Examples of great mathematical writing

Question

This question is basically from Ravi Vakil's web page, but modified for Math Overflow.

How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to name as many examples of "great writing" as possible. Asking for "the best article you've read" isn't reasonable or helpful. Instead, ask yourself the question "what is a great article?", and implicitly, "what makes it great?"

If you think of a piece of mathematical writing you think is "great", check if it's already on the list. If it is, vote it up. If not, add it, with an explanation of why you think it's great. This question is "Community Wiki", which means that the question (and all answers) generate no reputation for the person who posted it. It also means that once you have 100 reputation, you can edit the posts (e.g. add a blurb that doesn't fit in a comment about why a piece of writing is great). Remember that each answer should be about a single piece of "great writing", and please restrict yourself to posting one answer per day.

I refuse to give criteria for greatness; that's your job. But please don't propose writing that has a major flaw unless it is outweighed by some other truly outstanding qualities. In particular, "great writing" is not the same as "proof of a great theorem". You are not allowed to recommend anything by yourself, because you're such a great writer that it just wouldn't be fair.

Not acceptable reasons:

• This paper is really very good.
• This book is the only book covering this material in a reasonable way.
• This is the best article on this subject.

Acceptable reasons:

• This paper changed my life.
• This book inspired me to become a topologist. (Ideally in this case it should be a book in topology, not in real analysis...)
• Anyone in my field who hasn't read this paper has led an impoverished existence.
• I wish someone had told me about this paper when I was younger.

Answer 1

Anything by John Milnor fits the bill. In particular, "Topology from the differential viewpoint" made me feel that I understand what differential topology is about, and the "h-cobordism theorem" made me feel that it's beautiful. Many other books and papers by him are wonderful; the first that come to mind are "Characteristic Classes", "Morse Theory", lots of things in Volume 3 of his collected papers.

Answer 2

True story: When I was about to move to Stony Brook to start my PhD, one of my professors took me aside to tell me "You know, when I was a student Milnor was god, and Morse Theory was the bible." I found that nice and moved on, but a little later a younger professor took me aside to say "You know, when I was a student Milnor was god, and Introduction to Algebraic K-Theory was the bible." By then I knew that something was going on, but I was still taken by surprise when a more junior professor found me and said "You know, when I was a student Milnor was god, and Characteristic Classes was the bible."

Of course this was all planned. They succeeded in motivating me to take every opportunity to talk to and learn from the big names I met. But they made another point that I only recognized later, while writing my first paper: If you want to learn to write Mathematics well, read anything by Milnor.

When I was a student, Dynamics in One Complex Variable was the bible.

Answer 3

Canonical submission: Anything by J.-P. Serre (e.g., Local Fields, Trees, Algebraic Groups and Class Fields,...). Reasons:

• I can't get enough of Trees, chapter 2. I spent a year working on automorphic forms on function fields in part because of this book (it didn't work out well, but that's another story).
• Peer pressure: several people (including my Ph.D. advisor) have told me that if I were to choose a role model for writing style, I should choose him.
• Mundane reasons: His writing is incredibly clear and concise, but not so brief as to be confusing. He has a keen eye for what is important in a theory or construction. He doesn't waste words having a conversation with the reader or expounding on his philosophy of mathematical practice.

Answer 4

The book of Bott-Tu "Differential Forms in Algebraic Topology" was my door to enter the magic world of cohomology, Chern classes and similar topics. Moreover, it contains a wonderful (and in my opinion the best) exposition of spectral sequences with applications to the computation of some higher homotopy groups of the sphere. All that is presented in a self-contained way and in a magnificent style. A masterpiece!

Answer 5

Real Analysis by Elias Stein and Rami Shakarchi

I absolutely hated analysis until I read the Stein/Shakarchi analysis series (Fourier, Complex, Real Analysis). Now I find the subject to be very beautiful and full of deep ideas, and it is these books that really convinced me.

Answer 6

Michael Spivak's Calculus made me want to study analysis.

Answer 7

I think Algebraic Topology by Hatcher is one of my early favourites. It starts off being very basic but it manages to mention so much fascinating stuff, and I think the exposition is great. Definitely inspired me and got me interested in algebraic topology.

His book in progress "Vector Bundles and Characteristic Classes" is also very nice.

Answer 8

John Lee's Introduction to Smooth Manifolds. This book reminded me of all the mathematics I kinda learned in undergrad, prepared me for graduate school, and taught me differential topology. I feel like every undergrad should have this book and work through it on their own.

Answer 9

Just like in this thread, I am amazed that no one mentions Deligne. I think it was Illusie who said Grothendieck had a gift to build new theories and new language while Serre's talent was to find new things to do with old tools. Deligne got the generality, abstraction and theory building from Grothendieck and the clarity of exposition and the constant reference to older language/simple ideas from Serre. I think that's why he is sometimes overshadowed by his elders when someone asks this kind of question.

Here's a few examples. His "Théorie de Hodge I" explains the "yoga of weights" in just a few pages. The first sections of "La conjecture de Weil I" provide a great survey of both the theory of étale cohomology and Lefschetz theory for algebraic varieties almost from scratch. Another masterpiece is his "Le groupe fondamental de la droite moins trois points" where he builds a whole theory unifying several aspects of arithmetics, topology and differential equations but always comes back to very down to earth examples. Not to mention, his Bourbaki lectures or the uncountable number of private communications of his cited in the literature.

If you are looking for great examples of mathematical writing, you should definitely read some articles by Deligne.

Answer 10

Walter Rudin's Real and Complex Analysis has long been one of my favorites. Like Serre, Rudin seems to strike a nice balance for detail, and his proofs are always slick and fun to read; I became heavily interested in analysis after reading that one.

Answer 11

One of the math books I enjoy reading in most is Neukirch's book "Algebraic Number Theory". In my opinion, he presents the material beautifully and with a good degree of generality for a text book. Also, he manages to use language beautifully without losing mathematical rigor and without compromising clarity (this holds for the German version as well as for the English translation). When I have to look up some fact from algebraic number theory, Neukirch is usually the first book I try.

Answer 12

I love Grothendieck's Tohoku paper.

Answer 13

Without a doubt, Arnold's Mathematical Methods of Classical Mechanics is the book most responsible for me deciding to be a geometer. Only some papers of Atiyah were able to replicate the feeling of awe I had reading Arnold's classic as an impressionable green undergrad. Very few authors are able to convey to me the feeling of completely unconstrained thinking as Arnold's writings do. They continue to be the go to place whenever if feel stuck or stale in my research. A few pages from him still do the trick: they remind me why I became a mathematician.

Answer 14

I'm not sure whether "expository" writing counts, but I'll go with it anyway...

I don't think it would necessarily change the life of anybody who was already into mathematics enough to pay for it, but I very much wish the Princeton Companion had come out when I was younger. You don't get the chance to get your hands dirty with the details of any of the topics the PCM covers, but sometimes you're not looking to get your hands dirty, and there's not much else of any quality that can compare in terms of breadth.

Answer 15

Proofs from the Book, Martin Aigner, Günter M. Ziegler, 2000.

Anyone in [mathematics] who hasn't read this [book] has led an impoverished existence.

Answer 16

Éléments de géométrie algébrique (EGA) (full text available from numdam) continues to be an inspirational text for me. I wish I'd started reading it earlier.

It's come up in a few other places here on MO. To quote Jonathan Wise's answer to another question,

Virtually every page I've read of EGA/SGA has been useful to me, and almost every page I've skimmed I've later wished I'd read in more detail. The reputation for difficulty is, I think, unfounded. They are certainly abstract, but virtually every detail is present; in many ways, that makes EGA/SGA easier to read than other sources. Opening a volume and reading a sub-paragraph from the middle can be difficult because of all the back-references, but reading linearly can be very pleasant and rewarding. The French language may be a barrier for some, but one doesn't have to "learn French" to learn enough to understand EGA.

Answer 17

Silverman and Tate's "Rational Points on Eliptic Curves."

Answer 18

Silverman's The Arithmetic of Elliptic Curves got me interested in that area for some time, too. The exposition is fun to read, with both motivation and rigorous proofs.

Answer 19

Mumford's "The Red Book of Schemes and Varieties" was the first book trying to explain Grothendieck's new theory of schemes to the large public. It does this with a lot of examples from the 'real life' and even with drawings! It is far from complete, but it remains the best for communicating the love for the subject and for the clearness of the exposition. Another masterpiece!

Answer 20

This is not an example of great mathematical writing, but definitely something to know and probably even more useful: Paul Halmos' brilliant essay "How to write mathematics", which you'll find for example at http://i11www.iti.kit.edu/~awolff/lehre/scientific_writing/h-hwm-70.pdf.

