Riemann Surfaces to their Algebraic Equations

A Riemann surface can be recovered from its Riemann matrix, along with the Riemann theta function, as stated by Torelli's theorem. This talk focuses on the computational aspects of this statement, centring around applications of current trends in computational algebraic geometry. We begin with presenting types of machinery for the reconstruction to obtain polynomial equations that define the Riemann surface. In the second part of our presentation, we address how to locate Riemann surfaces in terms of their Riemann matrices. We will present some sets of polygons that can serve as informative starting points. From there, we will move on with the discrete Riemann surfaces to make the sources realizable for our procedures.