Perfect Harmony

'Harmonics,' by Cory Ench

When starting down a new path, I like to find the simplest book I can find on the subject at hand.

The eminent mathematician Edward Frenkel has provided a great service here in regard to the 'Langlands program,' which has been aptly described as the Grand Unified Theory (GUT) of mathematics.

I was quite excited to learn of a good number of intersections between that vast and lively body of work* and my own, including symmetry, projective geometry, spectral theory, Riemannian surfaces, quantum field theory (QFT), and harmonic analysis.

Although symmetry is arguably the most important theme here, gauge symmetries are fairly abstract, whereas harmonic analysis provides a trove of correlations between simple physical theory and what we directly experience in sight and sound.

Let's review.

The mathematician Fourier proved that any continuous function could be produced as an infinite sum of sine and cosine waves. His result has far-reaching implications for the reproduction and synthesis of sound. A pure sine wave can be converted into sound by a loudspeaker and will be perceived to be a steady, pure tone of a single pitch. The sounds from orchestral instruments usually consists of a fundamental and a complement of harmonics, which can be considered to be a superposition of sine waves of a fundamental frequency f and integer multiples of that frequency.

The process of decomposing a musical instrument sound or any other periodic function into its constituent sine or cosine waves is called Fourier analysis. You can characterize the sound wave in terms of the amplitudes of the constituent sine waves which make it up. This set of numbers tells you the harmonic content of the sound and is sometimes referred to as the harmonic spectrum of the sound. The harmonic content is the most important determiner of the quality or timbre of a sustained musical note.

Hyperphysics

OK, now here's Frenkel.

The roots of harmonic analysis are in the study of harmonics, which are the basic sound waves whose frequencies are multiples of each other. The idea is that a general sound wave is a superposition of harmonics, the way a symphony is a superposition of the harmonics corresponding to the notes played by various instruments. Mathematically, this means expressing a given function as a superposition of the functions describing harmonics, such as the familiar functions sine and cosine. Automorphic functions are more sophisticated versions of these familiar harmonics. There are powerful analytic methods for doing calculations with these automorphic functions. And Langlands' surprising insight was that we can use these functions to learn about much more difficult questions in number theory.

Frenkel

Well, this is just a taste, but that's enough for today.

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* The Langlands program has deep roots in number theory, but I've only scratched the surface of that sprawling topic. For the time being, here's a nice bridge in regard to theory vis-à-vis experience.

It was not until the advent of quantum mechanics in the twentieth century that absorbtion spectra were given a satisfactory theoretical explanation. They were shown to correspond with eigenvalues of appropriate Schrödinger operators. A given atom could absorb or emit light only at certain frequencies, corresponding to the energy levels of bound states represented by different eigenvalues. The mathematical spectra of differential operators thus carried fundamental information about the physical world, which even now seems almost magical.

The analogy with number theory is through spectra of other differential operators. These are Laplace-Beltrami operators (and variants of higher degree) attached to certain Riemannian manifolds. The spectra of these and other operators are expected to carry fundamental information about the arithmetic world, a possibility that also seems quite magical.

~Arthur