Birational Geometry and Rational Points

New York University, New York NY

Investigators

Abstract

This project is concerned with geometric and analytic approaches to the study of solutions of Diophantine equations, one of the oldest branches of mathematics. The study of Diophantine equations involves finding whole number solutions to polynomial equalities, a remarkably difficult problem -- it is not possible to devise a process with a finite number of operations that can decide whether a general Diophantine equation has a solution. This area of mathematics is finding applications in a range of fields in theoretical and practical computer science, from encryption protocols to error correcting and data management. It is also driving the development of sophisticated algorithms to test numerically some of the theoretical predictions. This project aims to deepen and extend knowledge in this important subject. Specifically, the project is concerned with cohomological obstructions to the existence of rational and integral points on algebraic surfaces, investigations of arithmetic implications of derived equivalence for K3 surfaces, geometric and analytic approaches to asymptotics of rational points of bounded height on higher-dimensional varieties, and with the study of variation of arithmetic properties in families of varieties.