Birational Geometry and Rational Points

这个项目是关于几何和分析的方法来研究丢番图方程的解决方案，数学的最古老的分支之一。丢番图方程的研究涉及到寻找多项式等式的整数解，这是一个非常困难的问题--不可能设计出一个有限数量的操作过程来决定一般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找到了应用，从加密协议到纠错和数据管理。它还推动了复杂算法的发展，以数字方式测试一些理论预测。该项目旨在深化和扩展这一重要学科的知识。具体而言，该项目涉及代数曲面上有理点和整点存在的上同调障碍，K3曲面导出等价的算术含义的研究，高维簇上有界高度有理点渐近的几何和分析方法，并与研究算术性质的变化在家庭的品种。

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.