Metrics whose Ricci tensor vanishes are called Ricci-flat, and many interesting examples of these metrics arise in the Kahler setting. I will present some background of K3 surfaces, which give fundamental examples of Ricci-flat Kahler metrics.   

I will present a result in combinatorial number theory due to Renling Jin:  if A and B are subsets of the natural numbers with positive Banach density, then their sum A+B is piecewise syndetic, a robust notion of largeness for subsets of the natural numbers.  The result bears a resemblance to a theorem of real analysis due to Steinhaus:  if C and D are subsets of the real line with positive Lebesgue measure, then their sum C+D contains an interval.  We will in fact see that these two theorems are both specific instances of a more general theorem using the language of nonstandard analysis.

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.