Abstract: In this talk, we will study the mean curvature flow in R^3.  We show that the intrinsic diameter of the surface along mean curvature flow is uniformly bounded as one approaches the first singular time, which confirms a conjecture of Haslhofer. In addition, we establish several sharp quantitative estimates of mean curvature flow. This is a joint work with Yiqi Huang (MIT).

Let $q,Q$ be two integral quadratic forms in $m < n$ variables. One can ask when $q$ can be represented by $Q$ - that is, whether there exists an $n \times m$-integer matrix $T$ such that $Q \circ T = q$. Naturally, a necessary condition is that such a representation exists locally, meaning over the real numbers and modulo $N$ for every positive integer $N$. In the absence of local obstructions, does a (global) representation of $q$ by $Q$ exist?

In this talk, we will discuss known results in this direction focusing on recent work with Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi in which such a local-global principle when n-m >2. Our proof establishes much more general effective equidistribution results for periodic orbits of semisimple groups on compact adelic quotients. We will discuss the connection between these topics as well as some new ideas going into the proof of our dynamical results.