Leonid Monin

Hyperelliptic curves with many automorphisms and triangular billiards

One can visualise a complex algebraic curve as a polygon in the complex plane with parallel equal sides identified by translations. Indeed, the complex structure of the plane descends to the resulting surface making it into a Riemann surface, or equivalently into an algebraic curve. But how one identifies which algebraic curve it is? Usually this question is impossible to answer. In this talk I will give a series of examples of such polygons for which one can explicitly determine the equations of corresponding algebraic curves. I will also explain how one can come up with these examples by studying billiards in triangles with rational angles.