Combinatorial Mutations of Fano Polygons

This talk will be an introduction to combinatorial mutations. These are transformations of lattice polytopes which play an important role in mirror-symmetric approaches to classification theory. Our primary focus will be on the case of surfaces, where the theory is particularly rich. It is conjectured that there is a one-to-one correspondence between mutation classes of Fano polygons and Q-Gorenstein deformation classes of Fano orbifold surfaces. Our main examples will be weighted projective planes, for which mutation classes of the associated polygons (triangles) are closely related to integer solutions of Markov-type Diophantine equations.