Answer 21

Gowers' Mathematics: A Very Short Introduction. This is in Oxford's series of "very short introductions" on a variety of topics (hieroglyphics, film, Rousseau,...), each of which I think is a tremendous challenge to the (invariably eminent) writer. Gowers dispenses with "anecdotes, cartoons, exclamation marks, jokey chapter titles," and instead plunges right into details without apology. The chapter on proofs is especially important for nonmathematicians to understand. The explanations of concepts (e.g., "dimension") are lucid, achieving clarity without compromising on technical accuracy. I read it in one sitting, which may dismay the author who must have labored over these small 160 pages, but which is a testimony of how smoothly he conveys his insights.

Answer 22

How about Munkres Topology? This book certainly made me want to be a (point-set) topologist. Turns out I came along a bit late for that field, but I'm sure this book helped push me into algebraic topology. Anyway, Munkres is full of fantastic examples and pictures, it treats all the major aspects of the field, and it seems to be the most popular book for courses in point-set topology all over the US.

Answer 23

Atiyah & Bott's paper "The Yang-Mills Equations on Riemann Surfaces" is probably the most satisfying thing I've read. The writing is great, and the ideas are all cool.

Answer 24

I'm surprised that nobody's mentioned almost anything by Emil Artin. His little monographs on Galois Theory and the Gamma Function are thrilling to read. They are so clear, and use the minimum necessary (but not more -- to paraphrase Einstein) I found them inspiring. Also his "Algebraic Numbers and Algebraic Functions" and "Geometric Algebra".

Another book, is G. H. Hardy's "Pure Mathematics". That's the book that I really learned analysis from (when I told that to Pat Gallagher he exclaimed that I was really lucky) when I was in high school. Reading that cemented my feeling that I wanted to be a mathematician.

Answer 25

Every differential geometer should read at least the first two volumes of Spivak's A Comprehensive Introduction to Differential Geometry. In particular, volume 2 is an absolute gem. Not only does it reprint (translations of) original papers by Gauss and Riemann, complete with very enlightening notes and commentary, but (if I remember correctly) Spivak presents about 5 or 6 different proofs that a Riemannian manifold is flat if and only if it is locally isometric to Euclidean space. This gives the reader the best, most intuitive grasp of the concept of curvature that I have seen anywhere. (I am a firm believer in learning by repetition...)

Answer 26

Gian-Carlo Rota's On the foundations of combinatorial theory I: Theory of Möbius Functions is an eye-opening gem. The same is true of practically every paper in Gian-Carlo Rota on Combinatorics, so consider this post a vote for the entire book. (If I become a combinatorialist, it will be because of this book.)

Answer 27

"Algebraic curves and Riemann surfaces" by Rick Miranda is one of my favorite books. It is full of concrete examples and is full of very clear explanations for everything from the basics of Riemann surfaces and their projective embeddings though sheaf cohomology. Also, it assumes little more than elementary complex analysis.

Answer 28

Since you wanted to learn writing from examples: J. Kock and I. Vainsencher's book "An Invitation to Quantum Cohomology" is wonderful reading, simply because of its incredibly friendly style. It gives you the feeling that the authors take you by the hand and lead you through their garden of wonders (always uphill of course). The achievement of the book is to give you lots of intuition - for moduli stacks, strategies for proofs in enumerative geometry, the necessity of a virtual fundamental class, how generating functions work... This is something very difficult to do in mathematical writing - in this respect you could compare it to John Baez's blog, only that it is a longer, coherent book on a single subject.

Answer 29

Paolo Aluffi's Algebra: Chapter 0 presents the usual algebra material with special emphasis on category theory and homological algebra. It's written in a somewhat informal style that spurs on the reader and motivates constructions really well. I found it much better than Hungerford (which is often touted as an example of good writing itself); it is responsible for my interest in algebra and (the rudiments of) algebraic geometry.

Answer 30

Dror's paper Khovanov's homology for tangles and cobordisms is one of the papers I loved back when I hated all math papers. In particular it's a paper that has a really good use of diagrams, a lot of papers use too few diagrams and suffer a lot for it.

Answer 31

Van der Waerden's Algebra. I became a mathematician because of this book.

Answer 32

Many of Atiyah's papers, especially those with Bott, are truly inspiring -- not only "The Yang-Mills Equations on Riemann Surfaces" but also "The Moment Map and Equivariant Cohomology" and "Convexity and Commuting Hamiltonians." He has a knack for writing in a style that, while not rigorous, allows the reader to fill in all rigorous details, and at the same time communicates high excitement.

His textbook Introduction to Commutative Algebra, written with Macdonald, is like a volume of poetry. I would guess (from internal stylistic evidence) that it was mostly written by Macdonald, but it's all great.

Answer 33

Some of my favourite authors are Serre, Mumford, Milnor, Fulton and Neukirch. These names are rather mainstream, I guess. But the most beautiful analysis textbook I ever read must be "Analysis now", by Gert Pedersen. Alas, the book is not as well known as it should be.

Answer 34

Well, for me Hartshorne was really the window into the brave new world — and yes, it fits several items from 'acceptable reasons'.

Though this prize should be shared with everyone else who was creating abstract algebraic geometry and scheme theory in the past century or so (I spare you the history, you already know it :) )

Answer 35

In my opinion Atiyah's paper "vector bundles over an elliptic curve" is a gem that everyone interested in algebraic geometry should read.

Answer 36

One of my all-time favorites for rigour and clarity:

Vistoli's Notes on Descent.

Answer 37

1. Milne's entire set of notes (algebraic number theory, class field theory, algebraic groups, complex multiplication, modular functions and modular forms, etc.), articles (abelian varieties, Jacobian varieties, Shimura Varieties, Tannakian Categories, etc.) and books (Elliptic Curves, Arithmetic Duality Theorems, Etale Cohomology etc.), available at www.jmilne.org/math/. They are indispensable for anyone who wishes to learn the fundamental concepts in arithmetic geometry. The Storrs lectures on Abelian Varieties and Jacobian Varieties are clear, succinct and give great references throughout. His notes on 'Class Field Theory' are superbly written. 'Etale Cohomology' is a standard reference for the subject, although I find his lecture notes on the same topic even more enjoyable. Finally, 'Arithmetic Duality Theorems' is quite possibly the only reference where one can find complete proofs of Tate's Duality Theorems as well as their generalizations using etale and flat cohomology.

2. Part 4 (particularly Chapter XX) of Lang's 'Algebra'. I may have learned (as little as I have) about homological algebra from Weibel or Gelfund-Manin as texts, but I always keep coming back to Lang's exposition. Not a lot of motivation, but it covers almost everything you need to know in a first course.

Answer 38

I'm shocked no one mentioned T.Y. Lam. He is perhaps the clearest expositor I've ever read, he motivates and gives history in every section, he fills his books with great examples and problems, and I have yet to find any errors. Indeed, his exercises are often finding counterexamples for errors in other published works, e.g. the following in Lectures on Modules and Rings:

In a ring theory text, the following statement appeared: "If $0\rightarrow C\rightarrow Q\rightarrow P\rightarrow 0$ is exact with $C$ and $Q$ finitely generated then $P$ is finitely presented" Give a counterexample.

Answer 39

Riemann's paper, "On the number of primes less than a given magnitude," is the reason why I decided to study mathematics (at the graduate level and beyond). I read the paper as an undergraduate and I was very impressed by the techniques that Riemann used to study the properties of the prime counting function. In particular, I was blown away by Riemann's use of complex analysis, Fourier analysis, and asymptotic analysis to study a problem in number theory, which I thought was a distant area of mathematics. This paper is truly a work of art and is less than 10 pages.

Answer 40

I've always really enjoyed the papers by Graeme Segal. They are short and I often feel like they have been distilled down into the essence of what is important. I keep going back to them and extracting new nuggets of beauty.

Answer 41

The American Mathematical Society awards the Leroy P. Steele Prize, in part for recognition of mathematical exposition:

The Leroy P. Steele Prize for Lifetime Achievement The Leroy P. Steele Prize for Mathematical Exposition The Leroy P. Steele Prize for Seminal Contribution to Research

These prizes were established in 1970 in honor of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein, and are endowed under the terms of a bequest from Leroy P. Steele. From 1970 to 1976 one or more prizes were awarded each year for outstanding published mathematical research; most favorable consideration was given to papers distinguished for their exposition and covering broad areas of mathematics. In 1977 the Council of the AMS modified the terms under which the prizes are awarded. Since then, up to three prizes have been awarded each year in the following categories: (1) for the cumulative influence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through Ph.D. students; (2) for a book or substantial survey or expository-research paper; (3) for a paper, whether recent or not, that has proved to be of fundamental or lasting importance in its field, or a model of important research. In 1993, the Council formalized the three categories of the prize by naming each of them: (1) The Leroy P. Steele Prize for Lifetime Achievement; (2) The Leroy P. Steele Prize for Mathematical Exposition; and (3) The Leroy P. Steele Prize for Seminal Contribution to Research. Each of these three US$5,000 prizes is awarded annually.

