Tutorial #2: Riemann surfaces: number, function theory, and geometry.

Caleb Ashley
Boston College

In our second tutorial we continue to investigate the rich connections of algebraic geometry and number theory via Riemann surfaces. The theory of Riemann surfaces is an extremely rich wonderland, although introduced by B. Riemann 160 years ago, mathematicians will be investigating Riemann surfaces for a least a hundred more years. Riemann surfaces are special complex manifolds in that they motivate more general phenomena for higher dimensional complex manifolds. We will pursue our connections along a main object of current research known as the moduli space of genus g Riemann surfaces, Mg. In particular, we emphasize the following themes: construction (especially topological, geometric, algebraic structures), the theory of equations (a la the 1st theorem of algebraic geometry), the theory of numbers (especially the role of underlying scalars defining our equations). Finally we frame the problem of distinguishing Riemann surfaces from one another.

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