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.

Bio. Prof. Fanghua Lin is Silver Professor of Mathematics at the Courant Institute of Mathematical Sciences, NYU. Professor Lin's research interests include classical and applied analysis, partial differential equations, geometric measure theory, and calculus of variations. His books include: Geometric Measure Theory: An Introduction, The Analysis of Harmonic Maps and their Heat Flows, Elliptic Partial Differential Equations. His research articles have been published in top journals of mathematics, including Annals of Mathematics, Acta Mathematica, Journal of the American Mathematical Society, Commutations in Pure and Applied Mathematics, etc.

Prof. Lin was awarded the Alfred P. Sloan Research Fellowship and Presidential Young Investigator in 1989, received the Outstanding Young Researcher award from NSF-China in 1998, the Bocher Memorial Prize of American Mathematical Society in 2002, the Chern Prize at ICCM in 2004, selected as ICM invited speaker in 1990, an Oversea Assessor for Chinese Academy of Sciences in 2002, a member of American Academy of Arts and Sciences in 2004, a fellow of American Mathematical Society in 2015.