The winners in the area of Mathematical Exposition, including prizes for specific books are listed here:

http://www.ams.org/prizes/steele-prize.html

Answer 42

John Hubbard's book on Teichmuller theory is clear, beautiful, inspiring, and (amazingly) essentially self-contained. He has a fantastic ability to take very technical and difficult results and make them seem clear and natural.

Bill Thurston wrote a preface for it which can be read here :

http://matrixeditions.com/Thurstonforeword.html

The money quote : "I only wish that I had had access to a source of this caliber much earlier in my career."

Answer 43

From more introductory texts, some of the most well-written textbooks I came across are

• Visual complex analysis, by Tristan Needham
• Differential equations, by Blanchard, Devaney, and Hall

Answer 44

Toric Varieties, about to be published by Cox, Little and Schenck is an unmeasurable amount of joy. It is impossible to get tired of it. Everything is well-bounded and it made me learn as much Algebraic Geometry as Toric Geometry itself. Its introductory sections to Algebraic Geometry before it develops the theory and shows you how to compute examples made me learn more than any dry full theoretic book in Algebraic Geometry. Definitely the best book I have read in two years.

Available at Cox's website (Wayback Machine: July 2011, August 2011, July 2015)

Answer 45

Lazarsfeld's book Positivity in Algebraic Geometry seems to fit the category ``Anyone in my field who hasn't read this paper has led an impoverished existence.''

I agree with Scott on `anything Serre.' His FAC and GAGA are gems; they will change your life.

Answer 46

I think Sipser's Introduction to the Theory of Computation deserves a mention. It is incredibly clear, full of valuable examples, and an absolute classic. I can't think of a better book for a mathematician who's interested in theoretical computer science. It also seems to serve computer scientists without a great deal of mathematical background by providing an introduction to proofs at the beginning. My favorite part: all theorems come with "Proof Idea" first and then proof after that. This helps the computer scientists who are not that familiar with proofs, but it's also great for a mathematician to get the main idea of the proof, fill in the blanks themselves, and then move on to the next result.

Answer 47

Probably the book(s) that most captivated me to study numerical analysis are the two by Forman Acton: "Numerical Methods That (usually) Work" and "Real Computing Made Real". The pithy and practical advice contained in both instilled in me the habits of trying to figure out just how structured the givens of a problem may be, among other things.

Answer 48

Donaldson and Kronheimer's "Geometry of 4-manifolds" - masters of the subject, they have a knack of explaining the crux of a difficult theorem in a concise and elegant way, and gauge theory has a lot of difficult theorems. After many years of reading, it still has new surprises.

Answer 49

I'm surprised I haven't seen Serge Lang. Some complain that he is too terse, but I really enjoy his style. Often times when i grab several books from the library on the same subject, it will be Lang's book I end up using the most. As for a single piece of writing, I think Lang's Algebra will do.

Answer 50

Harris and Fulton Representation theory is a nice book. Linear algebraic groups of Jim Humphreys is also very good one. On Riemann Surface the book of Forster's is really very good one.

Answer 51

Category Theory by Steve Awodey (Oxford Logic Guides 49) is a very clear exposition of the subject.

And I know my students hate it, but I really like The Way of Analysis by Robert Strichartz.

Answer 52

I'm astonished that nobody has mentioned I. G. Macdonald's classic Symmetric Functions and Hall Polynomials. It is more than 50 years old but fun to read, compact, self-contained, elegant, deep.

Answer 53

Another paper that I really like, because it makes a lot of things that are muddled in the literature very clear, is Sawin's Quantum Groups at Roots of Unity and Modularity. In particular the lesson to learn is that if it takes you a while to sort through which papers use which conventions or to find all the relevant constants attached to Lie algebras, then you really owe it to your readers to put that information in your paper where it's easy to find (preferably in a table).

Answer 54

A useful example of good writing style is how Thurston introduces his geometrization conjecture in

W. P. Thurston. Three dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.), 6(3):357–381, 1982.

Here is the excerpt:

Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the

1.1. CONJECTURE. The interior of every compact 3-manifold has a canonical decomposition into pieces which have geometric structures.

In §2, I will describe some theorems which support the conjecture, but first some explanation of its meaning is in order.

