Title: The geometry of singular Kahler-Einstein metricsAbstract: The existence of a Kahler-Einstein metric can be a powerful tool to study complex manifolds. To use similar methods in the study of singular varieties, we need to understand the geometry of singular Kahler-Einstein metrics. I will discuss what some of the basic questions are, as well as […]

Title: Kakeya sets in R^3 Abstract: A Kakeya set is a compact subset of R^n that contains a unit line segment pointing in every direction.  Kakeya set conjecture asserts that every Kakeya set has Minkowski and Hausdorff dimension n. We prove this conjecture in R^3 as a consequence of a more general statement about union […]