The Princeton Companion to Mathematics

I have decided to follow the excellent example of Terence Tao and start up a blog. For the moment I am too busy to do this properly, because, with the help of June Barrow-Green and Imre Leader, I am editing a book called The Princeton Companion to Mathematics. However, it is partly for that very reason that I want to set up the blog. It is somewhat hard to explain what the book is, but if you want to get quite a good idea, there is a substantial (though out of date) description of it, with several sample articles available here. You can get into this site with userid Guest and password PCM. Comments welcome. A sufficiently sensible comment could even influence what goes into the book, but I should warn that, because we are at a rather late stage of the editing process, I no longer have much room for manoeuvre. So I may end up having to say, “Yes, great point, but unfortunately it’s too late to do anything about it.”

Actually, I hope that the PCM blog will really come into its own after the book comes out. In particular, if you feel that there are unfortunate gaps (as there undoubtedly will be) then maybe it will be possible to do something about it online — I might even start up a wiki consisting of PCM supplements. (The distinction between that and the regular mathematics articles in Wikipedia would be some kind of certification that an article had reached PCM levels of comprehensibility. I’d probably be unwilling to put in the sort of editorial efforts I’ve been putting in over the last few years, but would try to distribute that task by using the blog medium. If you are reading this, maybe you will have a suggestion about how to go about it — in particular, I don’t yet know anything about the technicalities of this kind of thing.)

38 Responses to “The Princeton Companion to Mathematics”

  1. Terence Tao Says:

    Dear Tim,

    Welcome to the nascent maths blogging community! I think the idea of a PCM blog is a great idea, both to discuss the Companion itself, and also to experiment with “21st century” ways of doing, communicating, and teaching mathematics. There are some other experiments out there with “certified” wikis, such as Scholarpedia, though it is still too early to say how successful they will be in the long run. In the meantime, though, blogs are certainly much better than nothing. Judging from my own blog, there is certainly a lot of quality feedback to be tapped via this medium.

  2. PCM article: Harmonic analysis « What’s new Says:

    […] as the “official” blog for the Princeton Companion to Mathematics, as indicated by his first post which also contains links to further material (such as sample articles) on the Companion.  Tim is […]

  3. Kay Says:

    Hello Dr. Gowers,

    I am very glad to see your new blog. I will be checking it regularly. I also plan to purchase a copy of the PCM when it is available. I was thinking it is quite a contrast, at a brobdingnagian 1008 pages (according to Amazon) to the lillipution “Mathematics: A very short Introduction”! The blurb on Amazon says that it will answer questions like ‘what is the point of symplectic geometry?’ which is the kind of question I have quite often and I never know where to look for an answer.

    There are often many resources for finding out ‘What is it?’ or ‘How is it done?’ but not ‘What is the point of doing this?’ so this should be an awesome resource.

  4. Thomas Love Says:

    I enjoyed the articles I read but I noted that the Exponential and Log functions entry appeared twice and the Riemannian metric link opened to the Knots and Polynomials article.

  5. klien4g Says:

    Prof. Gowers,

    Thank you so much for the PCM link. The advice to young mathematicians by Sir Atiyah is really excellent (at least to this phd student).
    As for some of the other articles, I wish my undergrad days were supplemented with expository pieces like these (for example Cliff Taubes’ article on diff top).

  6. william of ockham Says:

    Welcome to the internet! I recommended the pages on sets at William Vallicella’s blog recently here

    There is also a discussion going on at Bill’s place about the foundations of mathematics and set theory – a set of links can be found here

    I don’t know how much you will be talking about the philosophy of mathematics here, however.

  7. william of ockham Says:

    Oops the link to my site is wrong. suspect because wordpress is removing something but I’ll try again.

  8. william of ockham Says:

    Yes, wordpress removes the @ sign from the link. should be

  9. B. Sari Says:

    Speaking of “… “21st century” ways of… teaching mathematics”, I am entertaining an idea of offering a course the Companion style. This could be for senior undergrads or first year grad students. The main goal would be to expose the students to major ideas/areas of mathematics, as much as possible. We all can relate to the difficulty of a first year grad student who is at the turning point of his/her life trying to decide which area to choose.

    As for the course, a collection of articles from the Companion could be assigned to the participants to study and speak on. The instructor could set up certain criteria for the presentations, for instance, that a major theorem in the area is requested to be proved. This will be harder for certain areas than others because of the amount of background required to understand the proof.

    I would appreciate everyones input on this. Would such a course be useful? If you had to do it, what articles would you choose?, and what would you expect from the participants in such a course?

  10. Jim Clarage Says:

    As a recovering physicist who still struggles to keep up w/ modern math, this book could be a godsend (or Pythagorean-send maybe).

    The general aim and scope states:

    As far as possible, the book should be comprehensible to a typical first-year undergraduate, or even mathematically interested lay reader.

    My only ernest advice is to please assure you achieve your aim. Which should still be possible if the book is in the editorial process. Though I know mathematics is not an empirical endeavor maybe you could “test” some of the articles on a few freshman and lay readers and see if they pass the test. I’ll try to pass along the portal you sent to some undergrads here at the University of Houston.

