Mathematics

 It is well-known that the classical Dirichlet problem for Laplace's equation in the ball with polynomial boundary data always has a polynomial solution. It is less well-known (but still a classical fact) that the same holds true for ellipsoids.   According to the Khavinson-Shapiro conjecture (1992) this property characterizes ellipsoids.  I will discuss this problem and ones related to it along with a control-theoretic interpretation of the general class of problems that ask for polynomial solutions to partial differential equations.