Rational Points & Rational Curves on Algebraic Varieties

该项目的重点是代数几何和数论的接口问题，涉及代数簇和有理点的全局几何不变量之间的连接。其中具体的问题要解决的是：有效的计算Brauer-Manin障碍的K3表面，研究潜在的密度合理和整点的高维品种超过数场和功能领域，并研究其渐近分布。算术几何从几何角度研究整系数多元多项式方程组的积分解，而算术几何则从几何角度研究整系数多元多项式方程组的积分解。最简单的问题之一是：找到三条边都是整数的直角三角形。这就引出了半径为1的圆上有理点的研究。高维几何对象显示出非常高的复杂性，寻找和描述它们上的有理点构成了巨大的理论和计算挑战。算术几何的发展在信息传输与存储、密码学和图论等领域有重要的应用。

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.