Computing with Maple on Riemann surfaces arising from algebraic curve

Our group is interested in computations on Riemann surfaces that arise from irreducible plane algebraic curves, and this talk will highlight some of the work that we have done. I will begin with a brief outline of how a Riemann surface is obtained from an algebraic curve, along the way introducing some methods that have already been implemented in the Maple package \verb!algcurves!. For example, procedures to compute bases of both the homology and cohomology, as well as the Riemann matrix of a Riemann surface originating from an algebraic curve. Subsequently I will discuss the Abel map, the vector of Riemann constants and algorithms to compute both; implementations of these algorithms will be included in future versions of the \verb!algcurves! package. The talk will conclude with Maple demonstrations of these procedures as time allows.