Many physical phenomema lead to tracking moving fronts whose speed depends on the curvature. The level set method has been tremendously succesful for this, but the solutions are typically only continuous. We will discuss results that show that the level set flow has twice differentiable solutions. This is optimal. These analytical questions crucially rely on understanding the underlying geometry. The proofs draws inspiration from real algebraic geometry and the theory of analytical functions. Further developing these geometric techniques gives solutions to other analytical questions like Rene Thom's gradient conjecture for degenerate equations. We believe these results are the first instances of a general principle: Solutions of many degenerate equations should behave as if they are analytic, even when they are not. If so, this would explain various conjectured phenomena. Finally, the techniques should have applications to other geometric flows. If time permits, then we will discuss results about this. This is joint work with Bill Minicozzi.