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

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

Answer 2

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 3

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 4

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

Answer 5

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 6

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 7

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

Vistoli's Notes on Descent.

Answer 8

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 9

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 10

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 11

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 12

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 13

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 14

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 15

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 16

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 17

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 18

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 19

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 20

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 21

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 22

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 23

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 24

"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 25

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 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

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 28

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 29

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

Answer 30

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).