CAREER: Rational Points via Asymptotics and Geometry

在科学和工程中感兴趣的广泛系统可以由多项式方程组建模。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组的解的集合来描述，换句话说，通过代数簇来描述。这个项目关注的是确定这样一个系统何时有一个所有坐标都是有理数的解。对于一个任意的方程组，很难确定是否存在这样的有理解。这个研究项目开发了一个新的角度对这个问题使用几何和渐近行为。本提案中的研究得到了教育和推广活动的补充，包括为职业生涯中后期的研究生举办的研讨会，重点是演讲技巧以及如何利用这些技能来发展和扩大研究计划。研究项目主要有两个方向。第一个集中在合理的点和障碍物的行为的变化，作为基地领域的变化。在这个方向上，该项目追求0-圈的存在与几何理性曲面上的理性点之间的直接联系，并探索新的渐近问题，这将揭示Colliot-Thélène关于0-圈的猜想。第二个重点是利用从一个品种X产生的显性理性地图，以获得有关的布劳尔群，理性点，和X的0-圈的信息。

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.