Friday, December 16, 2011

Math to think about

There are several interesting directions I'd like to investigate. So interesting, I have decided to let you in on it too!


No living man cannot deny interest in moonshine. Terry Gannon's "Monstrous moonshine and the classification of CFT" (arXiv:math/9906167) provides a great review.

The basic idea is that we have a way to associate to "algebraic stuff" (e.g., groups) some "modular stuff".

What's great is, the word "stuff" is used in the technical sense of the word.

This would require reviewing group theory, finite groups, as well as some ring theory (for the "algebraic stuff"). To discuss "modular stuff", we'd need to review complex analysis, functional analysis. The connection here would require studying conformal field theory, to some degree.

Robert Wilson's Finite Simple Groups is a wonderful reference for finite simple groups; and as always SPLAG is a good reference too.

Noncommutative Geometry

People mean many things by "Noncommutative Geometry", here I mean Connes' approach.

I suppose this first requires us to consider what "commutative geometry" is!

This requires knowledge of commutative algebra and differential Geometry. The idea is to model "differential calculus over a commutative ring" (as Wikipedia calls it), i.e., consider the algebraic "grammar" underlying differential geometry.

Commutative algebra describes this algebraic "grammar". Studying this model in commutative algebra is precisely what I mean by "commutative geometry" (where projective modules correspond to vector bundles, and so on).

Noncommutative geometry, on the other hand, generalizes this model to the noncommutative setting!

I still need to write up my notes on differential geometry, but there are no good references for "commutative geometry"! There are a few books on commutative algebra, though...

I should type up my notes on algebraic topology too, since spin bundles are a "principal Spin bundle". Although I have some notes written on the Spin group (see my Lie groups notes), I should review it some more. Michelson and Lawson's Spin Geometry is a wonderful book to consider...

Operator algebras need to be reviewed for considering spectral triples. The algebra we typically work with are von Neumann algebras which are related to C* algebras.

Some references for operator algebras:

  1. Kadison and Ringrose, Fundamentals of the theory of operator algebras vol. I and II
  2. Blackadar. Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences. Springer-Verlag, 2005.
  3. Yasuyuki Kawahigashi, "Conformal Field Theory and Operator Algebras" arXiv:0704.0097 (18 pages)
  4. Meghna Mittal, Vern Paulsen, "Operator Algebras of Functions." arXiv:0907.5184
  5. John M. Erdman Lecture Notes on Operator Algebras (129 pages)
  6. J. A. Erdos, C*-Algebras (51 pages).
  7. N.P. Landsman, "Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics" arXiv:math-ph/9807030 (89 pages).
  8. Jacob Lurie's Course Notes on Von Neumann Algebras, quite comprehensive!
  9. Wassermann, Operators on Hilbert space [ps] (70 pages)
  10. VFR Jones, von Neumann algebras [pdf] (150 pages)
  11. NP Landsman's Lecture Notes on Operator Algebras [pdf] (64 pages).
  12. John Hunter and Bruno Nachtergaele, Applied Analysis (free, legal ebook!)

And, of course, there is Connes' Noncommutative Geometry [pdf], as well as Connes and Marcolli's Noncommutative Geometry, Quantum Fields and Motives [pdf]

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