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

Canonical submission: Anything by 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 2

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 3

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 4

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 loosing 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 5

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 6

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 7

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

Answer 8

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

Answer 9

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 10

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 11

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 at http://retro.seals.ch/digbib/view?rid=ensmat-001:1970:16::59&id=browse&id2=browse2.

Answer 12

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 13

I love Grothendieck's Tohoku paper.

Answer 14

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 15

"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 16

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 17

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 18

Anyone in combinatorics who hasn't read Gian-Carlo Rota's Mobius function paper has led an impoverished life.

Edit: Well, really this applies to every paper in Gian-Carlo Rota on Combinatorics, so let me declare that this post is a vote for the entire book, since I now get to add the reason that, if I become a combinatorialist, it will be because of this book.

Answer 19

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

Answer 20

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 21

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 22

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 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

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 25

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 26

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 27

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 28

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 29

Some of my favourite authors are Serre, Mumford, Milnor, Fulton and Neukirch. There 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 30

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.