Representation Theory

What Hacker News says

1. sdenton4Aug 2015
   Sagan's book on the symmetric group is a great introduction to algebraic combinatorics, tracing the combinatorial approach to understanding the representation theory of the symmetric group. For more depth in that direction, Fulton & Harris' 'Representation Theory' builds the connections with Lie groups.

   Algebraic combinatorics dances in the intersection of a number of fields of mathematics, where some of the most fundamental and beautiful structures are to be found. I had a relatively large breadth of knowledge coming out of my phd as a result, which I'm pretty happy about.

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2. impendiaJul 2015
   It would take an extraordinary effort. I doubt there is anyone who could learn this material given anything less than a year of solid study -- and I am thinking of the leading researchers in the world, who are already experts in everything related. I am a professional mathematician myself, and I doubt I could manage it even within a year.

   This is dictated by the difficulty of the subject, and it is even more so since Mochizuki's papers are notoriously difficult to read. What I have heard from other research mathematicians is that Mochizuki doesn't make much of an effort to make himself comprehensible, to answer questions, or in general to explain his results to the community.

   Also please be warned that the consensus is generally that Mochizuki's theory is probably incorrect.

   That said, if you want to learn the stuff, there is no reference other than Mochizuki's papers themselves. If you would like to learn some general background theory I would recommend learning some algebraic geometry, for which see Ravi Vakil's book:

   http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pd...

   and maybe Milne on etale cohmology:

   http://www.jmilne.org/math/CourseNotes/LEC210.pdf

   If you find that incomprehensible, start with Dummit and Foote's Abstract Algebra (read it cover to cover), Neukirch's Algebraic Number Theory, Atiyah and Macdonald's Commutative Algebra, and Fulton and Harris's Representation Theory.

   If you indeed want to read all that, you may as well enroll in a Ph.D. program in math....

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3. impendiaMay 2011
   Some thoughts, as an academic mathematician who got his Ph.D. three years ago:

   First of all, the author is right in one aspect, in that bad writing -- and also bad speaking -- are too widely tolerated in mathematics. Most departments have a regular colloquium, where the entire point is to listen to a talk to mathematicians outside your specialty. Many speakers do a good job, but there are some who will start out with "Let M be a symplectic manifold, let omega be the associated bilinear form, let X be the ring of differentials, ..." [ugh.]

   And yet I think there is more pressure to write and speak well than there is to do so poorly. Certainly as a graduate student I was pressured by my advisor to explain things clearly, to add examples and exposition to my papers, to omit technical details from talks, and in general to keep in mind the perspective of the non-expert. The author complains, "Submit a paper with two consecutive sentences of exposition and watch how quickly the referee gets on you for it." But I have not experienced this myself, nor heard this complaint from anyone until this article.

   Moreover I can think of lots of well-known books that are loaded with exposition. Representation Theory by Fulton and Harris comes particularly to mind. Of course, there are many books which are notoriously terse as well. (Any HN'ers tried to hack their way through Baby Rudin?)

   Indeed, there are many mathematicians who make quite an effort to write and speak well, and largely succeed, and the author seems unable to identify any mathematical writing which he does approve of. If we are doing a bad job, please give us specific examples of what you would like to see instead.

   The fact is that learning new math is just damn hard, even from well-written papers. I think that we could do a much better job of explaining our work to others. But, IMHO, this article overstates the problem, and omits to propose any practical solutions.

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4. logicalmindJun 2010
   If anyone is interested, here are the textbooks for the course:

   Math 55a

   * Required: Axler, Linear Algebra Done Right, Springer, 1997.

   * Required: Artin, Abstract Algebra, Prentice-Hall, 1991. * Recommended: Halmos, Naive Set Theory, Springer-Verlag, 1974.

   * Also useful: Halmos, Finite-Dimensional Vector Spaces, Springer-Verlag, 1987.

   * Also useful: Fulton and Harris, Representation Theory: A First Course, Springer-Verlag, 1991. See especially Appendix B.

   * Also useful: Bott and Tu, Differential forms in Algebraic Topology, Springer-Verlag, 1982. See especially Chapter 1.

   Math55b

   * Required: Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.

   * Required: Marsden and Hoffman, Basic Complex Analysis, Freeman, 1999.

   * Recommended: Stein and Shakarchi, Fourier Analysis, an Introduction, Princeton University Press, 2003.

   * Also useful: Stein and Shakarchi, Complex Analysis , Princeton University Press, 2003.

   * Also useful: Bott and Tu, Differential Forms in Algebraic Topology , Springer, 1982. See especially 1.1-1.3.

   * Also useful: H. M. Schey, Div, Grad Curl and All That , Norton, 2005.

   * Also useful: Hubbard and Hubbard, Vector Calculus, Linear Algebra and Differential Forms, Prentice Hall 1999.

   * Also useful: T. Needham, Visual Complex Analysis , Oxford University Press, 1997.

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