CAREER: Rational Points via Asymptotics and Geometry

职业：通过渐近学和几何学有理点

• 批准号: 1553459
• 负责人: Bianca Viray
• 金额: $ 45万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1553459&HistoricalAwards=false
• 关键词: CAREER; Rational; Points; via; Asymptotics

A wide range of systems of interest in science and engineering can be modeled by systems of polynomial equations. Often, particularly in computer science and cyber security, important quantities can be described by the set of solutions to a system of polynomial equations with rational coefficients, in other words by an algebraic variety. This project is concerned with determining when such a system has a solution where all coordinates are rational numbers. For an arbitrary system of equations, it can be quite difficult to determine whether there exist any such rational solutions. This research project develops a new perspective on this problem using geometry and asymptotic behavior. The research in this proposal is complemented by educational and outreach activities, including workshops for mid-to-late career graduate students focusing on presentation skills and how these skills can be leveraged to develop and broaden one's research program. The research projects fall in two main directions. The first focuses on the change in behavior of rational points and obstructions as the base field varies. In this direction, the project pursues a direct connection between the existence of 0-cycles and rational points on geometrically rational surfaces and explores new asymptotic questions that will shed light on Colliot-Thélène's conjecture on 0-cycles. The second focuses on leveraging dominant rational maps emanating from a variety X to obtain information about the Brauer group, rational points, and 0-cycles of X.

科学和工程中广泛的系统都可以用多项式方程组来建模。通常，特别是在计算机科学和网络安全中，重要的量可以通过具有有理系数的多项式方程系统的一组解来描述，换句话说，可以通过代数变量来描述。这个项目是关于确定当这样一个系统有一个解，其中所有的坐标都是有理数。对于任意的方程组，要确定是否存在这样的有理解是相当困难的。本研究项目利用几何和渐近性为这一问题提供了新的视角。该提案中的研究还辅以教育和推广活动，包括为中后期职业研究生举办的研讨会，重点关注演讲技巧，以及如何利用这些技能来发展和扩大自己的研究项目。研究项目主要分为两个方向。第一个重点是随着基本场的变化有理点和障碍物的行为变化。在这个方向上，该项目追求0环的存在与几何有理曲面上的有理点之间的直接联系，并探索新的渐近问题，这些问题将有助于阐明colliot - th<s:1> l<e:1>关于0环的猜想。第二个重点是利用从各种X发出的主导有理映射来获取关于X的Brauer群、有理点和0环的信息。