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I've been studying math as a hobby, just for fun for years, and I had my goal to understand nearly every good undergraduate textbook and I think, I finally reached it. So now I need an another goal. I've just found a very nice book /S. Ramanan – Global Calculus/ from the "Graduate Studies in Mathematics" series and it looks nearly awesome:

  • Sheaves and presheaves
  • Differential manifolds
  • Lie groups
  • Differential operators
  • Tensor fields
  • Sheaf cohomology
  • Linear connections
  • Complex manifolds
  • Ricci curvature tensor
  • Elliptic operators

But it's only 316 pages and it seemed not very fundamental and detailed for me (but yes, it's still great). So here's my question: what huge complicated calculus textbooks like this one do you know that I should aim to understand? The Big Creepy Books, you know :) I'm very interested in algebraic and differential geometry, general and algebraic topology, Lie groups and algebras, pseudo- and differential operators. I don't know very much about all of this yet but I'm trying so hard to do, it's so exciting! ;)

I've already covered:

  • Algebra: Chapter 0 (Graduate Studies in Mathematics) by Paolo Aluffi
  • A Course in Algebra (Graduate Studies in Mathematics, Vol. 56) by E. B. Vinberg
  • Linear Algebra and Geometry (Algebra, Logic and Applications) by P. K. Suetin, Alexandra I. Kostrikin and Yu I Manin
  • Topology from the Differentiable Viewpoint by John Willard Milnor
  • Topology and Geometry for Physicists by Charles Nash and Siddhartha Sen
  • Mathematical Analysis I and II by V. A. Zorich and R. Cooke
  • Complex Analysis by Serge Lang
  • Ordinary Differential Equations by Vladimir I. Arnold and R. Cooke
  • Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics) by Sigurdur Helgason

So I'm looking for something like Ramanan's book, but maybe more detailed and fundamental.

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    $\begingroup$ If all you want is something very long and difficult to understand, I can recommend Woodin's The axiom of determinacy, forcing axioms, and the nonstationary ideal. Over 800 pages of difficult set theory, which will probably require you to first learn through the first 500 pages (or so) of other books! All in all, over a thousand pages of claims, proofs, and technical details! $\endgroup$ – Asaf Karagila♦ Feb 19 '13 at 10:59
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    $\begingroup$ This question really should be Community Wiki. I will convert it if you have no objection. $\endgroup$ – robjohn♦ Feb 19 '13 at 11:19
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    $\begingroup$ @Thomas It will be helpful if you specify which books you have already covered. $\endgroup$ – user14082 Feb 19 '13 at 11:36
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    $\begingroup$ Can I just ask: What's the point of going after the big and hairy books? To me, it seems like it's mostly about bragging or self esteem. I could point you to some really hairy books, but if you really wanted to learn the stuff, I would suggest something else. $\endgroup$ – Matsemann Feb 19 '13 at 18:09
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    $\begingroup$ Dear @Matsemann: what's the point of running a marathon? What's the point of climbing Mount Everest? Trying to learn difficult mathematics is a very laudable aim, not correlated to bragging. After all, laymen would be more impressed if I told them that I can solve any exercise in a calculus book than if I bragged that I can use the Hodge-de Rham spectral sequence for computing the algebraic de Rham cohomology of a scheme! $\endgroup$ – Georges Elencwajg Mar 2 '13 at 9:45

10 Answers 10

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As a cure for your desire I recommend a few pages per day of Madsen and Tornehave's From Calculus to Cohomology.
The book starts at the level of advanced calculus and introduces an amazing set of concepts and results: de Rham cohomology, degree, Poincaré-Hopf theorem, characteristic classes, Thom isomorphism, Gauss-Bonnet theorem,...

The style is austere but very honest: none of the odious "it is easy to see" or "left as an exercise for the reader" here!
On the contrary, you will find some very explicit computations rarely done elsewhere : for an example look at pages 74-75, where the authors very explicitly analyze the tangent bundles and some differential forms on numerical spaces $\mathbb R^n$, spheres $S^{n-1}$ and projective spaces $\mathbb R\mathbb P^{n-1}$.

All in all, a remarkable book that should appeal to you by its emphasis on algebraic topology done with calculus tools .

Edit:September 25, 2016
An even more complicated book is Nickerson, Spencer and Steenrod's Advanced Calculus, of which you can find a review by Allendoerfer here.
The book is an offspring of lecture notes given in Princeton around 1958 for an honours course on advanced calculus.
The book starts very tamely with an introduction to vector spaces in which students are requested to prove (on page 5) that for a vector $A$, one has $A+A=2A$.
On page 232 however (the book has 540 pages) the authors introduce the notion of graded tensor algebra, on page 258 Hodge's star operator, then come potential theory, Laplace-Beltrami operators, harmonic forms and cohomology, Grothendieck-Dolbeault's version of the Poincaré lemma and Kähler metrics.
I suspect that a teacher who tried to give such a course at the undergraduate level today would run a serious risk of being tarred-and-feathered.

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    $\begingroup$ Wow! It looks very nice! $\endgroup$ – Thomas Feb 19 '13 at 12:09
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The obvious "scary book" for linear partial differential operators (and pseudodifferential operators, and Fourier integral operators, and distribution theory and...) is

Hörmander, Lars, Analysis of linear partial differential operators, 1-4, Springer Verlag.

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    $\begingroup$ For general knowledge and education for those who do not know this: Hörmander is a Fields medalist. $\endgroup$ – Rudy the Reindeer Feb 19 '13 at 12:04
  • $\begingroup$ It looks very nice :)Is there something with more topology? I like topology ;) $\endgroup$ – Thomas Feb 19 '13 at 12:05
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    $\begingroup$ For even more general knowledge, Hörmander was my mathematical great grandfather ;) $\endgroup$ – mrf Feb 19 '13 at 12:09
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The five volume set by Spivak should keep you busy for a while.

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    $\begingroup$ You mean A Comprehensive Introduction to Differential Geometry (5 Volume Set), right? $\endgroup$ – Thomas Feb 19 '13 at 13:09
  • $\begingroup$ Yes thats correct. $\endgroup$ – nonlinearism Feb 19 '13 at 15:13
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Another ambitious book is Raghavan Narasimhan's Analysis on real and Complex Manifolds
It consists of three chapters.
Chapter 2 contains more or less standard material on real and complex manifolds, but the other two chapters are quite unusual:

Chapter 1 contains some hard analysis on $\mathbb R^n$.
You will find there some classical results like Sard's theorem but also tough results like Borel's theorem, according to which there exists a smooth (generally non-analytic) function with any prescribed set of coefficients as its Taylor development at the origin.
In other words, the morphism of $\mathbb R$-algebras $ C^\infty(\mathbb R^n) \to \mathbb R[[x_1,...,X_n]]$ from smooth functions to formal power series given by Taylor's formula is surjective .
The chapter also contains theorems of Whitney approximating smooth functions by analytic ones: these results are very rarely presented in books.

Chapter 3 is devoted to linear elliptic differential operators.
As an application of that theory, Narasimhan proves Behnke-Stein's theorem (it is the last result of the book) according to which every non-compact connected Riemann surface is Stein.

This is a difficult but great book by a great mathematician.

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Try Jean Dieudonné, Treatise On Analysis, 9 volumes.

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  • $\begingroup$ Thanks for advice, but it seemed too primitive for me :(I mean is there something like Ramanan's book (by difficulty) but more detailed and maybe fundamental? $\endgroup$ – Thomas Feb 19 '13 at 11:18
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    $\begingroup$ I thought of the English translation of his multi-volume opus "Traité d'Analyse". As far as I know, the "primitive" one is the first volume, but there are 8 more of them which -- certainly -- are not primitive. $\endgroup$ – azimut Feb 19 '13 at 11:23
  • $\begingroup$ Oh, I'm sorry. I saw only the first 3 volumes, and they're just like the second volume of Zorich, Cooke – Mathematical Analysis II. Maybe you're right about the complexity of other 5 volumes. $\endgroup$ – Thomas Feb 19 '13 at 11:35
  • $\begingroup$ @Thomas Yes and there are a lot of good exercises (with very few misprints because the treatise has been taught and performed in ENS and faculties many times) with bordering examples and counterexamples. $\endgroup$ – Duchamp Gérard H. E. Jan 1 '17 at 7:20
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I propose "Principles of Algebraic Geometry" by Phillip Griffiths and Joseph Harris. The "Foundational Material" Chapter 0, which includes a proof of the Hodge theorem, will hopefully satisfy your appetite for differential geometry and analysis of (pseudo-)differential operators. I don't know if the rest of the book really classifies as either "topology" or "differential geometry", as in my opinion "complex geometry" really has an entirely different flavour, but there is certainly a lot of immensely interesting material, which is at least highly connected to topology as well as differential geometry (Lefschetz theorem, Chern classes, fixed point formulas, ...)

Have fun!

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  • $\begingroup$ Thank you! You recommendation is really good, the book is very interesting. $\endgroup$ – Thomas Feb 19 '13 at 14:01
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Although it is not that huge (~350p) I still recommend to take a look at Differential Forms in Algebraic Topology by Raoul Bott & Loring W. Tu.
I personally like it a lot and I think that it might suit you well, especially since you've already covered Milnor's Topology from the Differentiable Viewpoint.

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I love Morita's "Geometry of Differential Forms". Don't remember whether it covers the linear algebra prerequisites (alternating tensor product) thoroughly or not and while those are just linear algebra, they aren't covered in most books and having them down really solidly makes a dramatic difference in how difficult or easy differential forms is. Regardless, I remember it as a beautiful book.

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    $\begingroup$ His description of the aim of differential geometry in the introduction to that book is just amazing. $\endgroup$ – James S. Cook Apr 14 '14 at 2:31
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I'd go for Rudin's "Real and Complex Analysis" and "Functional Analysis"

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    $\begingroup$ It's interesting, but there's too little topology, no differential forms, no manifolds etc. :( But thanks for advice! $\endgroup$ – Thomas Feb 19 '13 at 11:36
  • $\begingroup$ There are differential forms in his introductory real analysis. $\endgroup$ – superAnnoyingUser Feb 19 '13 at 11:45
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An interesting read that would give you considerable insight into some natural applications of mathematics in physics would be the three volume set Group Theory in Physics by John F. Cornwell. I think it is fair to say these contain some calculus which is probably not covered by the other beautiful books mentioned in answers here. In a similar vein, you might look at Quantum Fields and Strings: A Course for Mathematicians (2 Volume Set) (v. 1 & 2) by Pierre Deligne, David Kazhdan, Pavel Etingof, John W. Morgan, Daniel S. Freed, David R. Morrison, Lisa C. Jeffrey and Edward Witten. That'll keep you occupied for some time.

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