Alladi Ramakrishnan Hall

Rational points

Dinakar Ramakrishnan

Caltech

Since time immemorial, people have been trying to understand the
rational number solutions of systems of homogeneous polynomial equations
with integer coefficients (called a Diophantine system). It is more
convenient to think of them as rational points on associated projective
varieties X, which we wll take to be smooth. This talk will introduce the
various questions of this topic, and briefly review the reasonably well
understood one-dimensional situation. But then the focus will be on
dimension 2, and some progress for those covered by the unit ball will be
discussed. The talk will end by mentioning a program (with M. Dimitrov) to
establish an analogue of a result of Mazur.