September 17
Joint Colloquium with University of Toronto
Department of Mathematics,
3:30 p.m,
Room 230 Fields
Paul Vojta (UC Berkeley)
The ABCs of diophantine geometry
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 nonconstant
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).

