Solvable PDEs have measure zero

PDEs are very hard to solve.

I heard it claimed that it's very lucky, if you're someone who does mathematical finance, that the Black-Scholes partial differential equation, when used to price a call option, has a closed-form solution. (Well, if you include the error function in "closed form".)

The payoff of the thing being priced gives the initial condition for the Black-Scholes PDE and for a simple perturbation of that, there is probably not such a simple solution. And options markets probably would have developed very differently if there hadn't been an exact solution, because solving such an equation numerically, while possible, is a lot slower.

A friend of mine who knows more about PDEs than I do said that, basically, the set of exactly solvable PDEs has measure zero. Of course this isn't a theorem, but the point is that people who study PDEs don't expect there to be exact solutions.

Huygens' principle

A weird fact I came across yesterday: Huygens' principle for the wave equation.

When you throw a pebble on the surface of some water, circular waves propagate outward from that point. However, inside the circular wavefront there will still be some sort of disturbance.

Light, in three dimensions, doesn't work the same way. If I have a pulse of light that lasts one second, then one second after that pulse stops, then the wave resulting from that pulse will be supported on the annulus from one light-second to two light-seconds from me. There will not be any sort of wave going on closer than one light-second from me.

I doubt this was known to Huygens, simply because he lived in the seventeenth century when they didn't have partial differential equations. They had waves, though, and the page I linked to above implies that Huygens knew that the leading edge of a wave travels at a constant speed, usually denoted by c; the most important example of waves is light waves, so c is now reserved for the speed of light. (I suspect he was aware of ripples in water; it's less obvious that there aren't analogous ripples in three dimensions, though, simply because there don't seem to be three-dimensional phenomena where the wave travels at a reasonable speed that one could observe. So observation is not particularly useful here, and one simply has to trust the symbols.)

Okay, this seems reasonable... things in different numbers of dimensions behave differently. But in four, six, eight, ... dimensions, solutions to the wave equation behave like the two-dimensional case (the surface of water), in five, seven, nine, ... dimensions they behave like the three-dimensional case (like light). I wouldn't have been surprised to learn that the behavior is different in low-dimensional space than in high-dimensional space. Or even that the behavior was different in "medium"-dimensional space -- it's similar to how things work in the theory of manifolds, where one- and two-dimensional manifolds are basically trivial to study, five-dimensional and higher manifolds aren't that hard, but three and four dimensions are difficult. (I'm vastly oversummarizing here; this isn't my area, and I'm just going on things I hear in the halls.) But who would think it would depend on the parity?

It seems that the dependence on the parity falls out of the series solutions to the spherically symmetrized wave equation, and this equation includes the dimension as a coefficient. I'm having a bit of trouble parsing the page I linked to (although I'm not working too hard on it). The author comments:

It would be interesting to work out the connections between Huygens' Principle and the zeta function (whose value can only be given in simple closed form for even arguments) and the Bernoulli numbers (which are non-zero only for even indices).

This isn't something I would have thought of.