#### 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.

Done