16:49
  • $\begingroup$ It's a stimulating book (very much in the "Russian style"). At the end of Chapter 3 there is a brief intuitive discussion of Springer theory limited to the case of the symmetric group as Weyl group of the special linear group. Here the component groups of centralizers of unipotent elements are trivial, while both unipotent classes and Weyl group representations are parametrized by partitions. In Chapter 4 they show in an original way how to study finite dimensional representations of special linear groups in a similar spirit. Very nice but not the book I'd like on Springer theory. $\endgroup$
    – Jim Humphreys
    Commented Jan 31, 2011 at 21:01
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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.

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    $\begingroup$ You should read Spivak's 5 volume "A Comprehensive Introduction to Differential Geometry." In particular, the first 3 volumes. He makes sure to treat almost every single aspect 3 ways: in local coordinates (what you call the physicist's "tensor calculus"), with moving frames (the Cartan/Chern approach), and the modern "invariant" formulation. In my opinion, all differential geometers should be comfortable moving back and forth between all three, because they're all useful in various different situations. $\endgroup$
    – Spiro Karigiannis
    Commented Jan 24, 2011 at 13:14
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    $\begingroup$ I've read Spivak's 1st volume. I had good intentions of going further but never made it. What I have in mind is a little different from Spivak in that I'd like to see the comparisons from the beginning. Maybe start with geometry from the viewpoint of Schey's book "Div, Grad, Curl and All That" and show how the vast machinery of differential geometry makes these concepts rigorous. $\endgroup$
    – John D. Cook
    Commented Jan 24, 2011 at 13:51
  • $\begingroup$ I've never seen anything like what you have in mind (which would be awesome), but Hicks' "Notes on Differential Geometry" is pretty good about bridging the gap between the differential geometry of curves and surfaces and Riemannian geometry. $\endgroup$
    – arsmath
    Commented Jan 24, 2011 at 16:50
  • $\begingroup$ David Bressoud's "Second Year Calculus" combines a typical undergraduate calculus approach with differential forms. It's a start at what I have in mind, but it doesn't go very far. $\endgroup$
    – John D. Cook
    Commented Jan 25, 2011 at 11:07
  • $\begingroup$ This would be very nice, but the 'modern' view on differential geometry should probably also mention the synthetic approach (ncatlab.org/nlab/show/synthetic+differential+geometry) in an ideal book. $\endgroup$
    – Alec Rhea
    Commented Jun 20, 2019 at 3:20
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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.

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    $\begingroup$ I take it you already know Banyaga's "The Structure of Classical Diffeomorphism Groups"? $\endgroup$
    – Maxime Bourrigan
    Commented Jan 31, 2011 at 15:16
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    $\begingroup$ @Maxime. That's a good reference if you want to know about the group theory of Diff, but not if you want to know about its algebraic topology (e.g. what can one say about the homotopy-type of $Diff(S^n)$?). I think Daniel is right that we are missing a book on that topic. $\endgroup$
    – Tim Perutz
    Commented Jan 31, 2011 at 17:03
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    $\begingroup$ @Tim: "I think Daniel is right that we are missing a book on that topic". So do I (I would love this book to explain the links between the cohomology of these groups and foliations, à la Mather-Thurston and the few known results about these cohomology groups). $\endgroup$
    – Maxime Bourrigan
    Commented Jan 31, 2011 at 17:27
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    $\begingroup$ math.harvard.edu/~kupers/teaching/272x/book.pdf Maybe he listened? $\endgroup$
    – Thomas Rot
    Commented Jun 19, 2019 at 21:08
  • $\begingroup$ @ThomasRot This is very nice!! My peeve is that it doesn't seem to treat Cerf's work at all and merely outlines Hatcher's proof of the Smale Conjecture- but it looks wonderful nonetheless! $\endgroup$
    – Daniel Moskovich
    Commented Jul 7, 2019 at 14:52
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"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.

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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.

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    $\begingroup$ Working on it...! $\endgroup$
    – Laurent Berger
    Commented Jan 24, 2011 at 11:57
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    $\begingroup$ While waiting for Laurent's book (!) , you could try reading Modular Forms and Fermat's Last Theorem (Cornell, Silverman, Stevens eds.), which is a fantastic graduate level introduction to the subject. $\endgroup$
    – Emerton
    Commented Jan 24, 2011 at 12:37
  • $\begingroup$ For Laurent : will this book be an extended version of your Galois trimester notes or something else ? $\endgroup$
    – A M
    Commented Jan 24, 2011 at 18:26
  • $\begingroup$ @AM: it will only be the extended version of my Galois trimester notes. $\endgroup$
    – Laurent Berger
    Commented Jan 30, 2011 at 18:42
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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.

