most recent 30 from http://mathoverflow.net 2013-05-21T17:23:42Z http://mathoverflow.net/feeds/question/358 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing Anton Geraschenko 2009-10-12T17:12:16Z 2012-12-15T04:14:31Z

<p>This question is basically from <a href="http://math.stanford.edu/~vakil/" rel="nofollow">Ravi Vakil</a>'s web page, but modified for Math Overflow.</p> <p><em>How do I write mathematics well?</em> 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?"</p> <p>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). <strong>Remember that each answer should be about a <em>single</em> piece of "great writing", and please restrict yourself to posting one answer per day.</strong></p> <p>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.</p> <p>Not acceptable reasons:</p> <ul> <li>This paper is really very good.</li> <li>This book is the only book covering this material in a reasonable way.</li> <li>This is the best article on this subject.</li> </ul> <p>Acceptable reasons:</p> <ul> <li>This paper changed my life.</li> <li>This book inspired me to become a topologist. (Ideally in this case it should be a book in topology, not in real analysis...)</li> <li>Anyone in my field who hasn't read this paper has led an impoverished existence.</li> <li>I wish someone had told me about this paper when I was younger.</li> </ul>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/360#360 S. Carnahan 2009-10-12T17:30:01Z 2009-10-12T17:30:01Z

<p>Canonical submission: Anything by Serre (e.g., Local Fields, Trees, Algebraic Groups and Class Fields,...). Reasons:</p> <ul> <li>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).</li> <li>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.</li> <li>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.</li> </ul>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/361#361 Ilya Nikokoshev 2009-10-12T17:41:06Z 2009-10-12T17:41:06Z

<p>Well, for me <strong>Hartshorne</strong> was really the window into the brave new world &mdash; and yes, it fits several items from 'acceptable reasons'. </p> <p>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 :) ) </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/375#375 Chris Schommer-Pries 2009-10-12T20:13:07Z 2009-10-12T20:13:07Z

<p>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. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/376#376 Lars 2009-10-12T20:16:17Z 2011-04-20T15:07:52Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/377#377 Noah Snyder 2009-10-12T20:19:17Z 2009-10-12T21:59:20Z

<p>Dror's paper <a href="http://front.math.ucdavis.edu/math.GT/0410495" rel="nofollow">Khovanov's homology for tangles and cobordisms</a> 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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/380#380 John D. Cook 2009-10-12T20:30:07Z 2009-10-12T20:30:07Z

<p>Michael Spivak's <a href="http://www.amazon.com/gp/product/0914098918?ie=UTF8&amp;tag=theende-20&amp;linkCode=xm2&amp;camp=1789&amp;creativeASIN=0914098918" rel="nofollow">Calculus</a> made me want to study analysis. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/382#382 Noah Snyder 2009-10-12T20:53:10Z 2009-10-12T20:53:10Z

<p>Silverman and Tate's "Rational Points on Eliptic Curves."</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/387#387 Theo Johnson-Freyd 2009-10-12T21:30:43Z 2012-04-24T06:29:28Z

<p><a href="http://books.google.com/books?id=KvQr9l0wgf8C&amp;lpg=PP1&amp;dq=proofs%20from%20the%20book&amp;pg=PP1#v=onepage&amp;q=&amp;f=false" rel="nofollow">Proofs from the Book</a>, Martin Aigner, Günter M. Ziegler, 2000.</p> <p>Anyone in [mathematics] who hasn't read this [book] has led an impoverished existence.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/390#390 Noah Snyder 2009-10-12T21:59:56Z 2009-10-12T21:59:56Z

<p>Another paper that I really like, because it makes a lot of things that are muddled in the literature very clear, is Sawin's <a href="http://front.math.ucdavis.edu/0308.5281" rel="nofollow">Quantum Groups at Roots of Unity and Modularity</a>. 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).</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/392#392 Reiner Martin 2009-10-12T22:13:48Z 2009-10-12T22:13:48Z

<p>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 <a href="http://retro.seals.ch/digbib/view?rid=ensmat-001:1970:16::59&amp;id=browse&amp;id2=browse2" rel="nofollow">http://retro.seals.ch/digbib/view?rid=ensmat-001:1970:16::59&amp;id=browse&amp;id2=browse2</a>.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/491#491 David Zureick-Brown 2009-10-14T15:05:21Z 2009-10-14T15:05:21Z

