We will introduce the definition of -adic sheaves in algebraic geometry. The talk will begin by motivating why we might define such things and end with the rigorous definition of $\mathbb{Q}_\ell$-sheaves.

Abstract: I will give an introduction to the theory of convex projective structures on closed manifolds, and explain some of the ways they can be used to understand these manifolds at a group-theoretic level. Then we will go over a recent result of Zimmer, which states that the fundamental group of a closed convex projective manifold is either a nontrivial product, a lattice in a higher rank symmetric space, or "rank one" (i.e. has strong hyperbolic-like properties).

Let $G = G(n, \lambda / n)$ be an instance of the Erdos-Renyi random graph model on vertices, with edges present independently of one another with probability $p = \lambda / n$. When $\lambda1$, it is known that this random graph has a unique giant component with high probability as $n \to \infty$ (where giant means that it contains a positive fraction of all vertices). What can we say about the mixing time of random walk on this giant component? It turns out that the answer depends very much on the starting point of the walk. In this work we show that from a typical starting point the cutoff phenomenon takes place. The limiting order of the mixing time depends on the speed and the entropy of random walk on a Galton-Watson tree which is the Benjamini-Schramm limit of the giant component of a random graph. The result extends to graphs with prescribed degree sequences.

In this talk I will explain a categorification of sheaf theory in derived algebraic geometry using the framework of categories of correspondences, and describe its application to the construction of fully extended topological field theories.