Birational Geometry and Rational Points

这个项目涉及几何和分析方法来研究丢番图方程的解，丢番图方程是最古老的数学分支之一。丢番图方程的研究涉及寻找多项式等式的整数解，这是一个非常困难的问题--不可能设计出一个具有有限运算次数的过程来确定一般的丢番图方程是否有解。这一数学领域在理论和实践计算机科学的一系列领域中都有应用，从加密协议到纠错和数据管理。它还推动了复杂算法的开发，以便对一些理论预测进行数字测试。本课题旨在加深和扩展这一重要课题的知识。具体地说，该课题涉及代数曲面上有理积分点存在的上同调障碍，K3曲面导出等价的算术蕴涵的研究，高维变种上有界高度有理点的渐近性的几何和解析方法，以及变种族中算术性质的变异研究。

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.