On Diophantine Approximation and Holomorphic Curves

The theorems of Faltings, Siegel and the uniformization theorem for
Riemann surfaces show that for an algebraic variety of dimension one
defined over a number field K
the following conditions are equivalent:

(1) Every holomorphic map from the complex line into the associated
complex space is constant.
(2) The set of all integral points (= points with coordinates in
the ring of algebraic integers of the respective number fields)
on the variety is finite for every finite field extension of K.

It is conjectured by Lang and Vojta
that similar equivalences hold in higher dimensions.

The purpose of this talk is to give an introduction into these
conjectures.