#theta functions

[isomorphismes]

Equation (1) of www.math.ucla.edu/~wdduke/preprints/mocktheta.pdf reads ∏ 1 ÷ (1−qⁿ). I started expanding the denominator (in crayon) and found the pattern of + − + + − +’s easier to understand in colour.

Here is a sketch of the "combinatorial tetris" that results. I think the colours make the sources of the terms easier to track, and therefore to get a better intuitive feel for the algebra. (Anyone want to make a javascript app of this process?

Doodling with binomial expansion coefficients helped me understand the central limit theorem, Black-Scholes, and the Jones polynomial.

Check out also:

David J. Wright

Indra’s Pearls

diproton youtube video of Jacobi’s theta

Ardila combinatorics lectures (and arXiv), to get a sense of post-Rota combinatorics (which is less boring)

Vaughan Jones’ explainers 1 2 on his knot invariant (which is a polynomial)

Kapranov-Mazur-Reznikov knot-number correspondence

Ed Witten 1, Sir Michael Atiyah 1 2 3, Lou Kauffman 1 2 and Joan S Birman on how knots relate to physics. Briefly, when evaluating Feynman integrals, you consider all possible paths a particle could take. Some of these paths double over on each other, or loop around in strange ways. Witten’s idea (correct me if I’m wrong here) was to split these paths up into good=well-organised groups that are easier to sum-up. I think this somehow relates to gauge and Chern class.