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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 ?

LSpice
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Gonçalo Marques
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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
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    Yeah, but as most of you probably already know, JB won't write (2) nor his projected "higher algebra", because he thinks that he worked off his debt by writing all those expository papers... – Tim van Beek Jan 24 '11 at 10:51
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    Steve Lack wrote something approximating (2): http://arxiv.org/abs/math/0702535 – Tom Leinster Jan 24 '11 at 11:31
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    Thanks Dylan. Tim: it would be great indeed if JB would publish his "higher algebra" but his past expository work is already amazing. Tom: I am aware of that paper (and you also have some sections about it on your wonderful book) but I was thinking about a full textbook presentation that might have more examples and applications. – Gonçalo Marques Jan 24 '11 at 12:28
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    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
  • Thank you for your suggestions I will look for Katz and Deligne's articles. I am aware of Freitag and Kiehl's textbook, unfortunately it's a hard to find. I was thinking of a textbook that would use the Weil conjectures as a "leitmotiv" while introducing some of the more modern characters in algebraic geometry. But maybe it can't be done (at least at level I would understand...). – Gonçalo Marques Jan 24 '11 at 14:16
  • @Tom: Sadly Lack's account is hardly an introduction to 2-categories, skipping even the definitions (!) and all motivation. It would be even less suitable for a "working mathematician". – Andrea Ferretti Jan 24 '11 at 15:31
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    It boggles my mind that nobody has written 5) yet. Shouldn't one of Grothendieck's students be doing this or something? – Qiaochu Yuan Jan 24 '11 at 15:58
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    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
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    One more reference on the Weil conjectures: notes of Beilinson's lectures on the subject available at http://www.math.uchicago.edu/~mitya/beilinson/ – AFK Jan 26 '11 at 23:30
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    functor-of-points: besides Demazure-Gabriel, there is also the last chapter of Eisenbud-Harris "Geometry of Schemes" – Sean Rostami Jan 27 '11 at 04:36
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    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! – Joseph O'Rourke Feb 01 '11 at 00:07
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    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 01 '11 at 15:03
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    Mumford's "Lectures on curves on an algebraic surface" is a great solution to 5), different from and (for me) more geometrically appealing than Demazure and Gabriel. – inkspot Feb 05 '11 at 16:59
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    The Tropic of Calculus (as suggested by Tom Lehrer). 50 Shades of Gray Codes. Lady Chatterly's Prover. – Gerry Myerson Oct 09 '15 at 05:12
  • What about a question: "A mathematical theory you would like to see developed." Or perhaps "Mathematical theory you would like to learn (if somebody would just develop them...)" – Gil Kalai Jan 25 '21 at 20:40

47 Answers47

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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.

gowers
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    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...) – Gonçalo Marques Jan 24 '11 at 15:07
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    "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 – Vectornaut Jan 24 '11 at 15:36
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    +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. – Dylan Wilson Jan 24 '11 at 16:32
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    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. – arsmath Jan 24 '11 at 16:41
  • Perhaps "physics for mathematicians" is a little bit on the unmodest side; among the more specialized courses I like "Manifolds and mechanics" by Jones, Gray and Hutton, and "Group theory and physics" by Sternberg. – Franz Lemmermeyer Jan 24 '11 at 17:43
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    How about the two-volume series "Quantum Fields and Strings: A Course for Mathematicians" (eds. Deligne, Etingof, Freed, Jeffrey, Kazhdan, Morgan, Morrison, Witten)? – Greg Marks Jan 24 '11 at 17:53
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    Darryl Holm's two-volume "Geometric Mechanics" may be worth looking at. – Simon Lyons Jan 24 '11 at 20:08
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    I second this, for a book with emphasis on mathematics related to quantum optics. – Peter Shor Jan 24 '11 at 22:29
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    I have the new Spivak volume, and yes it is only about Classical mechanics, and yes it is written with Spivak's usual style and clarity. It is the first volume of a projected series, the next according to Mike will be on E&M. You can get the Book directly from Spivak (Publish or Perish Press) at a good price. – Dick Palais Jan 24 '11 at 22:36
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    I ought to say, in case anyone gets the wrong idea, that I like the articles in the Princeton Companion to Mathematics that deal with theoretical physics. However, there's a difference between a PCM article and an entire book. – gowers Jan 24 '11 at 22:59
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    My personal opinion is that unfortunately such a book cannot possibly exist. Physicists derive a lot of intuition their intuition from their rich experience doing computations (starting with their undergraduate homework, continued through graduate classes to basically every single paper they work on). Based on this intuition, they develop models, tweak them, expand them, generalize them, discard them. Now, for any specific model, and given a couple of years, we mathematicians can translate everything in this model into mathematics (and develop some great mathematics on the way). But... – Arend Bayer Jan 31 '11 at 18:04
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    ...when physicists tomorrow re-tweak or expand their model, then lacking their computational intuition, we at first again won't understand what they are doing, and we have to restart translating physics into mathematics. (And of course, as it took a couple of years to translate the previous model into mathematics, "tomorrow" will of course already have happened a while ago.) I don't think of this as a bad state of affairs though, it also means that the input from theoretical physics will continue to be surprising and interesting. – Arend Bayer Jan 31 '11 at 18:09
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    I really like Folland's book "Quantum field theory, A tourist guide for mathematicians". – Rob Harron Feb 02 '11 at 02:43
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    I also would like to ask about the IAS quantum fields and string that Greg mentioned. Also how is Penrose's book "a road to reality" for such a purpose? – Gil Kalai Feb 13 '11 at 14:26
  • "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. 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." I'd argue that this is partially true: physics is full of concepts that when learned transform your outlook on reality, and it's very hard to go back and talk like you don't know this stuff. Math is this way too. – John Baez Oct 23 '21 at 20:42
76

Counterexamples in scheme theory

Qfwfq
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    Maybe the examples chapter https://stacks.math.columbia.edu/tag/026Z in the Stacks project can be seen as an approximation of the book you would like to read. – pbelmans Jun 19 '19 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.

Martin Sleziak
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Johannes Ebert
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    Before we have a book "(Counter)examples in Algebraic Topology" can be the title for a good and useful MO question. – Gil Kalai Feb 13 '11 at 14:22
51

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.

Jim Humphreys
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    Does Chriss and Ginzburg, Representation Theory and Algebraic Geometry, 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. – David E Speyer Jan 31 '11 at 16:49
  • 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. – Jim Humphreys Jan 31 '11 at 21:01
49

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.

John D. Cook
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    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. – Spiro Karigiannis Jan 24 '11 at 13:14
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    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. – John D. Cook Jan 24 '11 at 13:51
  • 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. – arsmath Jan 24 '11 at 16:50
  • 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. – John D. Cook Jan 25 '11 at 11:07
  • This would be very nice, but the 'modern' view on differential geometry should probably also mention the synthetic approach (https://ncatlab.org/nlab/show/synthetic+differential+geometry) in an ideal book. – Alec Rhea Jun 20 '19 at 03:20
47

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.

Daniel Moskovich
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    I take it you already know Banyaga's "The Structure of Classical Diffeomorphism Groups"? – Maxime Bourrigan Jan 31 '11 at 15:16
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    @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. – Tim Perutz Jan 31 '11 at 17:03
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    @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). – Maxime Bourrigan Jan 31 '11 at 17:27
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    http://www.math.harvard.edu/~kupers/teaching/272x/book.pdf Maybe he listened? – Thomas Rot Jun 19 '19 at 21:08
  • @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! – Daniel Moskovich Jul 07 '19 at 14:52
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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.

Dirk Basson
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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.

Gunnar Þór Magnússon
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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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    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. –  Jan 24 '11 at 14:32
  • What do you think of Oh's book: http://www.math.wisc.edu/~oh/all.pdf – Orbicular May 28 '11 at 12:06
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    @Orbicular Dead link. This seems to be the current one: http://cgp.ibs.re.kr/~yongoh/all.pdf – Akiva Weinberger May 26 '17 at 00:24
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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)

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

Community
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darij grinberg
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    +1 for the third bullet point (I know a few more that fall into this chapter). – Theo Buehler Jan 24 '11 at 12:03
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    Sorry, it's the fourth one now ;) – darij grinberg Jan 24 '11 at 12:04
  • What do you mean by Quillen's K-Theory? His definition of Algebraic K-theory? – Sean Tilson Jan 25 '11 at 05:37
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    Yes, I just want his higher K-groups defined without classifying spaces. – darij grinberg Jan 25 '11 at 08:15
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    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. – Johannes Ebert Jan 25 '11 at 08:54
  • Neither have I. But I haven't heard of a good reason why it shouldn't exist either... – darij grinberg Jan 25 '11 at 09:38
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    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. – Jim Humphreys Jan 30 '11 at 18:14
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    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. – darij grinberg Jan 30 '11 at 21:25
  • Goodman-Wallach do cover some of the ground in classical invariant theory, but not all. Procesi's book is less tightly organized but focuses more strongly on invariant theory in modern guise. Weyl's classic isn't easy to read but is unlikely to be duplicated elsewhere, since people tend to get interested in many other aspects of representation theory and write books about these. – Jim Humphreys Jan 31 '11 at 23:12
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    One can define Quillen's K-theory without topological spaces but just with simplicial sets. I don't think, it is necessary (or even not misleading) to see simplicial sets as part of topology. – Lennart Meier Oct 24 '11 at 12:08
  • Lennart: Any place where this approach has been detailed? – darij grinberg Oct 24 '11 at 14:59
  • When you take the K-theory of a symmetric monoidal or a Waldhausen category, the output is automatically a simplicial set. You can associate to a ring $R$ the full subcategory of $R$-mod on the $R^n$ to recover usual K-theory. Have, for example, a look at http://www.math.rutgers.edu/~weibel/Kbook/Kbook.IV.pdf – Lennart Meier Dec 11 '11 at 18:51
  • Quillen's algebraic K-theory for rings can be defined in terms of non-abelian homological algebra. The only book-length presentation that I know is this:
    Hvedri Inassaridze: Non-abelian homological algebra and its applications.Kluwer, 1997. ISBN: 0-7923-4718-8

