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Heat equation and Theta, Parts I-III

  1. Jan 16, 2007 #1

    Chris Hillman

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    I am taking the liberty of collecting mathwonk's "short course" for some followup comments/questions, since this topic is IMHO more interesting than the context in which it first appeared. (Hope this is OK under PF rules!).

    Part I:

    Part II:

    Part III:

    How annoying, Part IV won't fit (is there really a 20000 word limit on the size of a post?), TBC...
    Last edited: Jan 16, 2007
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  3. Jan 16, 2007 #2

    Chris Hillman

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    Heat Equation and Theta, Part IV

    And here is Part IV of mathwonk's minicourse, followed by some suggested (broadly relevant) background reading:

    Suggested reading for general background (I'll probably add to this once I catch my breath!):

    Algebraic geometry generally:

    Miles Reid, Undergraduate Algebraic Geometry, LMS Student Texts 12, Cambridge University Press, 1988.

    Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer, 1992. IVA for short. One of the best books of all time! Lots of pictures and computational tools.

    Hal Schenck, Computational Algebraic Geometry, LMS Student Texts 58, Cambridge University Press, 2003. See chapter 7 for the "homological signature" of configurations of points. (We are interested in something spiritually analogous, only involving "embeddings" of the next complicated thing after points. Sort of...)

    Joe Harris, Algebraic Geometry, Springer, 1992. Better emphasis of the role symmetry than Hartshorne, IMO. (Hartshorne isn't everywhere dense, but I am trying very hard not to frighten off the undergraduates... this is meant to be an invitation, not an initiation, as in "rite of hazing"!)

    Complex curves and abelian varieties:

    C. G. Gibson, Elementary Geometry of Algebraic Curves, Cambridge University Press, 1998. If you read no other book this year, read this! (Or IVA, cited above.) Not to be missed: "Cramer's paradox" is almost discussed in section 17.1.

    Frances Kirwan, [I}Complex Algebraic Curves[/I], LMS Student Texts 23, Cambridge University Press, 1992. Riemann surfaces, holomorphic differentials, Riemann-Roch theorem.

    Gareth Jones and David Singerman, Complex Functions, Cambridge University Press, 1987. Elliptic curves, the modular group, the Lorentz group, and so on.

    Herb Clemens, A Scrapbook of Complex Curve Theory, Plenum, 1980. Offers a bit about moduli spaces of cubic curves.

    Waldschmidt et al., From Number Theory to Physics, Springer, 1995, has a number of chapters dealing with abelian varieties, theta functions, and so on.

    Zeta functions (cause, you know, "zeta" rhymes with "theta") and all that:

    Jameson, The Prime Number Theorem, LMS Student Texts 53, Cambridge University Press, 2003. No seriously, great background for the intersection of algebra, number theory, and analysis.

    Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, LMS Student Texts 50, Cambridge University Press, 2001. Ditto, density theorems in chapter 16, plus it has an appendix on Fourier Analysis if you don't know that that is.

    Bedford et al, Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces, Oxford University Press, 1992. See the wonderful short introduction "Ergodic theory and subshifts of finite type" by Michael Keene, plus chapters by Series, Pollicott, and Mayer related to dynamical zeta functions.

    Lie theory, representation theory, and invariant theory (cause everyone should know some)

    Carter et al., Lectures on Lie Groups and Lie Algebras, LMS Student Texts 32, Cambridge University Press, 1995. Great stuff on the classification of complex Lie algebras, Weyl groups, Lie groups, and algebraic groups.

    Bernd Sturmfels, Algorithms in Invariant Theory, Springer, 1993. More great stuff on invariants of finite groups and how they can be computed. Goes beyond the chapter in IVA.

    Brian J. Cantell, Introduction to Symmetry Analysis, Cambridge University Press, 2002. One of many fine books on Lie's theory of differential equations which I might mention. Everyone should know that Lie theory arose as the background needed for Lie's attempt to pursue the idea that whenever you can solve a differential equation, you can do so because of some underlying symmetry.

    (This might be a good place to mention that when Lie met Klein in Berlin c. 1869, he soon had to confess that he hadn't paid enough attention when he took a course on finite group theory from none other than Sylow! So if anyone is feeling a tad daunted, well, many have felt that, and the successful math students work through the terror! In 1869, Klein and Lie both found Galois theory [recently rediscovered] and the work of Darboux in differential geometry and differential equations a bit daunting. So they traveled to Paris and promptly got caught by the outbreak of the Franco-Prussian war. For some reason Lie thought this would be a good time to enact his lifelong ambition of hiking from France to Norway, via the Italian Alps. In the nude. You can guess what happened next!...fortunately, the gendarmes didn't shoot him, they put him on the train with a note that they were deporting a Norwegian lunatic.)

    Peter W. Olver, Equivalence, Invariants, and Symmetry, Cambridge University Press, 1995. More about Lie groups and invariant theory.

    Fulton and Harris, Representation Theory, Springer, 1991. For some basic stuff about representations of the symmetric group, Young diagrams, etc.

    Number Theory Generally:

    Burr et al., The Unreasonable Effectiveness of Number Theory, American Mathematical Society, 1991. Articles by leading researchers, surveying beautiful connections between number theory and dynamical systems, Rogers-Ramanujan identities in statistical mechanics, diffraction gratings, acoustics of concert halls, coding theory, random number generators, computer graphics, and more. The article by Rogers should be of particular interest because this is another place where theta functions arise.

    Thomas M. Thompson, From error-correcting codes through sphere packings to simple groups, MAA, 1983. Great background for the next listed item:

    Conway andSloane, Sphere Packings, Lattices and Groups, Springer, 1998. More neat applications of theta functions.

    Baezetics (cause John Baez is a subject all his own):

    John Baez, TWF (Weeks 62-66, 157,178-188, 193, 201, 213-218, 230, 241, 243): http://www.math.ucr.edu/home/baez/TWF.html. Incidence geometry, Schubert cells, cohomology of homogeneous spaces, ADE for Lie algebras, simply laced Dynkin diagrams, Young diagrams, nonabelian Hodge theory, quantum calculus and q-deformations, the MacKay correspondence, the Monster, cohomology of groups, Eilenberg-Mac Lane spaces, categorification, Klein's quartic, zeta and L-functions, Langlands program, ADE for catastrophes, cobordism, Cartan geometry, and much more, such as hamsters in physics. (Note: if the webglimpse search tool is not working, try googling with "baez TWF site:ucr.edu keyword" where keyword is a phrase like ADE.)

    Note that the LMS Student Texts are designed for UG students and are as inexpensive as possible--- mathwonk, have you ever considered writing one of these? I think moduli spaces of curves and geometric invariant theory would make a wonderful and timely topic!
    Last edited: Jan 16, 2007
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