Zeta functions are central topics in number theory and arithmetic algebraic geometry.
They can be viewed as generating functions for counting "points" of polynomial equations.
In this lecture, we will explain various zeta functions from this point of view, including
the Riemann zeta function, the Hasse-Weil zeta function, and zeta functions over finite fields.

Nonlinear elliptic PDEs play a central role in many geometric and physical applications. To construct solutions and determine their qualitative behavior, our key tool is the maximum principle. We will discuss the role of elliptic PDE and the maximum principle in the solution of two important problems (the plateau problem, and optimal transport), and mention some interesting open questions.