    It seems that this approach never got very popular. The book seems to be little known.

     – M Mueger Nov 22 '12 at 11:08
  • There is also a newer approach due to Grayson and it is possible to prove the basic theorems in this framework: http://intlpress.com/site/pub/files/_fulltext/journals/hha/2015/0017/0001/HHA-2015-0017-0001-a013.pdf – Lennart Meier Jul 19 '15 at 18:10
  • @LennartMeier: Link is broken for me. – darij grinberg Jul 19 '15 at 18:17
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    @darijgrinberg That's my paper behind an HHA paywall. An earlier version is on the arxiv: http://arxiv.org/pdf/1311.5162v1.pdf The published version has a much simpler proof of the cofinality theorem in it, but not much else is too different. I should note that Lennart is a little optimistic: I don't prove all of the basic theorems in that paper, just additivity, resolution, and cofinality. – Tom Harris Dec 16 '15 at 23:58
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The construction of galoisian representations associated to primitive cuspidal eigenforms. I hope the user BCnrd gets the hint.

Chandan Singh Dalawat
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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.

Dylan Wilson
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    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?" – Dylan Wilson Jan 24 '11 at 10:34
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    have you seen http://www.amazon.com/Motivic-Homotopy-Theory-Nordfjordeid-Universitext/dp/3540458956/ref=sr_1_1?s=books&ie=UTF8&qid=1295922831&sr=1-1? – Sean Tilson Jan 25 '11 at 02:35
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    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!! – Dylan Wilson Jan 25 '11 at 04:07
  • 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. – Sean Tilson Jan 25 '11 at 05:37
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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.

Chandan Singh Dalawat
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    Well, you are one of the lucky ones! There are now two volumes recently published by de SMF in the Documents Mathématiques series. – F Zaldivar Jul 13 '15 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.

John D. Cook
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    I think Steve Awodey's "Category theory" might be just right for you. – Gonçalo Marques Jan 24 '11 at 14:42
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    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). – Qiaochu Yuan Jan 24 '11 at 15:55
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    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 – Yemon Choi Jan 24 '11 at 19:32
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    There is a book like this: Emily Riehl's Categories in Context. It s the best Category book I have ever read. – Nico Nov 23 '21 at 14:16
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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.:)

Mikhail Bondarko
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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.

Mark
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  • Compare to my "Weyl's Classical Groups made readable". – darij grinberg Feb 02 '11 at 11:02
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    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. – Jim Humphreys Feb 02 '11 at 15:57
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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.

