Diophantine problems are those stemming from attempts to solve systems of polynomial equations, while allowing the variables to take on only integral or rational values. Diophantine geometry is the study of diophantine problems using the methods and language of algebraic geometry. It has been noticed by C. F. Osgood, S. Lang, and the speaker, that theorems and conjectures in diophantine geometry often correspond closely to similar statements for holomorphic maps to complex varieties (Nevanlinna theory). For example, there are no non-constant holomorphic maps from the complex line to a Riemann surface of genus $\ge 2$ (Picard's theorem), and a smooth algebraic curve of genus $\ge 2$ over a number field has only finitely many rational points (Faltings' theorem on Mordell's conjecture).

Not much is known about why this is so, but it has led to sweeping conjectures in diophantine geometry and has spurred more work in Nevanlinna theory. This semester's thematic program at the Fields Institute will study the interplay between these two areas, as well as with Arakelov theory (a key tool in diophantine geometry which relies heavily on tools from several complex variables).