A recent breakthrough of Wu and Yau asserts that a compact projective Kahler 
manifold with negative holomorphic sectional curvature must have ample 
canonical line bundle. In the talk, we will talk about some of the recent 
advances along this direction. In particular, we will discuss the case 
where the manifold is a noncompact Kahler manifold. We will also discuss 
the case when the Kahlerity is a priori unknown. Part of these are joint 
work with S. Huang, L.-F. Tam, F. Tong.

This is the fifth in a series of lectures on naive descriptive set theory based on an expository paper by Matt Foreman. We continue the discussion of universality properties of Polish spaces and subspaces of Polish spaces.

A single soap bubble has a spherical shape since it minimizes its surface area subject to a fixed enclosed volume of air.  When two soap bubbles collide, they form a “double-bubble” composed of three spherical caps.  The double-bubble minimizes total surface area among all sets enclosing two fixed volumes.  This was proven mathematically in a landmark result by Hutchings-Morgan-Ritore-Ros and Reichardt using the calculus of variations in the early 2000s.  The analogous case of three or more Euclidean sets is considered difficult if not impossible.  However, if we replace Lebesgue measure in these problems with the Gaussian measure, then recent work of myself (for 3 sets) and of Milman-Neeman (for any number of sets) can actually solve these problems.  We also use the calculus of variations.  Time permitting, we will discuss an improvement to the Milman-Neeman result and applications to optimal clustering of data and to designing elections that are resilient to hacking.  http://arxiv.org/abs/1901.03934

The Allen-Cahn equation appears in the study of phase-transitions for a fluid with two-stable phases. It has been known from the work of Modica and Mortola that the level sets of the solution behave at large scales as minimal surfaces. This fact suggests that global solutions to the Allen-Cahn equation have the same rigidity properties as global minimal surfaces. In particular De Giorgi conjectured that the Bernstein theorem for minimal graphs is valid for the Allen-Cahn equation. I will discuss the history of this conjecture together with some of its nonlocal counterparts.