Mathematics

Prof. Dinakar Ramakrishnan
California Institute of Technology

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 will take to be smooth. This talk will introduce the various questions which arise in this topic, and briefly recall the reasonably well understood one-dimensional situation. But then the focus will be on dimension 2, some conjectures relating to the topology and geometry of X(C), and some progress for those covered by the unit ball. The talk will end by mentioning a program (joint with M. Dimitrov)  to establish an analogue of a result of Mazur