  For a monotonically advancing front, the arrival time is the time when the
front reaches a given point. We show that it is twice differentiable everywhere
with uniformly bounded second derivative. It is smooth away from the critical
points where the equation is degenerate. We also show that the critical set has
finite codimensional two Hausdorff measure.
  For a monotonically advancing front, the arrival time is equivalent to the
level set method; a priori not even differentiable but only satisfies the
equation in the viscosity sense. Using that it is twice differentiable and that
we can identify the Hessian at critical points, we show that it satisfies the
equation in the classical sense.
  The arrival time has a game theoretic interpretation. For the linear heat
equation, there is a game theoretic interpretation that relates to
Black-Scholes option pricing.
  From variations of the Sard and Lojasiewicz theorems, we relate
differentiability to whether or not singularities all occur at only finitely
many times for flows.

Differential Geometry;Analysis of PDEs