Markoff numbers are positive integers that occur in some positive integer triple (x,y,z) satisfying x2+y2+z2 =3xyz.
In this talk I will explain how Markoff numbers are related to triangulated polygons, the numerical frieze patterns of Conway and Coxeter, the Farey / Stern-Brocot tree, and superbases of Z2. It will be shown that each Markoff number has enumerative significance: it counts the number of perfect matchings in a suitable graph.
Markoff numbers also carry geometrical meaning, and combinatorial methods can be applied to geometric questions. In particular, one can use "Markoff polynomials" (a generalization of Markoff numbers) to show that for a generic hyperbolic structure on the once-punctured torus, there is at most one simple closed geodesic of any given length.
This is joint work with Dylan Thurston and with (former or current) Boston-area undergraduates Gabriel Carroll, Andy Itsara, Ian Le, Gregg Musiker, Gregory Price, and Rui Viana.