Yifeng Yu - University of California, Irvine (UCI), Mathematics The existence of homogenization of Hamilton-Jacobi equations in the periodic setting has been established by Lions, Papanicolaou and Varadhan in 80's. However, it remains a major open problem to understand deep properties of the effective Hamiltonian that governs the homogenization limit. In this talk, I will focus on the mechanical Hamiltonian H(p,x)=12|p|2+V(x) and present some recent progress together with close connections with the Aubry-Mather theory.