In my last few posts, I described a pair of matrices that can form a group, that I referred to as the generators for such groups. From here, we can use those matrices to perform linear transformations that can get us to any group member.
For instance, with the modular group, we have two matrices that correspond roughly to rotation (specifically reflection) and scaling. In the case of the modular group, we generate the Lie group with a Lie algebraic structure, with their symmetries defined up to isomorphism.
Yet these groups can be distinct from each other in some topological manner, such as smooth manifolds in R^n, the modulo groups in R, or periodic waves in H. Each form quotient groups from members of their respective spaces: (non-)Euclidean geometry (e.g. S^1 from R/2πZ), the real number line (Z/nZ for some integer n), or the complex upper-half plane. Each one of these subgroups are called cosets, and they differ by the elements (so topological spaces) they are indexed to. So for example, {…,0,1,2,0,1,2,…} and {…,0,1,2,3,0,1,…} each form a coset, with the union of cosets forming a quotient group.
This heterogeneous indexing is what I argued in my last piece generates the distinction between algebra vs geometry (with geometry dealing in the realm of algebraic polynomials in terms of points, lines, and so on1). Yet fundamentally, they are linked to the same generators, so with the modular group you end up with circle rotation and wave oscillation being governed by the same periodic symmetry. That is how you link circles to music!
Angular rotation in R^2 has periodicity 2π, as R(θ+2π) = R(θ). Let SO(2) be plane rotations generating the Lie group S^1. R(θ) is a linear 2×2 matrix (in SL_2(Z)) of sines and cosines. Apply e^θJ to apply an isomorphic linear transformation with such generator matrix to travel to complex space. Then there exists an isomorphism between modular group operations in Euclidean space generating smooth manifolds, and the periodic waves in H, when f(z+1)=f(z). This holds precisely when the rotations repeat themselves over 2π. Sines and cosines are then periodic over intervals of π given f(-1/z)=(z^k)f(z) must hold, with Fourier series in general repeating themselves over factors of π.
If all this seems confusing, it makes more sense to me visually:
This is why I find Hilbert's nullstellensatz so elegant. For any algebraic set, you decompose each into a series of polynomial products (called rings), and find the solutions for which each polynomial equals zero (its zero set).
For instance, when you decompose sin(x) into x(x-π)(x+π)(x-2π)… so then you have (1/x)sin(x) = Πn(1-(x^2)/π^2(n^2)) with this being a polynomial ring forming an ideal over sin(x). You can express sin(x) as any set of zero polynomials with π solutions. Sin(x) is said to be indexed to nπ with n being a positive integer.
Then from there you can derive its generators. You look for the higher-order polynomials consistent with that zero set, so via your standard calculus operations you find the transformations where invariance holds, which give you your ideal. That produces other cosets within that quotient group. Hence not only do all algebraic (and geometric!) spaces contain some first principles deduction, but they can also be reverse-engineered.
Whilst finding the zero set of your algebraic ideal, you may discover hidden and unexpected isomorphisms across other elements. For instance, take the Riemann zeta function. Given that this yields ζ(2) = Σn(1/n^2), we have an expansion in sin(x) giving 1-(1/π^2)xζ(2)+… so ζ(2)=(1/6)π^2 when we equate with its Taylor series. Moreover, consider restricting the codomain to the real half-plane s>1. Then we can form a coset and eliminate evens from our series:
(1/2^s)ζ(s) = 1/(2^s)+1/(4^s)+…,
(1 - 1/2^s)ζ(s) = 1/(3^s)+1/(5^s)+…
and continue this indexation strategy until we are left with only primes. For p being prime numbers, Πp(1-1/p^s)ζ(s) = 1. Therefore we get the Euler products that can express ζ(s) in terms of primes:
ζ(s) = Π_p=2{1/(1-1/p^s)}
In other words, the same symmetries linking circles to music also establish connections to the prime numbers. The Hilbert nullstellensatz generalises this procedure to algebraic indexing in general. Is a subscription to uncover the hidden patterns in reality worth enjoying?
Basically general topological space without counting numbers.