<p>The survey paper <em><a href="http://www-math.mit.edu/~poonen/papers/aws2003.pdf" rel="nofollow">Hilbert's Tenth Problem over rings of number-theoretic interest</a></em> 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'. </p> <p>In general Bjorn's writing is extremely clear and I have tried to model my own writing on his.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/495#495 Andy Putman 2009-10-14T15:44:27Z 2009-10-14T15:44:27Z

<p>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.</p> <p>Bill Thurston wrote a preface for it which can be read here : </p> <p><a href="http://matrixeditions.com/Thurstonforeword.html" rel="nofollow">http://matrixeditions.com/Thurstonforeword.html</a></p> <p>The money quote : "I only wish that I had had access to a source of this caliber much earlier in my career."</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/501#501 Mikael Vejdemo-Johansson 2009-10-14T18:08:17Z 2009-10-14T18:08:17Z

<p>Actually a book that ended up changing my life: Kaczynski, Mischaikow, Mrozek: Computational Homology</p> <p>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.</p> <p>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.</p> <p>I'm not certain I'd peg it for <em>great writing</em> as such, but the criteria above <em>did</em> state "changed my life".</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/514#514 B. Bischof 2009-10-14T21:49:22Z 2009-10-14T21:49:22Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/538#538 Akhil Mathew 2009-10-15T01:20:18Z 2009-10-15T18:01:09Z

<p>Walter Rudin's <em>Real and Complex Analysis</em> 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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/579#579 Harrison Brown 2009-10-15T08:04:23Z 2009-10-15T08:04:23Z

<p>I'm not sure whether "expository" writing counts, but I'll go with it anyway...</p> <p>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 <em>Princeton Companion</em> 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. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/586#586 Jason Starr 2009-10-15T10:10:23Z 2009-10-15T10:10:23Z

<p>I love Grothendieck's Tohoku paper.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/596#596 known google 2009-10-15T13:52:17Z 2009-10-15T13:52:17Z

<p>Lazarsfeld's book <em>Positivity in Algebraic Geometry</em> seems to fit the category ``Anyone in my field who hasn't read this paper has led an impoverished existence.'' </p> <p>I agree with Scott on `anything Serre.' His FAC and GAGA are gems; they will change your life.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/608#608 AH 2009-10-15T16:22:52Z 2009-10-15T16:22:52Z

<p>Real Analysis by Elias Stein and Rami Shakarchi</p> <p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/622#622 Akhil Mathew 2009-10-15T18:01:50Z 2009-10-18T00:36:06Z

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

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/632#632 Qiaochu Yuan 2009-10-15T19:00:52Z 2011-05-15T12:24:31Z

<p>Gian-Carlo Rota's <a href="http://www.springerlink.com/index/W041888J1J528605.pdf" rel="nofollow">On the foundations of combinatorial theory I: Theory of Möbius Functions</a> is an eye-opening gem. The same is true of practically every paper in <a href="http://www.amazon.com/Gian-Carlo-Rota-Combinatorics-Introductory-Mathematicians/dp/0817637133/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1255633479&amp;sr=8-1" rel="nofollow">Gian-Carlo Rota on Combinatorics</a>, so consider this post a vote for the entire book. (If I become a combinatorialist, it will be because of this book.)</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/644#644 Ilya Grigoriev 2009-10-15T20:02:56Z 2009-10-15T20:02:56Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/654#654 Grétar Amazeen 2009-10-15T20:39:03Z 2009-10-15T20:39:03Z

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

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/658#658 userN 2009-10-15T20:56:21Z 2009-10-15T20:56:21Z

<p>Atiyah &amp; 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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/701#701 Jon Yard 2009-10-16T02:14:31Z 2009-10-16T02:14:31Z

<p>This paper changed my life:</p> <p>Vaughan Jones, "Hecke algebra representations of braid groups and link polynomials"</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/704#704 Jon Yard 2009-10-16T02:25:06Z 2009-10-16T02:25:06Z

<p>"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. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/736#736 Thomas Riepe 2009-10-16T12:46:44Z 2009-10-16T12:46:44Z