To state the conjecture formally, Thurston would have had to first define his 8 geometries. Instead, he wisely spends the introduction to explain where the conjecture comes from, in particular mentioning that it would imply the Poincaré conjecture. He introduces the now famous 8 geometries in the subsequent pages, along with illuminating examples and figures. All in all, a deep technical conjecture that would take pages to state formally becomes an enjoyable read even for the non-expert.

This example teaches us how one can provide a rough overview of a result before getting to the technical details.

Answer 55

The survey paper Hilbert's Tenth Problem over rings of number-theoretic interest by Bjorn Poonen (my advisor) is one of my favorites. He has several survey papers on his web page. My first year of grad school I read many of these and decided I wanted to work with him, so in a sense this paper did `change my life'.

In general Bjorn's writing is extremely clear and I have tried to model my own writing on his.

Answer 56

books I enjoyed very much: Mumford "Lectures on Curves on an Algebraic Surface"; Mumford "Curves and their Jacobians"; Mumford "Basics of Torus Embeddings; Examples of the Theory": Chapter 1 in "Smooth Compactification of Locally Symmetric Varieties"; Koblitz "Introduction to Elliptic Curves and Modular Forms"; Deligne SGA 4 1/2

In general I found Bourbaki-seminar texts often very helpfull and readable. Unfortunately many of the newer issues seem not to be free available on the web.

Answer 57

"A panoramic view of Riemannian Geometry" by M. Berger is an example of excellent mathematical writing to my taste. This book is great to learn what are the questions of interest in the field, and what are the main results. Although you will not find detailed proofs of the results, the main ideas are often explained in an intuitive way.

Answer 58

I think I went through all questions without seeing Bourbaki. I think that his Algèbre Commutative is simply a dream to read; every time I open it, I found myself keeping on reading it about perfectly useless subjects (compared to what I was looking for) just for the pleasure, as I was used to do as a kid with some of my parents' books. The same happens to me with Topologie Générale

Besides this, I must also add two litte gems by John Tate, namely his Rigid Analytic Spaces and a small paper where he studies residues on curves in an adelic language, Residues of differentials on Curves, Ann. Sci. Ecole Norm. Sup. (1968).

Answer 59

Anything and everything (including papers) by Marty Isaacs. I am more familiar with his "Finite Group Theory" book, but from the little I have seen, the same holds for his "Algebra" textbook. Of course, his book on Character Theory is the canonical text in that area.

The power of Isaacs' writing lies primarily in the clarity of his arguments. You are never left wondering where a certain bit came from, and he manages that without over-repeating himself, or over-explaining.

Then it's the content itself. In the first 70 pages of his FGT book, one already learns "exotic" stuff like the Chermak-Delgado measure, and the Theorems of Horosevskii, Lucchini, and Zenkov.

Another merit of his books is his choice of exercises. He really can't be commended enough on this. All are carefully chosen to supplement and strengthen the material presented in a flawless way.

A huge bonus is that nearly all arguments presented are his, and are as close to "from first principles" as possible.

Answer 60

The book "Infinite loop spaces" by J.F. Adams is one of my favorite mathematical texts. He introduces the subject with enough technicality to make it rigorous, but not so much that one loses sight of the forest for the trees. The added humor also makes it a delight to read.

Answer 61

1. The book "Linear Algebra" by Greub; I've always thought his writing here was gorgeous, if a bit Spartan.

2. Most of John Stillwell's books.

Answer 62

Simon Donaldson - Riemann Surfaces Great writing, deep understanding. I believe that noone have mentioned it because this topic is older than the book.

Answer 63

Poincaré's `Sur les solutions périodiques et le principe de moindre action'.

in Comptes Rendus des Séances de l'Académie des Sciences, t. 123, p 915-918, 1896

This paper humbled me. It made me realize how much the founders of the subject(s) I work in really knew, and how far ahead they could see. I am in awe of how Poincaré could give such a detailed trip report of his investigations without the formal language we use today being in place. (In many ways, formal language often gets in the way.) In these 2 and a half pages, Poincaré does most everything I did in a 13 page paper 102 years later. He does it more clearly and elegantly.

Answer 64

Lou Kauffman's book "On Knots" inspired me to become a topologist. It conveys the feel of the way topologists think with copious hand-drawn pictures. It also gets into deep waters without losing a playful touch. It would actually be nice to have a similar book that covers the recent developments in knot theory as well.

Answer 65

Steven Strogatz's "Nonlinear Dynamics and Chaos" book is written in a manner that almost allows one to kick back in a recliner and enjoy. The style is one that draws one into the material on nonlinear ODEs... probably the best undergrad text book that I used.