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    $\begingroup$ I should say that very recently such a book has been written by Audin and Damian - "Théorie de Morse et homologie de Floer", which is a beautiful and comprehensive introduction to the easiest parts of Floer homology. My only complaint with this book is that it doesn't go quite far enough - I guess I'm thinking more of a book the size of McDuff and Salamon's wonderful "J-holomorphic curves and symplectic topology" - but written specifically for Floer theory. $\endgroup$
    – user8594
    Commented Jan 24, 2011 at 14:32
  • $\begingroup$ What do you think of Oh's book: math.wisc.edu/~oh/all.pdf $\endgroup$
    – Orbicular
    Commented May 28, 2011 at 12:06
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    $\begingroup$ @Orbicular Dead link. This seems to be the current one: cgp.ibs.re.kr/~yongoh/all.pdf $\endgroup$
    – Akiva Weinberger
    Commented May 26, 2017 at 0:24
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  • "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)

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  • $\begingroup$ I think people are working on the second one... I hope they are at least. $\endgroup$
    – Sean Tilson
    Commented Jan 25, 2011 at 2:39
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    $\begingroup$ This looks like a readable text on Falting's p-adic Simpson correspondence: arxiv.org/abs/1102.5466 $\endgroup$
    – Thomas Riepe
    Commented Mar 1, 2011 at 17:53
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    $\begingroup$ @Sean: Much of the subject since the 1970s could be viewed as "Morava explained". $\endgroup$
    – Tyler Lawson
    Commented Oct 25, 2011 at 23:24
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You forestalled some of what I would have posted...

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    $\begingroup$ +1 for the third bullet point (I know a few more that fall into this chapter). $\endgroup$
    – Theo Buehler
    Commented Jan 24, 2011 at 12:03
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    $\begingroup$ Sorry, it's the fourth one now ;) $\endgroup$
    – darij grinberg
    Commented Jan 24, 2011 at 12:04
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    $\begingroup$ Does such a definition of higher K-groups without topology actually exist? I have never heard about that, so it sounds more like an ambitious research project. $\endgroup$
    – Johannes Ebert
    Commented Jan 25, 2011 at 8:54
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    $\begingroup$ Concerning Weyl's The Classical Groups, an argument can be made in favor of either modern text: Goodman & Wallach Symmetry, Representations, and Invariants (2nd ed., Springer GTM 255, 2009) and Procesi Lie Groups (Springer Universitext, 2007). I won't try to make the argument, since what you mean by asking for the same proofs as in Weyl's book might need further discussion. $\endgroup$
    – Jim Humphreys
    Commented Jan 30, 2011 at 18:14
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    $\begingroup$ I don't believe Goodman-Wallach can really supersede Weyl. For example, where are Capelli's identities in Goodman-Wallach? I only see Theorem 5.7.1, which neither gives an explicit form nor applies to the classical case (Goodman-Wallach require $V=S^2(\mathbb C^n)$ or $V=\wedge^2(\mathbb C^n)$, which lead to the Turnbull rsp. Howe-Umeda-Kostant-Sahi identities rather than the actual Capelli ones), let alone an explicit proof "from the definitions". Procesi's text could do the trick indeed. $\endgroup$
    – darij grinberg
    Commented Jan 30, 2011 at 21:25
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The construction of galoisian representations associated to primitive cuspidal eigenforms. I hope the user BCnrd gets the hint.

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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.:)

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    $\begingroup$ I have heard once that Yuri Manin had translated SGA (and EGA?) into Russian. If anyone, he knews if that is correct and if translations of BBD exist too. $\endgroup$
    – Thomas Riepe
    Commented Jan 26, 2011 at 20:34
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    $\begingroup$ ... and, of course, French is a very nice language! $\endgroup$
    – Thomas Riepe
    Commented Jan 26, 2011 at 20:35
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    $\begingroup$ Mikhail, you will be better off learning the little French required to read EGA, SGA, FGA, BBD, DPP, GAGA, SAGA, etc., than waiting for English or Russian translations to appear. $\endgroup$
    – Chandan Singh Dalawat
    Commented Jan 27, 2011 at 5:03
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    $\begingroup$ The problem is that these books would be complicated reading for me even in English. $\endgroup$
    – Mikhail Bondarko
    Commented Jan 27, 2011 at 10:07
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Algebraic Geometry from a Homotopical Viewpoint: For the topologist who really wants to like geometry but doesn't know where to start.