<p>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 </p> <p>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. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/771#771 hurzilipochtli 2009-10-16T18:03:37Z 2009-10-16T18:03:37Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/789#789 Gerald Edgar 2009-10-16T19:56:42Z 2009-10-16T19:56:42Z

<p><em>I wish someone had told me about this paper when I was younger.</em></p> <p>I have had this feeling a few times. For example: Gillman &amp; Jerison, <em>Rings of Continuous Functions</em>.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/1001#1001 Sam Derbyshire 2009-10-18T04:44:26Z 2009-10-18T04:44:26Z

<p>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.</p> <p>His book in progress "Vector Bundles and Characteristic Classes" is also very nice.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/1033#1033 Anna Varvak 2009-10-18T13:45:09Z 2009-10-18T13:45:09Z

<p>From more introductory texts, some of the most well-written textbooks I came across are</p> <ul> <li>Visual complex analysis, by Tristan Needham</li> <li>Differential equations, by Blanchard, Devaney, and Hall</li> </ul>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/1476#1476 timur 2009-10-20T19:45:59Z 2009-10-20T19:45:59Z

<p>Van der Waerden's Algebra. I became a mathematician because of this book.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/2482#2482 Wanderer 2009-10-25T15:29:50Z 2009-10-25T17:17:49Z

<p>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 <a href="http://www.amazon.com/Analysis-Now-Graduate-Texts-Mathematics/dp/0387967885/ref=sr%5F1%5F1?ie=UTF8&amp;s=books&amp;qid=1256484405&amp;sr=8-1" rel="nofollow">"Analysis now"</a>, by Gert Pedersen. Alas, the book is not as well known as it should be.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/3898#3898 Anton Geraschenko 2009-11-03T04:04:19Z 2009-11-03T04:04:19Z

<p>Éléments de géométrie algébrique (EGA) (full text available from <a href="http://www.numdam.org" rel="nofollow">numdam</a>) continues to be an inspirational text for me. I wish I'd started reading it earlier.</p> <p>It's come up in a few <a href="http://mathoverflow.net/questions/2446/best-algebraic-geometry-text-book-other-than-hartshorne/2540#2540" rel="nofollow">other places</a> here on MO. To quote <a href="http://mathoverflow.net/questions/3041/the-importance-of-ega-and-sga-for-students-of-today/3102#3102" rel="nofollow">Jonathan Wise's answer</a> to another question,</p> <blockquote> <p>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.</p> </blockquote>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/3983#3983 James 2009-11-03T18:22:36Z 2009-11-03T18:22:36Z

<ol> <li><p>The book "Linear Algebra" by Greub; I've always thought his writing here was gorgeous, if a bit Spartan.</p></li> <li><p>Most of John Stillwell's books.</p></li> </ol>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/9577#9577 Kevin Lin 2009-12-23T02:11:47Z 2009-12-23T02:11:47Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/9617#9617 Anirbit 2009-12-23T13:55:17Z 2009-12-23T13:55:17Z

<ol> <li><p>"Differential calculus on Normed Linear Banach Spaces" by Prof.Kalyan Mukherjee</p> <p>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. </p></li> <li><p>"Topology, Geometry and Gauge Fields" by Gregory Naber. (2 Volumes) </p> <p>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. </p> <p>Naber's are books that changed my career decision. </p></li> <li><p>"Global Calculus" by S.Ramanan (in the AMS series)</p> <p>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. </p> <p>Prof.S.Ramanan's lectures in my alma mater CMI, affected my career choices almost as much as Naber's books did.</p></li> <li><p>"Calculus on Manifolds" by Spivak</p> <p>Its treatment of Fubini's theorem and related issues are great. </p></li> <li><p>The writings on group theory by a college senior of mine called Vipul. His wiki "groupprops" is an amazing repository on finite group theory.</p> <p>His extensive efforts into mathematical writing also inspired me into periodically LaTex-ing up interesting things in mathematics as I learn. </p></li> </ol> <p>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. </p> <p>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. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/9826#9826 Joseph Malkevitch 2009-12-26T21:08:16Z 2009-12-26T21:08:16Z

