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.

可以通过多项式方程系统对科学和工程的广泛感兴趣系统进行建模。通常，尤其是在计算机科学和网络安全方面，重要的数量可以通过对具有理性系数的多项式方程系统的解决方案来描述，换句话说，代数品种。该项目涉及确定该系统何时具有所有坐标为有理数的解决方案。对于任意方程式系统，很难确定是否存在任何此类合理解决方案。该研究项目使用几何形状和不对称行为对这个问题产生了新的观点。该提案中的研究是由教育和推广活动完成的，包括专注于演讲技能的中级职业生涯研究生的研讨会以及如何利用这些技能来开发和扩大自己的研究计划。研究项目属于两个主要方向。第一个侧重于理性点和对象作为基础场的变化。在这个方向上，该项目在几何理性表面上存在0循环的存在与理性点之间的直接联系，并探讨了新的不对称问题，这些问题将揭示Colliot-Thélène在0-Cycles上的猜想。第二个重点是利用从品种X发出的主要有理图，以获取有关X的Brauer组，理性点和0循环的信息。