Answer 66

This paper changed my life:

Vaughan Jones, "Hecke algebra representations of braid groups and link polynomials"

Answer 67

Kontsevich's 1994 paper "Homological Algebra of Mirror Symmetry" has been very inspiring to me. It is full of tantalizing ideas and speculations, and brings so many different aspects of mathematics (and physics!) together into one beautiful tapestry.

Answer 68

Lam's Serre's Problem on Projective Modules. It contains everything: The big picture, the proof details, interesting techniques and the links between different methods. I wish more books were like this.

Answer 69

I loved reading Zelevinsky's 'Representations of finite classical groups'. It gave a totally different perspective to everything that I knew before about representation theory.

Answer 70

Actually a book that ended up changing my life: Kaczynski, Mischaikow, Mrozek: Computational Homology

I read it while working on my master's thesis in computational homological algebra, in order to see what they had to say about efficient implementations of simplicial homology.

After reading it, I first realized that algebraic topology has applications far outside what I had seen thus far - and now, a doctorate later, I'm active in the field of Applied Algebraic Topology and Topological Data Analysis.

I'm not certain I'd peg it for great writing as such, but the criteria above did state "changed my life".

Answer 71

I'll mention WL Burke's "Applied Differential Geometry." It's written for physicists, it will not be to the liking of the majority of mathematicians, but it changed this engineer's view of geometric methods forever.

The book changed the direction of my research because it presented a point of view that is not readily accessible if you follow the control and optimization literature. Becoming familiar with the differential geometry literature is an investment that a controls person is unlikely to make without a general idea of where the complete set of tools leads to. In this sense, the mathematics literature can present an obstacle. Burke's exposition is intuitive, though quite informal, and led me to read Spivak, Milnor, and other books, some mentioned here, which I would not have read if I had started with the math literature.

Answer 72

I believe that the papers on $A_\infty$-structures by Bernhard Keller are extremely well written: they provide the reader with an overview of the state of the art of research in the topic(s), applications and even understandable proofs. I suggest in particular

A-infinity algebras, modules and functor categories

Answer 73

Imre Lakatos: "Proofs and Refutations. The logic of mathematical discovery" is a fantastic read!

Answer 74

I couldn't find Disquisitiones Arithmeticae listed as an answer, and find this strange. (It has been translated from the Latin). The book is a delight to read, and the proofs always seem to be exactly the right ones.

Answer 75

I an algebraic geometer, so the book I'm going to propose is about as far from my subject as it can be. Still I think that Steele's book on stochastic calculus is one of the best written mathematical books I know. It really makes you enjoy probability, starting from the simplest examples of random walks and building a lot of theory, like martingales, Brownian motion and Ito's integral. I almost wanted to change my subject when I was reading it! :-)

Answer 76

Mathematical logic has at least a couple of great writers. The canonical example is Bruno Poizat, especially the French originals; I would put Hodges in the same league. Both are emphatically not concise writers (at least in their most famous books). Their use of the full capabilities of language is very didactic, and often poetic. I greatly admire them both.

Answer 77

For anyone studying Stable Homotopy Theory, a fantastic text is Doug Ravenel's "green book" (although the second edition is red) Complex Cobordism and Stable Homotopy Theory

Not only is this book full of useful results for those in the field (making it an incredible reference for those starting out), it is also written in a very clear style and it's completely self-contained. I cannot think of a better book on how to do computation in homotopy theory. This definitely fits under "I wish someone had told me about this when I was younger" and "anyone in my field who hasn't read this is leading an impoverished existence"

Answer 78

1. "Differential calculus on Normed Linear Banach Spaces" by Prof.Kalyan Mukherjee

   This book gave me a very hands-on explorable window into the world of manifolds and Lie groups. Like it shows explicit calculations of derivative of matrix multiplication and determinant maps and also about computing tangents to curves inside Lie groups.

2. "Topology, Geometry and Gauge Fields" by Gregory Naber. (2 Volumes)

   Its an exciting book which got me motivated into topology when it explained to me very simply the Heegard decomposition of S^3 and hence Hopf Fibration and how that relates to Dirac Monopoles! Before I read this book I had no clue that I would find mathematics exciting. Especially this revived my childhood interest in geometry.

   Naber's are books that changed my career decision.

