Eigenvalue Problems for the Laplacians are among the most studied ones in classical analysis, partial differential equations, calculus of variations and mathematical physics. In this lecture I shall discuss some recent progress on a couple extreme problems involving the Dirichlet eigenvalues of the Laplacian, including the well-known Polya's Conjecture. These problems have origins in shape optimization, pattern formation, ..., and even data science. It is also related to the theory of harmonic maps into singular spaces and free boundary value problems involving vector valued functions.