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    $\begingroup$ The title is supposed to suggest that this would be about A^1 homotopy theory... starting from scratch and assuming roughly zero exposure to algebraic geometry. Just an answer to the question, "How can we think about algebraic geometry from the perspective of homotopy theory?" $\endgroup$
    – Dylan Wilson
    Commented Jan 24, 2011 at 10:34
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    $\begingroup$ have you seen amazon.com/Motivic-Homotopy-Theory-Nordfjordeid-Universitext/dp/…? $\endgroup$
    – Sean Tilson
    Commented Jan 25, 2011 at 2:35
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    $\begingroup$ One of the great things about this question is that it secretly allows us to ask/answer a bunch of questions of the form "Is there a book like blah about bleh?" Thanks Sean, that book looks great!! $\endgroup$
    – Dylan Wilson
    Commented Jan 25, 2011 at 4:07
  • $\begingroup$ I have looked at it a little, I found that I still don't know enough. PS a lot of that book is on the websites of various authors. $\endgroup$
    – Sean Tilson
    Commented Jan 25, 2011 at 5:37
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As I have been telling many people involved in mathematical publishing, the one book I would like to read is The Serre-Tate correspondence.

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    $\begingroup$ Well, you are one of the lucky ones! There are now two volumes recently published by de SMF in the Documents Mathématiques series. $\endgroup$
    – F Zaldivar
    Commented Jul 13, 2015 at 23:37
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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.

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    $\begingroup$ I think Steve Awodey's "Category theory" might be just right for you. $\endgroup$
    – Gonçalo Marques
    Commented Jan 24, 2011 at 14:42
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    $\begingroup$ Gonçalo: agreed. Awodey is written specifically for computer scientists and other people who don't have much exposure to algebraic topology and the like, so it develops all the necessary examples from scratch (the two most prominent, I think, being posets and monoids). $\endgroup$
    – Qiaochu Yuan
    Commented Jan 24, 2011 at 15:55
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    $\begingroup$ Actually, as a non-categorist, I think Mac Lane's title is apt if treated as an introduction to the theory rather than as a handbook for practical reference. But each to their own $\endgroup$
    – Yemon Choi
    Commented Jan 24, 2011 at 19:32
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    $\begingroup$ There is a book like this: Emily Riehl's Categories in Context. It s the best Category book I have ever read. $\endgroup$
    – Nico
    Commented Nov 23, 2021 at 14:16
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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.

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  • $\begingroup$ Compare to my "Weyl's Classical Groups made readable". $\endgroup$
    – darij grinberg
    Commented Feb 2, 2011 at 11:02
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    $\begingroup$ The subject is definitely hard to present rigorously in book form without making it look too abstract, especially if you want to work over fields of prime characteristic. I've never tried to teach an actual graduate course using a book like mine or Springer's. The Russian approach is an interesting alternative, especially for characteristic 0 theory parallel to Lie groups: MR1064110 (91g:22001), Onishchik, A. L.; Vinberg, È. B., Lie groups and algebraic groups. Translated from the Russian and with a preface by D. A. Leites. Springer, 1990. $\endgroup$
    – Jim Humphreys
    Commented Feb 2, 2011 at 15:57
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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.

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    $\begingroup$ "We shall now shew..." for instance (W&W, p.13 and many other places.) $\endgroup$
    – Stopple
    Commented Jan 24, 2011 at 20:50
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    $\begingroup$ Exactly. I had no idea anyone wrote "shew" in the 20th century until I saw that. $\endgroup$
    – John D. Cook
    Commented Jan 24, 2011 at 21:44
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    $\begingroup$ But what about Watson's 1944 2nd Edition or the reprinted version: books.google.com/… perhaps these are typographically still similar (or the same) as the 1922 versions, but don't know if the copyright lapse applies anymore? $\endgroup$
    – Suvrit
    Commented Jan 25, 2011 at 10:21
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    $\begingroup$ @John D. Cook: There's also George Bernard Shaw. $\endgroup$
    – Nate Eldredge
    Commented Jan 27, 2011 at 5:30
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    $\begingroup$ @Stopple I guess that you are hoping for the taming of the ‘shew’? $\endgroup$
    – LSpice
    Commented Jul 15, 2011 at 14:21
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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.

