Rational Points & Rational Curves on Algebraic Varieties

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This project focuses on problems at the interface of algebraic geometry and number theory, concerning connections between global geometric invariants of algebraic varieties and rational points. Among specific problems to be addressed are:effective computationsof Brauer-Manin obstructions on K3 surfaces, the study of potential density of rational and integral points on higher-dimensional varieties over number fields and function fields, and the study of their asymptotic distribution. One of the key issues is the investigation of rational curves on these varieties and their deformations.Arithmetic geometry studies integral solutions of polynomial equations in several variables with integral coefficients from a geometric point of few. One of the simplest questions is:find rectangular triangles with all three sides integers. This leads to the study of rational points on a circle of radius 1. Higher-dimensional geometric objects display a very high degree of complexity; finding and describing rational points on them poses tremendous theoretical and computational challenges. Advances in arithmetic geometry have important applications in the area of information transmission and storage, cryptography and graph theory.