Birational Geometry and Rational Points

双有理几何和有理点

• 批准号: 1601912
• 负责人: Yuri Tschinkel
• 金额: $ 18万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601912&HistoricalAwards=false
• 关键词: Birational; Geometry; Rational; Points

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.

该项目涉及用于研究二氧化苯胺方程解决方案的几何和分析方法，这是数学最古老的分支之一。对苯胺方程的研究涉及找到多项式平等的整数解决方案，这是一个非常困难的问题 - 不可能设计具有有限数量的操作的过程，这些过程可以决定一般的二磷剂方程是否具有解决方案。该数学领域是在理论和实用的计算机科学领域中找到应用程序，从加密协议到错误纠正和数据管理。它还推动了复杂算法的开发，以数字测试一些理论预测。该项目旨在加深和扩展这一重要主题的知识。该项目与代数表面上的理性和组成部分存在的共同体学障碍有关，研究对K3表面的算术含义的算术含义的调查，几何学和分析方法对高度界定范围界面的数量和分析方法的变化，以及范围的变化界面，以及范围的变体，以及范围的变体，以及多样化的多样性，以及多样化的多样性。品种家族的特性。