Martin Sleziak
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Dmitri Pavlov
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    Also: I thought it was acknowledged that while you can (and to some extent, should) set up linear algebra without coordinates and bases and matrices, getting things done in functional analysis rather often needs you to choose bases, etc. (Cf. the difference between categories of Hilbert spaces and categories of RKHS) – Yemon Choi Jan 24 '11 at 19:23
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    Might I also suggest separating off your suggestions about higher categories and writing them as a separate answer? They seem to have a different flavour from your other "down with coordinates! define everything in terms of the coordinate ring!" suggestions – Yemon Choi Jan 24 '11 at 19:26
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    Finally: having taken some time to read the exposition in Connes NCDG part II of how to work out the cyclic cohomology of $C^\infty(M)$ ... I am puzzled as to how this would be made clearer by refusing to use charts to work in local coordinates. You'd presumably have to use a double-complex argument and resolve by acyclics, but how does one check these are acyclic without doing some kind of local computation? – Yemon Choi Jan 24 '11 at 19:29
  • (Last question honest, not rhetorical. Maybe you can do it in a chart-free way, but it's not clear to me what the benefits would be.) – Yemon Choi Jan 24 '11 at 19:30
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    Does a localic approach to commutative algebra actually exist (albeit not written down in textbook form)? I'm puzzled at the suggestion that such an approach could avoid maximal ideals: aren't these part of the subject? (Note that I barely even know what a locale is...) – Pete L. Clark Jan 24 '11 at 22:47
  • @Yemon: I am somewhat confused by your question about the bidual of c_0. The dual of c_0 is l^1 and the dual of l^1 is l^∞. I don't think that passing to locales affects the validity of any of these statements. – Dmitri Pavlov Jan 25 '11 at 04:42
  • @Yemon: Could you please be more specific about “getting things done in functional analysis”? What kind of theorem do you have in mind? I can guess that RKHS means reproducing kernel Hilbert space, but I have no idea what kind of result you are referring to. – Dmitri Pavlov Jan 25 '11 at 04:46
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    @Yemon: Only 2 out of 8 proposals mention coordinates, not 7. My point is that since there are so many books that teach you how to do all kinds of things using coordinates, it would be nice to have at least one book that would teach you how to do things in a coordinate-free way. I find that coordinate-free proofs often enhance my intuition. – Dmitri Pavlov Jan 25 '11 at 04:57
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    @Pete: You only need to consider maximal ideals as long as you insist that the spectrum of a commutative ring consists of actual points. Once you pass to the localic framework you only need to keep track of open sets, which correspond to prime ideals. There are plenty of papers in this area (for example, Thierry Coquand has lots of papers on his homepage: http://www.cse.chalmers.se/~coquand/algebra.html), but I am not aware of any books written in this language. – Dmitri Pavlov Jan 25 '11 at 05:27
  • @Dmitri: fair point about just wanting some coordinate-free texts, given that "working in coordinates" approaches are indeed well-covered. Also, yes, I rather stupidly wrote "bidual of c_0" when I meant "bidual of l^1". – Yemon Choi Jan 25 '11 at 06:12
  • @Yemon: My confusion extends to this example also. For a set A the dual of l^1(A) is l^∞(A), l^∞(A)=C^0(B), where B is the Stone-Čech compactification of A, and the dual of C^0(B) is the space of Radon measures on B. In other words, passing to the framework of locales again does not change anything. Note that the Stone-Čech compactification of locales does not require any form of the axiom of choice. – Dmitri Pavlov Jan 25 '11 at 06:36
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    @Dmitri: thanks for responding. Part of my point is that maximal ideals are part of the core subject matter of commutative algebra in the sense that many of the key results refer to them (e.g. the Nullstellensatz). So I don't understand how it could still be commutative algebra without them. But about prime ideals...why are they so much better from a constructive point of view? The existence of a prime ideal in any ring still requires at least the Boolean Prime Ideal Lemma, right? – Pete L. Clark Jan 25 '11 at 11:20
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    @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: http://www.ams.org/journals/proc/1980-079-01/S0002-9939-1980-0560591-4/S0002-9939-1980-0560591-4.pdf – Dmitri Pavlov Jan 25 '11 at 16:42
  • @Dmitri: How do you plan to do linear algebra, beyond the first definitions, without the notion of a basis? – Johannes Ebert Jan 25 '11 at 18:58
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    @Johannes: Take a look at Bourbaki's book to see how linear algebra can be done without bases, coordinates, and matrices. If you have a concrete theorem in mind, feel free to ask. – Dmitri Pavlov Jan 25 '11 at 19:14
  • @Dmitri -1 for bullet point number one. Locales are not sufficient for any thoroughly general treatment of topology. No mater what ncat may have you believe (sorry ncat fans, just know I do respect the site). Just because you can squint your eyes and make it look like a topology does not mean it can actually handle all of the constructions from topology. – Not Mike Jan 25 '11 at 23:11
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    @Michael: Could you please be more precise? What kind of theorem or definition do you have in mind? – Dmitri Pavlov Jan 26 '11 at 02:42
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    @Dmitri: here are a couple of theorems that I need whenever I find a vector space on my desk: 1. any short exact sequence of $K$-vector spaces is split; 2. any linear functional on a subspace can be extended to the whole space; 3. Two vector spaces of the same dimension are isomorphic. How can these results be shown without bases (or an equivalent notion)? – Johannes Ebert Jan 26 '11 at 20:43
