Given a fractal set $E$ on the plane and a set $F$ of directions, can we find one direction $\theta\in F$ such that the orthogonal projection $\Pi_{\theta} E$ is large?

We will survey some classical and modern projection theorems and discuss their applications.

We discuss some open problems and some new results about the topology of arithmetic locally symmetric spaces. Among the new results is a proof of a conjecture of Gelander stating that the topology of these manifolds can be bounded just in terms of the volume. The tools in the proofs include an arithmetic version of the classical Margulis lemma about discrete subgroups of Lie groups and  makes use of the existence of certain curious word maps called almost laws.

All notions will be explained. Based on joint projects with M. Fraczyk, J. Raimbault, Lvzhou (Joe) Chen and Homin Lee.

Abstract:

Mean curvature flow can be continued through singularities using a weak solution to the flow. Two major weak solutions are Brakke flow and level set flow. Brakke flow is defined with an inequality which makes it tantamount to a subsolution to mean curvature flow, whereas level set flow may attain positive measure and is properly thought of as a supersolution. In this talk, we will discuss these weak solutions, and we will relate uniqueness problems to multiplicity problems for mean curvature flow. In particular, we will discuss how Brakke flows with only generic singularities achieve equality in their defining inequality. This uses an analysis of worldlines in Brakke flows, analogous to the theory of singular Ricci flows. This talk will be aimed at those without background in mean curvature flow.