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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 Lp-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.

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    $\begingroup$ @Pete: Sorry, I got it all wrong. Replace “maximal” by “prime” and “prime” by “radical”. I am not sure what I was thinking about when I wrote my previous comment, but points in spectra correspond to prime ideals and open sets correspond to radical ideals. Nullstellensatz can be reformulated in such a way that it does not refer to maximal or prime ideals. A proof of such a localic version can be found in this paper: ams.org/journals/proc/1980-079-01/S0002-9939-1980-0560591-4/… $\endgroup$
    – Dmitri Pavlov
    Commented Jan 25, 2011 at 16:42
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    $\begingroup$ @Michael: Could you please be more precise? What kind of theorem or definition do you have in mind? $\endgroup$
    – Dmitri Pavlov
    Commented Jan 26, 2011 at 2:42
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    $\begingroup$ @Yemon: One size does not fit all. You and darij seem to subtly imply (or at least this is my feeling when I read your comments) that for any mathematical theory there is the best way to expose it, whereas I am more inclined towards diversity of expositions. Some people (like me) like coordinate-free expositions, while others prefer bases and matrices. There are plenty of linear algebra textbooks written using bases and matrices, but very few or none are written in a coordinate-free way. That's why I included linear algebra in my list. $\endgroup$
    – Dmitri Pavlov
    Commented Jan 27, 2011 at 19:05
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    $\begingroup$ @Dmitri: I think I for one have not learned anything "literally in a few minutes". Do you have experimental evidence for this claim? Anyway, I agree with you that most people take a lot longer to learn the abstract approach. But moreover, for many people understanding concrete examples is a necessary route to abstraction. If you're going to teach people about dualizable objects in categories, you can go ahead and teach them about bases and matrices first, I think, without wasting anyone's time. $\endgroup$
    – Pete L. Clark
    Commented Jan 31, 2011 at 4:24
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    $\begingroup$ Having thought things through a bit more, I wish to affirm the principle that the author of a math book ought not to be required to include any material beyond that which is of firm personal interest to herself. (Diligent application of this principle could lead to better books.) So I don't want to discourage anyone from writing this particular take on linear algebra. Rather what I mean to say is that such a book should be used for good rather than ill: raising a generation of mathematicians for whom bases and matrices are no more than an afterthought would be nothing to be proud of. $\endgroup$
    – Pete L. Clark
    Commented Jan 31, 2011 at 15:46
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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.

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  • $\begingroup$ There is Joe Taylor's article in a conference proceedings, circa 1972?? $\endgroup$
    – David Handelman
    Commented Mar 17, 2020 at 18:08
  • $\begingroup$ @DavidHandelman I may have read or at least skimmed Taylor's article -- maybe it was in the Algebras and Analysis confernce proceedings 1972? -- but as my answer was meant to imply, there has been lots of stuff since then that is only folklore or known to experts. Besides, if I recall correctly from my PhD days, Taylor's articles have had very little to say about the peculiarities of the Banach setting rather than the Frechet setting $\endgroup$
    – Yemon Choi
    Commented Mar 17, 2020 at 19:33
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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.)

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    $\begingroup$ Have you looked at Moriwaki's book from the Translations of Mathematical Monographs series? It has a focus on "birational" aspects (line bundles with appropriate analogues of "positivity), and generalizes to arbitrary arithmetic varieties, but certainly covers the curves/surfaces cases quite well, together with many of the results you mentioned. It doesn't explicitly give the classical algebraic results side-by-side, however. $\endgroup$
    – peterx
    Commented Oct 9, 2015 at 23:59
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I'm surprised that nobody has expressed the desire to read Bourbaki's Théorie des nombres.