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    "Linear algebra without bases" really means "module theory over commutative rings". Which is why it is probably a bad idea to write a full book without even once using a basis: Many properties of modules require finitely generated modules, i. e. generating sets, and the step from generating set to basis isn't that huge. Also, matrices are extremely important pretty much everywhere including the most abstract algebra (maps between direct powers of modules are written as matrices of maps, for example). But I agree that it would be better to START a linear algebra course with generic modules. – darij grinberg Jan 26 '11 at 21:42
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    @Johannes: For (1) use Zorn's lemma to find the maximal subspace that is disjoint with the image of the first map in the exact sequence. This maximal subspace is the image of the splitting map. For (2) use Zorn's lemma to find the maximal extension of your linear functional. This maximal extension is defined on the whole subspace. Note that the use of some form of the axiom of choice is inevitable because these statements are equivalent to some weak forms of the axiom of choice. – Dmitri Pavlov Jan 27 '11 at 04:34
  • @Johannes: (3) is trivial for infinite dimensional spaces if you use the usual definition of dimension for infinite-dimensional spaces (dim V=n if V is isomorphic to k^n). For finite-dimensional spaces you have much better definitions (with traces or exterior powers). In this case choose a maximal pair of isomorphic subspaces (no need to use Zorn's lemma here). By maximality the subspaces must coincide with the entire spaces. You also don't need Zorn's lemma in (1) and (2) if all spaces are finite-dimensional. – Dmitri Pavlov Jan 27 '11 at 04:38
  • @darij: I prefer to characterize finitely generated projective modules as dualizable objects in the category of modules and finitely generated modules as factorobjects of dualizable objects. I don't think that maps between direct sums of modules are that important to justify the introduction of matrices. – Dmitri Pavlov Jan 27 '11 at 05:00
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    @Dmitri's answers to Johannes' questions: that's really great! That would be an interesting book... – Dr Shello Jan 27 '11 at 05:04
  • @Dmitri: (1) and (2) are elegant arguments (but they use bases in disguise). I am not satisfied with (3). Your definition of dimension is really the same as the standard one (there exists a basis of length $n$), and the work to be done is then that each finitely generated vector space has a dimension (or, equivalently, that any subspace of $k^n$ is isomorphic to some $k^m$). The definition of the dimension via the trace is circuitous since you have to know that the natural map $V^* \otimes V \to End(V)$ is an isomorphism in order to define the trace without matrices. – Johannes Ebert Jan 27 '11 at 13:29
  • @Johannes: I fail to see how (1) and (2) use bases in disguise. The definition via traces is not circuitous and I don't need to show that the map V*⊗V→End(V) is an isomorphism. By definition, a finite-dimensional vector space is a dualizable object in the category of vector spaces, hence its endomorphisms possess traces, which are defined here: http://ncatlab.org/nlab/show/trace – Dmitri Pavlov Jan 27 '11 at 17:48
  • This discussion betwen Johannes and Dmitri touches on one of my misgivings. If one wants to train as an operator theorist - not an operator algebraist - and hence do messy things with particualr operators, does Dmitri's presentation still possess conceptual or pedagogical advantages over the "conventional" one? I don't claim it would be worse, because I haven't though about this, but would it really be better? In my experience functional analysis can be learned quite well by going from the particular to the general and back again, although this requires more time from the teachers – Yemon Choi Jan 27 '11 at 18:42
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    @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. – Dmitri Pavlov Jan 27 '11 at 19:05
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    @Dmitri: nothing could be further from the truth, and I apologize if that is the impression you got. I misread you as arguing for the "rightness" of this approach over others, rather than (as you clarify) arguing for its "rightness" alongside others. Sometimes I want to take a basis or an approximating net and get my hands dirty; sometimes I want to think of the proj. tensor product of Banach or operator spaces as left adjoint to the internal Hom. I think that functional analysis, or at least the corner I happen to have landed in, benefits from both perspectives – Yemon Choi Jan 27 '11 at 19:57
  • @Dmitri: checking the details of (1) and (2) (i.e., that the maximal objects do what they should do) amounts to the same (simple) arguments that are used in the traditional approach. Ad (3): the trace of the identity is not a good definition of dimension, since it is wrong in positive characteristic. In characteristic $0$: how do I know that the trace is a nonnegative integer? And how do I check that a finitely generated vector space or a subspace of a finite-dimensional space is dualizable? – Johannes Ebert Jan 27 '11 at 21:23
  • I think once you try to write down all the details of that approach, you will quickly start to appreciate bases as a convenient theoretical tool for setting up the coordinate-free theory. – Johannes Ebert Jan 27 '11 at 21:27
  • @Johannes: I think we already went through this story in the case of integration on smooth manifolds. I don't think that the situation here is different. As for positive characteristinc, the definition with exterior powers works perfectly in all cases. – Dmitri Pavlov Jan 28 '11 at 21:08
  • @Dmitri: I heartily agree with you that for an important subject in mathematics, there will not be one approach which is most satisfactory to all clients. In fact, I would take it a step further: for sufficiently basic and important subjects -- like linear algebra! -- even one person needs to learn multiple approaches, so here both the basis/matrix viewpoint and the coordinate-free viewpoint. IMO a mathematician who really does not know about matrices will be at a serious professional disadvantage:... – Pete L. Clark Jan 30 '11 at 21:06
