Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G, A))$ of the Cayley graph $\Gamma(G, A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding diam$(G) := max_A$ diam$(\Gamma(G, A))$.
It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$. In 2011, Helfgott and Seress gave a quasipolynomial bound, namely, $O\left (e^{(log n)^{4+\epsilon}}\right )$. We will discuss a recent, much simplified version of the proof.[-]

The study of groups often sheds light on problems in various areas of mathematics. Whether playing the role of certain invariants in topology, or encoding symmetries in geometry, groups help us understand many mathematical objects in greater depth. In coarse geometry, one can use groups to construct examples or counterexamples with interesting or surprising properties. In this talk, we will introduce one such metric object arising from finite quotients of finitely generated groups, and survey some of its useful properties and associated constructions.[-]