Here is a surprising application of billiard-related mathematics.
The story starts quite far, but rest assured that it will end up with problems related to billiard trajectories.

Consider the practical problem of finding out the position dependent electrical conductivity of an object by making voltage and current measurements at the boundary.
This is known as Calderón's problem, and it can be formulated more precisely as follows.
If the object is a domain $M\subset\mathbb R^3$, conductivity is modeled by a function $\gamma:M\to(0,\infty)$ that satisfies $\log\gamma\in L^\infty$ (measurable, bounded away from infinity and zero).
If $u:M\to\mathbb R$ is the electric potential, the current density is by Ohm's law $-\gamma\nabla u$ and it follows from Kirchhoff's law that $\operatorname{div}(\gamma\nabla u)=0$.
Measuring voltage at the boundary means measuring $u|_{\partial M}$ and measuring current amounts to measuring $\nu\cdot\gamma\nabla u|_{\partial M}$, where $\nu$ is the unit normal.
The problem is then this: Given the voltage and current data
$$
\{(u|_{\partial M},\nu\cdot\gamma\nabla u|_{\partial M});u \in H^1(M),\operatorname{div}(\gamma\nabla u)=0\},
$$
reconstruct the function $\gamma$.

If all of $\partial M$ is available for measurements, the problem is quite well understood — although much of the progress is very recent.
In dimension two the problem can be solved (Astala–Päivärinta 2006) and also in higher dimensions with the additional assumption that $\gamma$ is Lipschitz (Caro–Rogers 2014).

In practical situations it usually happens that one cannot make perfect measurements.
One case of imperfect data is a partial data problem where only part of $\partial M$ is available for measurements.
On the rest of $\partial M$ the potential is assumed to vanish (grounded surface) and current cannot be measured.
Suppose $M$ is contained in a cylinder $\mathbb R\times\Omega$ for a domain $\Omega\subset\mathbb R^2$ so that $M$ contains a cylindrical part $[0,L]\times\Omega$ and two caps at the ends to make it bounded.
Assume the part inaccessible to measurements is contained in the cylindrical part, more precisely of the form $[0,L]\times R$ for $R\subset\partial\Omega$.
It turns out that in this setting one can reconstruct a $C^2$ conductivity in $M$ if any function $f\in C(\Omega)$ can be recovered from its integrals over all billiard trajectories in $\Omega$ which reflect on $R$ and have endpoints on $\partial\Omega\setminus R$ (Kenig–Salo 2013).

This billiard problem (known as the broken ray tomography problem) is far from being completely understood, but the reconstruction is possible for example if $R$ is a subset of a cone (Ilmavirta 2014).
Broken ray tomography provides open problems related to billiards and connected with practical applications in nondestructive imaging.

verybad manners to ask a pure mathematician "what are the applications of your work"! :p $\endgroup$ – Federico Poloni Feb 6 '15 at 21:013more comments