    Having said that, I’m ready to pre-order. And ready to finally, hopefully, understand what/how/why areas such as algebraic geometry are the modern glory I keep hearing they are.

  11. Jim Clarage Says:

    I read the PCM proof [printer’s not mathematician’s] of Kollar’s Algebraic Geometry article. The level exposition is perfect!

    Two strengths I hope survive the editorial process:

    Please keep the figures– some of us still think about mathematics at that crude level of our senses. Kollar’s article has four. Wonderful. (I noticed the Moduli Spaces article didn’t have any… which is a shame, esp since e.g., Z2 lattices are the same dim as paper.)

    Please keep the humor. E.g.,

    Finally, if we marry a scheme to an orbifold, the outcome is a stack. Their study is strongly recommended to people who would have been flagellants in earlier times.

  12. Richard Gowers Says:

    You are funny, daddy!

  13. gowers Says:

    Thanks Richard — best comment yet!

  14. Jonathan Katz Says:

    Given this, I hope you gave the section on cryptography (as well as the section on computer science) a more balanced tone, taking into account the views of actual computer scientists instead of mathematicians only.

  15. gowers Says:

    Dear Jonathan,

    I think the theoretical computer science and cryptography communities can feel that their perspectives are very well represented in the PCM: in particular, there are long articles by Goldreich and Wigderson on complexity, Sudan on coding and communication, Clifford Cocks on crytography (which is also discussed in the Goldreich-Wigderson article), Kleinberg on algorithms, and Pomerance on computational number theory. If they said anything controversial, it was not obviously controversial to an outsider like me: I learned a lot from editing their articles and expect them to be widely read and enjoyed.

  16. Jonathan Katz Says:

    Glad to hear it!

  17. Jason Dyer Says:

    I’ve been working my way through all the samples (I especially like the one on Arithmetic Geometry, and your short entry on Metric Spaces is sparkling) and I have a comment of the “it’s not going to change but I just wanted to say it” variety. In the determinants article you comment that the actual terms picked seem a bit mystical, then justify it with a linear map. Unfortunately, I think as written it’s still a bit mystical. What I find most natural to those learning linear algebra is refer to Gaussian reduction. If you apply it to the a matrix with variables as entries the denominators of the fractions will be the determinant, and the actual process of permutation becomes very clear from the early steps of the reduction. Also, what happens when the determinant = 0 becomes clear in the prohibition of division by zero.

  18. gowers Says:

    It would be perfectly possible at this stage to change the determinants article, and it’s not one that I am especially happy with (or unhappy — just neutral). So let me see if I understand what you are proposing. Thinking about what you wrote above leads me to the following way of introducing determinants, which I think is essentially what you are suggesting. One asks the question, “How can we decide what a linear map, or matrix, does to (appropriately signed) volumes?” One then remarks that shears leave volumes unchanged, reflections in a hyperplane multiply them by -1, and (more generally) expansion in a single direction by a constant lambda multiplies them by lambda. Then one points out that Gaussian elimination shows us how to express every matrix as a diagonal matrix with 0s and 1s down the diagonal multiplied by a whole lot of very simple shears, reflections and one-dimensional enlargements. Perhaps less clear to me is how the formula via even and odd permutations arises. The proof that comes to my mind is the rather boring one of checking that that formula does the right thing when you apply elementary row operations, but you seem to be suggesting the much more satisfactory idea that the formula can be guessed in advance rather than merely checked afterwards.

    I think you are probably wrong in one respect: it seems to me that I should have started the article with the idea that what-a-matrix-does-to-volumes is an important invariant (because of the multiplicative property, which is intuitively clear) and only later said anything about a formula; and therefore the article will probably change.

  19. Jason Dyer Says:

    That is essentially what I mean (although I was focusing specifically on getting the formula). I am suggesting the the process of row operations can be generalized in a way that the machinery makes it obvious how the formula is composed.

    I agree with starting with the motivation rather than the formula.

  20. Jason Dyer Says:

    Sorry, not meaning to leave things hanging! I’m going to work on a more thorough writeup this weekend.

  21. PCM article: The Schrodinger equation « What’s new Says:

    […] 2nd, 2007 in Companion, math.AP, math.MP I’m continuing my series of articles for the Princeton Companion to Mathematics with my article on the Schrödinger equation – the fundamental equation of motion of quantum […]

  22. Princeton Companion to Mathematics « Nerd Wisdom Says:

    […] Princeton Companion to Mathematics From Terence Tao’s excellent blog, I learned about the upcoming Princeton Companion to Mathematics (PCM), a roughly 1000-page survey of and introduction to mathematics at the undergraduate level. It looks like the PCM editors have lined up a distinguished cast of mathematicians to write comprehensible articles covering all of mathematics. One of the editors is Timothy Gowers, who like Tao is a Fields Medalist, and who has very recently started his own blog. […]

  23. PCM article: The Fourier transform « What’s new Says:

    […] Fourier transform, moduli spaces, symmetry I’m continuing my series of articles for the Princeton Companion to Mathematics by uploading my article on the Fourier transform. Here, I chose to describe this transform as a […]

  24. PCM article: Compactness and compactification « What’s new Says:

    […] continuing my series of articles for the Princeton Companion to Mathematics with my article on compactness and compactification. This is a fairly recent article for the PCM, […]

  25. Vishal Says:

    Just wish to let you know how enlightening the articles on PCM are. For an undergraduate student coming from a small regional university, I learn mathematics mostly on my own, and those articles certainly contain a wealth of information. At times, while working on problems or studying some new field/topic, it is so important to get a global view of the same, and the PCM articles, among other things, precisely provide that.

