Varieties of Markoff Type: Arithmetic, Combinatorics, Dynamics

The Markoff equation x2+y2+z2=3xyz, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss (briefly) asymptotics of integer points, and (in some detail) recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical.