**Title: Billiards and Ramified Covers of Curves**

Thomas Schmidt, Oregon State University

**Abstract: **Given a Euclidean polygon, one can glue together copies (made
by reflections through its various sides) of the polygon, to construct a
surface. Straight-line paths with reflection off of the sides of the polygon can
then be studied as geodesics on the constructed flat surface. Moreover, the
complex structure of the plane induces a complex structure on the surface; **dz**
induces an abelian differential.

Conversely, integration of an abelian differential
ω on an algebraic curve X induces a flat structure. Since SL(2, **R**)
acts on the plane, there is a natural action of this group on the space of
abelian diffentials on algebraic curves of fixed genus. The orbit of any fixed
pair (X, ω) projects to the Riemann moduli
space of algebraic curves. On rare occasions, this projection is itself an
algebraic curve.

We report on joint work with Pascal Hubert on the projections into moduli space
of the orbit of pairs (X, ω) when
ω is the pull-back of an abelian
differential by way of a ramified covering.

PDF of slides