3. "Global Calculus" by S.Ramanan (in the AMS series)

   This is a hard book to read initially but it excites the reader a lot and it was great to read alongside when Prof.Ramanan taught me topology and differential geometry. Anyway Prof.S.Ramanan is a great expositor. He could teach topics like modular forms and algebraic curves to a bunch of undergrads in their first complex analysis course in Chennai Mathematical Institute (CMI), India! He really pushes up the possible limits of exposition.

   Prof.S.Ramanan's lectures in my alma mater CMI, affected my career choices almost as much as Naber's books did.

4. "Calculus on Manifolds" by Spivak

   Its treatment of Fubini's theorem and related issues are great.

5. The writings on group theory by a college senior of mine called Vipul. His wiki "groupprops" is an amazing repository on finite group theory.

   His extensive efforts into mathematical writing also inspired me into periodically LaTex-ing up interesting things in mathematics as I learn.

Can anyone here tell about nice expository writings on topics like Gromov-Witten theory or Reshetkhin-Turaev and Rozansky-Khovanov stuff and how these relate to QFT? Something which shows a lot of examples and may be also explicit calculations.

Most sources on Quantum Groups that I have tried looking at start off a bit harshly for the newcomer. I would be greatly interested to read of "great mathematical writing" in these areas.

Answer 79

Hugo Steinhaus' book "Mathematical Kaleidoscope"

Answer 80

Harder's Algebraic Geometry 1 is a beautiful example of explaining why an abstract subject makes sense. The book has a conversational style without wasting words, and focuses on providing intuition for the subject.

When I get disheartened, this is the book I turn to for inspiration.

Answer 81

‘Galois Cohomology’ by Larry Washington in Cornell–Silverman–Stevens - Modular forms and Fermat's last theorem is my one stop reference for the eponymous topic. In about twenty pages (and with minimal prerequisites), he introduces Galois cohomology groups, explains Tate Local Duality Theorem and Euler Characteristic, shows the connection between extensions, deformation and cohomology groups, introduces generalized Selmer groups and proves a result that appears in Wiles' proof. Along the way, he also fully explains the Poitou-Tate nine-term exact sequence! Terrific stuff.

Answer 82

Weil's "Basic Number Theory" is my favorite book. Norbert Schappacher once said "If you learn number theory from this book, you will never forget it."

Answer 83

I wish someone had told me about this book when I was younger: A.I. Mal'cev, Algorithms and Recursive Functions.

The exposition is simple, thought provoking and rigorous. You are teased to delve into many strains when reading it.

Answer 84

I wish someone had told me about this paper when I was younger.

I have had this feeling a few times. For example: Gillman & Jerison, Rings of Continuous Functions.

Answer 85

I want to mention two papers (related).

• The very famous Complex analytic connections in fibre bundles by Michael Atiyah.

• The not so famous The Atiyah bundle and connections on a principal bundle by Indranil Biswas.

Answer 86

Comparison Theorems in Riemannian Geometry by Jeff Cheeger and David Ebin.

Why it is great:

• ... this is a wonderful book, full of fundamental techniques and ideas. (Robert L. Bryant)
• Cheeger and Ebin's book is a truly important classic monograph in Riemannian geometry, with great continuing relevance. (Rafe Mazzeo)

Answer 87

Algebraic Functions and Projective Curves by David M. Goldschmidt; October 2000

Beautiful proofs; Chapter 6 (Zeta Functions), 6.3 Riemann Hypothesis, nice mathematics.

Anyone in David's field who has not at least sampled this book is leading an impoverished existence.

Falco

Answer 88

I am surprised nobody has recommend the two-volume series The Chauvenet Papers printed by the MAA.

The work is a collection of prize-winning expository papers on mid-to-advanced undergraduate topics in mathematics; I have read most of them myself and would say they are invaluable surveys into various topics of mathematics (examples include broad areas such as harmonic analysis or the history between mathematics and logic, problem-specific articles such as one on Hilbert's 10th problem, and more miscellaneous works like Zalcman's "Real proofs of complex theorems (and vice-versa).")

Answer 89

despite the numerous replies here is masterpiece yet to be mentioned :

Alain Connes' Noncommutative Geometry

This is a most brilliant introduction and overview of many modern aspects in all of the areas

• Analysis (in particular algebraic)
• Geometry (in particular algebraic, non commutative and infinite dimensional)
• Topology (in particular algebraic and analytic)
• Statistical Physics and Quantum Theory

It is simply just perfect in every aspect - the pedagogical style, graphical design and illustrations, historical side notes .... but most of all ... the originality of the mathematical thought!