  • ...arguments and computations with matrices come up in very sophisticated branches of modern number theory, and not of course because these number theorists do not know how to think in a coordinate-free way. But even if someone absolutely never finds it helpful to think about matrices, nevertheless he needs to know about them in order to communicate with other people. Thus I think a linear algebra text entirely without matrices creates the wrong impression, and could tempt zealous youngsters to literally never learn about the basis/matrix approach. That would be very bad. – Pete L. Clark Jan 30 '11 at 21:09
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    @Pete: I guess my point here is that once you know coordinate-free linear algebra, you can learn bases, coordinates, and matrices literally in a few minutes. The reverse transition is much more difficult to make. I have seen lots of students in my linear algebra discussion section struggling with “abstract” vector spaces. The same applies to manifolds: Charts and atlases are very easy to learn once you have mastered the coordinate-free approach. Also, if you have a coordinate-free proof it is very easy to turn it into a coordinate proof. The reverse process is highly non-trivial. – Dmitri Pavlov Jan 30 '11 at 23:55
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    @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. – Pete L. Clark Jan 31 '11 at 04:24
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    Let me put it another way: at what age and point in one's development does a person realize that she is going to be the kind of mathematician that wants to think in an especially categorical / abstract / basisfree way? Many if not most American students are not even exposed to these concepts until grad school. But my (American) high school had a course in which we multiplied matrices, solved linear systems and so forth. I presume that most of the people in this course are not now pure mathematicians. So how could I have avoided learning about matrices first? (And why? What was the harm?) – Pete L. Clark Jan 31 '11 at 04:30
  • @Pete, your comment speaks to the inadequacy of highschool mathematics in the United states to teach the skills necessary in higher mathematics. I actually took out the book from the famous New Math conference in 1959 where Dieudonné set out his original ideas, and they're not at all what you would expect. In particular, the major setpiece of his programme is removing Euclidean geometry and replacing it with more depth in algebra. Before you jump to the critics' side, I suggest you read it for yourself to see if it sounds so crazy. – Harry Gindi Jan 31 '11 at 14:07
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    The reference is New Thinking in School Mathematics, and it is the proceedings from a conference. Dieudonné's paper is included in full on page 32. Anyway, the reason why I bring this up is that this sort of thinking should be taught in highschool, which makes Dmitri's approach much easier to swallow. – Harry Gindi Jan 31 '11 at 14:09
  • @Dmitri: if your students struggle with the notion of an abstract vector space, do you really think they'd struggle less with the notion of a fully dualizable object in a symmetric monoidal category? I somehow doubt it. – Johannes Ebert Jan 31 '11 at 15:24
  • @Harry: "your comment speaks to the inadequacy of high school mathematics in the United states to teach the skills necessary in higher mathematics." It does? I was educated entirely in the US, and I think I learned all the necessary skills for higher mathematics. (At any rate, right after high school I learned higher mathematics. Isn't that the point? Are you truly arguing that my learning about matrix multiplication at age 16 was an impediment to learning about abstract vector spaces at age 17 and transfinite constructions in linear algebra at 18? I was there, and I'm not buying it.) – Pete L. Clark Jan 31 '11 at 15:31
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    Also: anyway, as a general rule I would regard suggestions for textbooks to be written after a complete overhaul of American high school mathematics to be somewhat hypothetical, if not actually counterfactual. – Pete L. Clark Jan 31 '11 at 15:32
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    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. – Pete L. Clark Jan 31 '11 at 15:46
  • @Pete: There's a very nice way to think of matrices category theoretically (as I'm sure you know), where an $I\times J$ matrix is given by a family of morphisms $f_ij:A_i\to B_j$, and the induced map is given $f:\coprod_i A_i \to \prod_j B_j$. – Harry Gindi Feb 01 '11 at 10:31
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    @Harry: sure, but what is the point? Once you understand both, matrices, bases, linear map and categories, coproducts, products, this is a triviality. If you think there is a pedagogical or mathematical advantage of defining matrices in this manner, what is it? – Johannes Ebert Feb 01 '11 at 11:44
  • @Johannes: It extends one's abilities to use matrices immensely! The "basis" is really a choice of decomposition, not just a sequence of elements. – Harry Gindi Feb 04 '11 at 06:36
  • @Harry: I taught linear algebra last year. Once bases (as sets of vectors) were understood, the passage to bases as arbitrary families did not cause many problems. Moreover, after several weeks of practice with matrices, writing linear maps into the entries of a matrix wasn't a big problem, either. Writing polynomials as entries (to define the characteristic polynomial), however, is a much more subtle thing to grasp, at least if you wish to do it correctly. – Johannes Ebert Feb 06 '11 at 15:44
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    The book http://www.amazon.com/Frames-Locales-Topology-Frontiers-Mathematics/dp/303480153X/ref=sr_1_1?s=books&ie=UTF8&qid=1339202318&sr=1-1&keywords=Frames+and+Locales by Picado and Pultr is a general topology book written in the language of locales. Against your wishes this book does mention topologies. Much of this material is to point out differences between topologies and locales. Since the likely audience for this book will likely have some knowledge of topology I don't think one can expect the authors to do otherwise. – Jay Kangel Jun 09 '12 at 00:46
20