<p>The American Mathematical Society awards the Leroy P. Steele Prize, in part for recognition of mathematical exposition:</p> <p>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</p> <p>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.</p> <p>The winners in the area of Mathematical Exposition, including prizes for specific books are listed here:</p> <p><a href="http://www.ams.org/prizes/steele-prize.html" rel="nofollow">http://www.ams.org/prizes/steele-prize.html</a></p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/14327#14327 kakaz 2010-02-05T23:47:34Z 2010-02-05T23:47:34Z

<p>Hugo Steinhaus <a href="http://en.wikipedia.org/wiki/Hugo_Steinhaus" rel="nofollow">http://en.wikipedia.org/wiki/Hugo_Steinhaus</a> book "Mathematical Kaleidoscope" </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/14328#14328 Harry Gindi 2010-02-05T23:57:23Z 2010-02-05T23:57:23Z

<p>One of my all-time favorites for rigour and clarity:</p> <p>Vistoli's Notes on <a href="http://homepage.sns.it/vistoli/descent.pdf" rel="nofollow">Descent</a>. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/14949#14949 Victor Miller 2010-02-10T21:26:37Z 2010-02-10T21:26:37Z

<p>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".</p> <p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/14958#14958 Ian Durham 2010-02-11T00:56:32Z 2010-02-11T00:56:32Z

<p><em>Category Theory</em> by Steve Awodey (Oxford Logic Guides 49) is a very clear exposition of the subject.</p> <p>And I know my students hate it, but I really like <em>The Way of Analysis</em> by Robert Strichartz.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35002#35002 trying 2010-08-09T14:16:26Z 2010-08-09T14:16:26Z

<p>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!</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35012#35012 Michael Thaddeus 2010-08-09T15:12:38Z 2010-08-10T01:27:59Z

<p>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.</p> <p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35014#35014 J. M. 2010-08-09T15:23:30Z 2010-08-09T15:23:30Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35016#35016 Spiro Karigiannis 2010-08-09T15:30:41Z 2010-08-09T15:30:41Z

<p>Every differential geometer should read at least the first two volumes of Spivak's <em>A Comprehensive Introduction to Differential Geometry</em>. 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...)</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35019#35019 A B 2010-08-09T16:05:14Z 2010-08-09T16:05:14Z

<p>Paolo Aluffi's <em>Algebra: Chapter 0</em> 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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35073#35073 Joseph O'Rourke 2010-08-10T03:06:14Z 2010-08-10T03:06:14Z

<p>Gowers' <em><a href="http://ukcatalogue.oup.com/product/9780192853615.do" rel="nofollow">Mathematics: A Very Short Introduction</a></em>. 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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35074#35074 Falco 2010-08-10T03:13:35Z 2010-08-10T03:13:35Z

<p>Algebraic Functions and Projective Curves by David M. Goldschmidt; October 2000</p> <p>Beautiful proofs; Chapter 6 (Zeta Functions), 6.3 Riemann Hypothesis, nice mathematics.</p> <p>Anyone in David's field who has not at least sampled this book is leading an impoverished existence.</p> <p>Falco</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35080#35080 Jesus Martinez Garcia 2010-08-10T04:48:15Z 2010-08-10T05:11:39Z

<p><em>Toric Varieties</em>, 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. Definetely the best book I have read in two years.</p> <p>Available at <a href="http://www.cs.amherst.edu/~dac/toric.html" rel="nofollow">Cox's website</a></p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35081#35081 Pooja 2010-08-10T05:05:12Z 2010-08-10T05:05:12Z

<p>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. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35214#35214 trying 2010-08-11T11:20:53Z 2010-08-11T11:20:53Z

<p>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!</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/35216#35216 Andrea Ferretti 2010-08-11T12:23:58Z 2010-08-11T12:23:58Z

<p>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 <a href="http://www.amazon.de/Stochastic-Calculus-Financial-Applications-Mathematics/dp/0387950168/ref=sr_1_1?ie=UTF8&amp;s=books-intl-de&amp;qid=1279114527&amp;sr=8-1" rel="nofollow">Steele's book on stochastic calculus</a> 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! :-)</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/36446#36446 Marco Caminati 2010-08-23T13:54:12Z 2010-08-23T13:54:12Z