    I should also add that the articles are very well-written; not too hard to comprehend and not too easy to be trivial. And the exposition is most wonderful. Just the thing that undergraduate students would want! Thanks a ton for your efforts as well the efforts of all the contributors!

    (I wonder if the PCM book is going to be available online once it’s finished.)

  26. PCM article: Phase space « What’s new Says:

    […] space, semiclassical analysis, Zeno’s paradoxes I’m continuing my series of articles for the Princeton Companion to Mathematics with my article on phase space. This brief article, which overlaps to some extent with my article […]

  27. PCM article: Differential forms « What’s new Says:

    […] combinatoric, several variable calculu I’m continuing my series of articles for the Princeton Companion to Mathematics through the holiday season with my article on “Differential forms and integration“. […]

  28. Frederick Ross Says:

    I’m actually a biologist these days (as in, I wear the big white suit and work on dangerous pathogens), despite my original training somewhere between math and physics. PCM looks like a godsend for me, for two reasons…

    First, I still do a lot of theory since there’s no one else to do it, and biology is in desperate need of it outside of the couple of accepted areas. After I get an idea nailed down to the point where I can make it into a mathematical structure, then I generally spend time looking if the mathematicians have already done the work on such structures for me (usually they have). The problem is always being about to look far enough and fast enough, and I’ve had to throw away a lot of ideas. From my random reading, it looks like PCM might extend my reach and be a useful research tool.

    Second, I indulge myself by reading mathematics in the evenings when I escape from the lab. This book is going to kill my sleep schedule if the examples online are representative.

  29. PCM article: Distributions « What’s new Says:

    […] number theory, distributions, factoring I’m continuing my series of articles for the Princeton Companion to Mathematics through the winter break with my article on distributions.  These “generalised […]

  30. PCM article: Generalised solutions « What’s new Says:

    […] solution, Leo Corry, proof, weak solution I’m continuing my series of articles for the Princeton Companion to Mathematics ahead of the winter quarter here at UCLA (during which I expect this blog to become dominated by […]

  31. PCM article: Function spaces « What’s new Says:

    […] norms, Peter Cameron From Tim Gowers, I hear the good news that the editing process of the Princeton Companion to Mathematics is finally nearing completion. It therefore seems like a good time to resume my own series of […]

  32. PCM article: Ricci flow « What’s new Says:

    […] Brian Osserman, Ricci flow, Weil conjectures I’m closing my series of articles for the Princeton Companion to Mathematics with my article on “Ricci flow“. Of course, this flow on Riemannian manifolds is now […]

  33. Cornucopia of Math « quantblog Says:

    […] but not least the superb ‘Princeton Companion to Mathematics‘ […]

  34. The Princeton Companion to Mathematics. « Is all about math Weblog Says:

    […] The Princeton Companion to Mathematics.  Princeton University Press just published the Princeton Companion to Mathematics. I learned about this book while I was reading a blog post on Timothy Gower’s  first blog post. […]

  35. hari Says:

    its really nice to see that many mathematicians coming forward to disseminate their knowledge in abstract mathematics even to non mathematicians. we , engineers, are so benefited from this initiative. please try to keep a balance between the abstract and applied aspects of the subjects.

  36. Das Buch der Bücher! DAS Buch für die einsame Insel « Analyse + Aktion Says:

    […] Herausgeber des Buchs hat einen eigenen Blog. Achja, ich vergaß zu erwähnen: Er ist Professor der Mathematik. Das Buch heißt: “The […]

  37. k Says:

    PCM is cool, it took me three months of focus whilst doing field work over summer to enjoy it fully; my only complaint (apart from it being too short) is (paradoxically) that it is too big! one needs popeye-arms just to hold it up and trying to read it in bed (in a tent); so please, when the next version comes out (or when you pop a collector’s edition) spare a thought for we of the skinny-arm brigade and ship it in a box containing slim volumes each of which covers a suitable segment; such an edition would be a real treat; keep up the good work; k

  38. Hitoshi Ishii Says:

    I am a member of a group who are trying to translate PCM into Japanese. I think, PCM is a great contribution to those who love mathematics. Here I would like to point out an erroneous statement
    I found in it:
    p. 203, column 1, line 13-15:
    It says “The sum above will be unchanged if we add a multiple
    of N to r , so we now care only about the values
    of f at points of the form n/N,” but, I think, the conclusion
    “so we now care only about the values
    of f at points of the form n/N” does not seem to be
    a consequence of the statement
    “The sum above will be unchanged if we add a multiple
    of N to r.”

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