Real life applications of Maass wave forms

Question

Explaining my work on Maass wave forms to friends and family (all non-mathematician) typically earns me blank faces. So I wonder whether there is some good example to explain their meaning to laymen. I am aware of the inner-mathematical importance of Maass wave forms, but what are real life applications of Maass wave forms?

Answer 1

I don't think you will find real-life applications of Maass forms in particlar, but you can explain what they are (in some sense) to a non-mathematician.

First, you explain that if you have a surface made of some skin stretched tight, you can beat it and set up vibrations of different frequencies (but not arbitrary frequencies; the possible standing waves are constrained by the geomery of the drum skin). These are the Laplacian eigenvectors; the frequencies are (essentially) the Laplacian eigenvalues.

You can explain that this can happen even if the drum skin is not stretched over a boundary (like a typical drum), but is wrapped up without a boundary (a vague description of a compact surface without boundary).

Now you can explain that if you for example have a guitar string and then one end comes loose (e.g. the string breaks), then (a) it can basically vibrate at any frequency (the constraints have gone away because one end is free), and (b) it won't make much of a tune, because all the energy in the wave will leak out at the free end (familiar to anyone who tries to pluck a broken string, broken rubber band, ... ; it just loses its vibration almost straight away).

Same if part of a drum skin separates from the rim --- you can't drum on it properly any more; the separated edge will just flap around and all the energy in your drumming will dissipate there.

So if you a surface like the upper half-plane mod a congruence subgroup of $SL(2,\mathbb Z)$, which is a punctured surface, you might think that there are no interesting vibrational modes --- that any vibration you try to set up will just leak out through the cusps. This can happen --- there is a continuous spectrum of the Laplacian (the Eisenstein series) --- and for a typical punctured Riemann surface this is all that happens; there is no discrete spectrum.

But for these beautiful, amazing, congruence quotients of the upper half-plane, we know (thanks to Maass and Selberg) that there is still a non-trivial discrete spectrum --- we can still set up standing waves in them. These are the Maass wave forms!

If your non-mathematical listener is patient enough, this kind of story might give them some sense of why Maass wave forms are unusual, interesting, and special.