**Dinakar Ramakrishnan**

Caltech, USA

December 29, 2016

**Rational Points**:
Since time immemorial, people have been trying to understand the rational number solutions of systems of homogeneous polynomial equatons 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 will take to be smooth. This talk will introduce the various questions of this topic, and briefly recall the
reasonably well understood one-dimensional situation. But then the focus will be on dimension 2, some conjectures involving the geometry and topology of $X(CC)$, and some progress for those covered by the unit ball. The talk will end with a program (joint with Mladen Dimitrov) to establish an analogue of a result of Mazur.