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    $\begingroup$ Some might be surprised that anyone has the desire to read Bourbaki at all ;-) $\endgroup$
    – Johannes Hahn
    Commented Jan 27, 2011 at 14:42
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    $\begingroup$ @Chandan: I honestly think that my current musings about "abstract algebraic number theory" are highly in the spirit of the text you name above. For instance, one of the points is the generalization of the Dirichlet Unit Theorem to a wider class of rings, and in this regard there is indeed a Samuel Unit Theorem. In general, Samuel's little book on the algebraic theory of numbers often feels like a little coda to Bourbaki. There are exceptions: for some reason I feel confident that Nicolas would die before mentioning the Minkowski Convex Body Theorem. (Perhaps he has.) $\endgroup$
    – Pete L. Clark
    Commented Jan 30, 2011 at 21:22
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    $\begingroup$ Johannes, =p! $\endgroup$
    – Harry Gindi
    Commented Jan 31, 2011 at 15:12
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Introduction to algebraic cycles.

With lots of examples...

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    $\begingroup$ How distant is Eisenbud and Harris book "3264 and All That: A Second Course in Algebraic Geometry" to your desired text? $\endgroup$
    – Leo Alonso
    Commented May 30, 2017 at 15:33
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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.

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    $\begingroup$ Dear Lorenzo, What is your objection to SGA 4.5 (which is my personal favourite of the SGAs)? $\endgroup$
    – Emerton
    Commented Jan 25, 2011 at 3:36
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    $\begingroup$ Dear Emerton, wasn't someone (I think Verdier) originally assigned by Grothendieck the project of replacing the spectral-sequence-laden arguments of SGA4.5 with simpler arguments using derived categories, but it was never finished? $\endgroup$
    – Harry Gindi
    Commented Jan 31, 2011 at 15:17
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"Cardinal Arithmetic: The New Corrected Edition (including index)" by Saharon Shelah...

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    $\begingroup$ This would be a wonderful book! $\endgroup$
    – Alberto Levi
    Commented Oct 24, 2013 at 20:02
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Algebraic Number Theory for Algebraic Geometers.

I want a book which explains algebraic number theory to somebody who is fluent in algebraic geometry (or maybe at least has read Hartshorne) but knows very little number theory to start with. In particular, such a book would not be seeking to minimize the prerequisites required of the reader, but rather adapt itself to a very particular set of prerequisites. At a bare minimum, some of the peculiar terminology of number theory -- "conductor", "discriminant", ... -- would be defined in terms of algebro-geometric concepts.

I asked a question about this here. Somebody suggested Neukirch's book, which is better than most in this regard, but even Neukirch has a different audience in mind, and confines his algebro-geometric discussions to impressionistic side-chapters rather than integrating them fully into the text.

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    $\begingroup$ I'd like to see the converse, algebraic geometry for number theorists. I am not a number theorist, but I maintained my interest in it, after having taken many undergraduate and graduate course in the subject, about 40 years ago. $\endgroup$
    – David Handelman
    Commented Mar 17, 2020 at 18:01
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"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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    $\begingroup$ 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.). $\endgroup$
    – Emerton
    Commented Jan 24, 2011 at 12:46
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    $\begingroup$ There are also the DDT notes, now available on Darmon's website, along with other related material. $\endgroup$
    – Chandan Singh Dalawat
    Commented Jan 24, 2011 at 14:16
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    $\begingroup$ 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. $\endgroup$
    – David Hansen
    Commented Jan 25, 2011 at 16:53
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    $\begingroup$ 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 ... $\endgroup$
    – Emerton
    Commented Jan 27, 2011 at 4:33
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    $\begingroup$ ... 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 ... $\endgroup$
    – Emerton
    Commented Jan 27, 2011 at 4:39
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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.

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"Quiver varieties with a wealth of examples" ?

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    $\begingroup$ I'd vote for nearly any book title containing the phrase "with a wealth of examples." $\endgroup$
    – John D. Cook
    Commented Jan 24, 2011 at 19:13
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    $\begingroup$ "with a variety of examples" would be better, though. $\endgroup$
    – darij grinberg
    Commented Jan 24, 2011 at 20:18
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    $\begingroup$ Some example varieties are not wealthy. Go for the wealth. Gerhard "Ask Me About System Design" Paseman, 2011.01.24 $\endgroup$
    – Gerhard Paseman
    Commented Jan 24, 2011 at 21:53
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    $\begingroup$ What about the book by Kirillov Jr ? bookstore.ams.org/gsm-174 $\endgroup$
    – Nicolas Hemelsoet
    Commented Jun 6, 2020 at 14:21
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