Given a polynomial P(x,y) in two variables we can ask
about all the rational solutions to P(x,y) = 0. The Mordell conjecture
(proved by Faltings) says that there are only finitely many solutions
so long as the genus of the curve P=0 is two or more.
I will explain joint work with Brian Lawrence that relates this statement to the geometry of period mappings. (The period mapping is an invariant of any family of complex algebraic varieties: I will review it in the talk.)
This leads to new proofs of some cases of Mordell's conjecture and
-- if one assumes a conjecture of purely topological nature, concerning
"big monodromy" -- it can be used to prove all cases of it.
Potentially, this approach might generalize to other situations. This leads to some questions about the transcendence properties of period mappings, which I will discuss.