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.

John D. Cook
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20

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.

Martin Sleziak
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Marko Amnell
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  • A recent commentary by Edward Frenkel on Gelbarts article: http://www.ams.org/journals/bull/0000-000-00/S0273-0979-2011-01347-7/S0273-0979-2011-01347-7.pdf – Thomas Riepe Jul 25 '11 at 14:45
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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.

Yemon Choi
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  • There is Joe Taylor's article in a conference proceedings, circa 1972?? – David Handelman Mar 17 '20 at 18:08
  • @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 – Yemon Choi Mar 17 '20 at 19:33
19

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.)

Ariyan Javanpeykar
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    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. – peterx Oct 09 '15 at 23:59
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I'm surprised that nobody has expressed the desire to read Bourbaki's Théorie des nombres.

Chandan Singh Dalawat
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    Some might be surprised that anyone has the desire to read Bourbaki at all ;-) – Johannes Hahn Jan 27 '11 at 14:42
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    @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.) – Pete L. Clark Jan 30 '11 at 21:22
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    Johannes, =p! – Harry Gindi Jan 31 '11 at 15:12
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Introduction to algebraic cycles.

With lots of examples...

Andreas Holmstrom
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    How distant is Eisenbud and Harris book "3264 and All That: A Second Course in Algebraic Geometry" to your desired text? – Leo Alonso May 30 '17 at 15:33
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"Cardinal Arithmetic: The New Corrected Edition (including index)" by Saharon Shelah...

Todd Eisworth
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15

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.

Lorenzo
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    Dear Lorenzo, What is your objection to SGA 4.5 (which is my personal favourite of the SGAs)? – Emerton Jan 25 '11 at 03:36
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    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? – Harry Gindi Jan 31 '11 at 15:17
14

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

David Hansen
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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
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    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
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    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
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    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 04:33
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    ... 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 04:39
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    ... go further with the theory; they are not really simplifications (as opposed to modifications) of the argument (in my view). If one is at the point of wanting to understand Hecke algebras in a hands-on way via Jacquet--Langlands (although I'm not sure exactly what you mean by this; I've always found Hecke algebras on spaces of modular forms to be pretty hands-on things), then you have gone beyond an amateur interest, and should just learn the subject the way you learn any other subject, beginning with the available existing text-book sources and then moving onto the research literature. – Emerton Jan 27 '11 at 04:42
  • Dear Emerton: Then I will do that. Many thanks for your insight and perspective! – David Hansen Jan 27 '11 at 05:16
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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.

Tim Campion
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    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. – David Handelman Mar 17 '20 at 18:01
13

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.

  • There is also the book Emerton referred to in a comment above. It's called "Mathematical Developments Arising from Hilbert Problems" and it is published by the AMS. – Gonçalo Marques Mar 06 '11 at 15:02
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    Irving Kaplansky wrote a review of the AMS volume mentioned by G. Marques: https://projecteuclid.org/euclid.bams/1183538903; see his list of references for more. Kaplansky also wrote his own volume on the Hilbert problems: http://www.amazon.com/Hilberts-problems-mathematics-University-Mathematics/dp/B00073D4F8, but I don't know the status of that work or how it was received. – Todd Trimble Oct 09 '15 at 22:31
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The book I would love to read (and own) is called:

HIGHER MATHEMATICS FOR THE IMPATIENT. (HMFTI)

Description: the book appeals to all math aficionados who have at least a solid ground in basic advanced math (say at the level of standard qualifying math department examinations in the USA).

Now, it is sort of similar to the Princeton Companion of Mathematics, but without the biographies, and in which the articles are full-fledged chapters, with a core introduction (main idea, main results) and a LOT of worked out key examples (take note of the word key, I would like to have only examples which help me grab the essence of the field, and nothing else).

So, just to give you a flavor, take Algebraic K-theory. At some point I kinda knew what it is, but I would love to grab HMFTI, get myself a glass of brandy, a churchwarden pipe, a notebook and a pen, and read the chapter, doing the exercises just enough to have a full sense of the field. Same for the other chapters, say Simplectic Geometry, Orbifolds, Finite Geometry, Higher Categories, etc.

