One of the major open problems in homogenization of Hamilton-Jacobi equations is to under deep properties of the effective Hamiltonian.  In this talk,  I will present some recent progress. In particular, consider the effective Hamiltonian associated with the mechanical Hamiltonian H(p,x)=(|p|^2)/2+V(x). We can show that for generic V, the effective Hamiltonian is piecewise 1d in a dense open set in two dimensions using Aubry-Mather theory. So the homogenization process has dramatically changed the local behavior of the original Hamiltonian.

Zoom meeting: Link