<p>I wish someone had told me about this book when I was younger: A.I. Mal'cev, Algorithms and Recursive Functions.</p> <p>The exposition is simple, thought provoking and rigorous. You are teased to delve into many strains when reading it.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/36454#36454 Pietro KC 2010-08-23T15:43:37Z 2010-08-23T15:43:37Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65030#65030 Saikat Biswas 2011-05-15T08:02:16Z 2011-05-15T20:33:03Z

<ol> <li><p>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 <a href="http://www.jmilne.org/math/" rel="nofollow">www.jmilne.org/math/</a>. 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.</p></li> <li><p>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.</p></li> </ol>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65046#65046 Pait 2011-05-15T15:40:07Z 2011-05-15T15:40:07Z

<p>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.</p> <p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65054#65054 Tommy Occhipinti 2011-05-15T18:26:05Z 2011-05-15T18:26:05Z

<p>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. </p> <p>When I get disheartened, this is the book I turn to for inspiration. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65061#65061 David White 2011-05-15T20:05:50Z 2011-05-15T20:05:50Z

<p>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 <em>Lectures on Modules and Rings</em>:</p> <p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65063#65063 David White 2011-05-15T20:10:56Z 2011-05-15T20:10:56Z

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

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65064#65064 Jim Conant 2011-05-15T20:18:28Z 2011-05-15T20:18:28Z

<p>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. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65078#65078 Jonny Evans 2011-05-15T21:10:40Z 2011-05-15T21:10:40Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65087#65087 Andrew 2011-05-15T21:58:32Z 2011-05-15T21:58:32Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65576#65576 David White 2011-05-20T18:13:38Z 2011-05-20T18:13:38Z

<p>How about Munkres <a href="http://www.amazon.com/Topology-2nd-James-Munkres/dp/0131816292" rel="nofollow">Topology</a>? 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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65577#65577 David White 2011-05-20T18:20:20Z 2011-05-20T18:20:20Z

<p>I think Sipser's <a href="http://www.amazon.com/gp/product/0534950973/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&amp;pf_rd_s=lpo-top-stripe-1&amp;pf_rd_t=201&amp;pf_rd_i=053494728X&amp;pf_rd_m=ATVPDKIKX0DER&amp;pf_rd_r=01DHD5J3R6YH4N045239" rel="nofollow">Introduction to the Theory of Computation</a> 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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65586#65586 Matthew Sartwell 2011-05-20T20:00:49Z 2011-05-20T20:00:49Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65592#65592 Saikat Biswas 2011-05-20T21:07:12Z 2011-05-20T21:07:12Z

<p>'Galois Cohomology' by Larry Washington in Cornell-Silverman-Stevens 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 Charateristic, 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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/65595#65595 YBL 2011-05-20T21:21:35Z 2011-05-20T21:21:35Z

<p>Just like in this <a href="http://mathoverflow.net/questions/5499/which-mathematicians-have-influenced-you-the-most" rel="nofollow">thread</a>, 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 langage 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 langage/simple ideas from Serre. I think that's why he is sometimes overshadowed by his elders when someone asks this kind of question. </p> <p>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 etale 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 litterature. </p> <p>If you are looking for great examples of mathematical writing, you should definitly read some articles by Deligne. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/90917#90917 Maciej Starostka 2012-03-11T17:21:13Z 2012-03-11T17:21:13Z

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

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/93347#93347 Mustafa Said 2012-04-06T18:38:19Z 2012-04-06T18:38:19Z

<p>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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/93367#93367 gauss 2012-04-06T23:14:32Z 2012-04-06T23:14:32Z

<p>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. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/95007#95007 Thomas Richard 2012-04-24T09:37:59Z 2012-04-24T09:37:59Z

<p>"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. </p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/95010#95010 Liviu Nicolaescu 2012-04-24T10:30:51Z 2012-04-24T10:30:51Z

<p>Without a doubt, Arnold's <em>Mathematical Methods of Classical Mechanics</em> 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.</p>

http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/116427#116427 Rodrigo A. Pérez 2012-12-15T04:14:31Z 2012-12-15T04:14:31Z

<p>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 <i>Morse Theory</i> 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 <i>Introduction to Algebraic K-Theory</i> 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 <i>Characteristic Classes</i> was the bible."</p> <p>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: <i>If you want to learn to write Mathematics <b>well</b>, read anything by Milnor</i>.</p> <p>When I was a student, <i>Dynamics in One Complex Variable</i> was the bible.</p>