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

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.)

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 ?

• I really like this question... hopefully someone will take a hint and write number (5) and (2) sometime soon! Commented Jan 24, 2011 at 10:30
• 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. Commented Jan 24, 2011 at 12:44
• Qiaochu: Demazure and Gabriel wrote a book using the functor of points approach over 3 decades ago. Some people love this book, while others... Commented Jan 24, 2011 at 17:55
• 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! Commented Feb 1, 2011 at 0:07
• 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. Commented Feb 1, 2011 at 15:03

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

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.

Inter-Universal Geometry for dummies.

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.

AD${}^+$ by Hugh Woodin.

• Actually, I'd vote for the "The collected unpublished works of the California Set-theorists" multiple times, if it were allowed! Commented Oct 25, 2011 at 20:50

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.

• 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. Commented Dec 31, 2015 at 21:25

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

• An English translation? Commented Jan 27, 2011 at 14:25
• I think Seamus was ironic ;-) Commented Feb 2, 2011 at 12:30
• That was the easy part. But what's the joke about? Commented Feb 3, 2011 at 17:02

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

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

1. Continuum Hypothesis Part I and II with a chapter headed The Art of Forcing
2. Five Pillars of 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)

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:

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.

A Bourbaki book on Category Theory.

(Counter)examples in scattering theory

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.

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.

Variational PDE's and Lagrangian field theory for mathemticians

Also

Hodge theory of asymptotic expansions of oscillatory integrals and resurgence

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)

• There is a forum for this: roadtoreality.info But probably you know that Commented Feb 2, 2011 at 11:02

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.