Abstract: Constructing interesting vector bundles in geometry is typically difficult. I will discuss how the addition of a unipotent structure to a vector bundle, which is a special tape of flag, surprisingly makes this much easier. I will then discuss some applications to an additive analog of results of Gabber—de Jong on the Brauer group and group actions with wild stabilisers. This is joint work with Bragg and Mathur. 

Abstract: In this lecture, we focus on a new online algorithm for vector balancing. Given any sequence of vectors, we first randomly and independently flip the signs of every vector. Then we design an online algorithm that discards only a small fraction of the signed vectors so that all partial sums of the remaining ones remain confined inside a given convex body. Combined with the partial coloring method (which we review), this algorithm proves weak forms of the Komlós conjecture. We will explain this simple randomized procedure and the natural geometric constant that controls its performance.