PS This book would be -I think- REALLY nice, but ain't easy to write, in fact extremely hard. You know why? Because it is much much easier writing a text in a field stuffed with all the latest results than a brief introduction which conveys the essentials and nothing but the essentials, all the while being no popularization

Mirco A. Mannucci
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"Quiver varieties with a wealth of examples" ?

Yann Palu
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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.

Holzinger Raphael
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9

Inter-Universal Geometry for dummies.

user261406
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9

Nonlinear Differential Galois Theory

This would provide an account of the ideas of Umemura and Malgrange, and their relationship with monodromy-preserving deformations & nonlinear pde.

Failing that, I'd settle for an updated edition of Pommaret's Differential Galois Theory.

Phil Harmsworth
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AD${}^+$ by Hugh Woodin.

Andrés E. Caicedo
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    Actually, I'd vote for the "The collected unpublished works of the California Set-theorists" multiple times, if it were allowed! – Todd Eisworth Oct 25 '11 at 20:50
8

    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.

Incnis Mrsi
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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 H Dec 31 '15 at 21:25
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An English translation of Curtis and Reiner, Methods of representation theory with applications to finite groups and orders would be nice.

Seamus
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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 Mathematical 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)
praeseo
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The algebra, geometry, and combinatorics of Total Positivity

This is an area which has seen a ton of interest in the last 30 or so years, but there is no canonical textbook.

Possible topics include:

Sam Hopkins
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As a counterpoint to gowers' answer, here's a different book -- or rather book series -- on physics which I'd like to see.

For the series, one would select a series of widely-used physics textbooks, and for each one, would write comprehensive mathematical footnotes, appendices, and references to more complete mathematical texts.

This way, one could skim the physics textbook at a first pass to get an idea of how physicists think about the subject matter, and then come back to the footnotes and appendices to get a sense for

  • at which points there are mathematical details / inconsistencies being glossed over (e.g. boundary conditions, path space measures, existence & uniqueness of various representations, ...)

  • what the larger context is for some of the methods being used (e.g. Hamlitonion / Lagrangian / Legendre transform, ladder operators and theory of weights, spectral theory, differential geometry, ...)

  • where conventions differ between physicists and mathematicians (terminology and bases for Lie algebras, differential geometry, etc.)

  • explanations and justifications of what is meant when a physicist says things like "the most general expression you can write down satisfying certain constraints (some of them implicit) is ____"

  • etc.

For these purposes, I think it would be best to select physics textbooks which are minimally rigorous / truly introductory: in my experience with physics textbooks, often when they make some attempts at mathematical rigor, they just bog down the storyline and end up doing the mathematics badly anyway. I think it would be preferable to just see the physicist do things their own way, and when more mathematical aspects come up, to just punt them to an actual mathematician for further explanation.

Tim Campion
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A Bourbaki book on Category Theory.

A413
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(Counter)examples in scattering theory

2

Perhaps books along the lines of the Oxford series of "very short introductions" to various topics. This includes the masterful Very Short Introduction to Mathematics by Timothy Gowers. But one could imagine narrowly focussed, specific topics in mathematics, perhaps no more than 150 pages, on many of the topics that have been suggested in response to this post. The closest similar suggestion is Mirco Mannucci's "higher mathematics for the impatient."

Alas, unlikely to be commercially viable.

Joseph O'Rourke
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Pitfalls in infinite dimensional (differential) geometry

Nowadays there exists a plethora of results on Riemannian, symplectic and Poisson geometries on infinite dimensional manifolds. While classical texts cover much of the theory up to Banach spaces (e.g Langs Fundamentals of diff geom.) and some nice survey articles exist (the articles by Bruveris on Riem. Geom. for example) a comprehensive guidebook would be nice. Some of the material is of course covered in Kriegl, Michors big book on convenient calculus, but the frequent questions on MO on the topics (let alone the discussions I had with graduate students over the years) indicate that this topic deserves a book.

Alexander Schmeding
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Variational PDE's and Lagrangian field theory for mathemticians

Also

Hodge theory of asymptotic expansions of oscillatory integrals and resurgence

Saal Hardali
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I think for graduate courses, it would be nice to have a graduate textbook on "stochastic partial differential equations" at the same exposition level as the usual graduate pde's textbook by Evans. So many examples and lots of instructive exercises.

For example, as mentioned here: Good books on stochastic partial differential equations? the main one is the Stochastic Equations in Infinite Dimensions by da Prato and Zabczyk. But it doesn't have any lists of exercises.

There are many good monographs eg. the one by Hairer An Introduction to Stochastic PDEs.

I also like the book "a course on rough paths" which goes into SPDEs too.

Thomas Kojar
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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)

Anonymous
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