CAREER: Arithmetic of Surfaces

PI将研究在全局域上定义的代数曲面上有理点的存在性和分布。拟议的研究有两个主要组成部分项目。第一个重点是K3曲面：在PI和他的合作者早期工作的基础上，从算术应用的角度出发，PI将追求系统的、概念性的和实用的方法，以明确地在K3曲面上构造表示超越Brauer类的非分支Azumaya代数。第二个项目集中在del Pezzo曲面上：证明了colliot - th<s:1> l<e:1>和sanso猜想的新情况，即Brauer-Manin障碍足以解释del Pezzo曲面上局域到全局现象的失败，低次曲面上这些障碍的有效计算，以及Hasse原理失败的统计。算术几何的一个重要主题是研究多变量多项式方程组，其解的坐标必须是有理数或整数（例如，11/17或-9）。当不存在这样的解决方案时，人们试图理解解决方案缺乏背后的现象。与多项式系统相关的几何结构可能会阻碍解的存在，而本项目试图在多项式方程系统定义曲面的某些情况下，使这种直觉变得精确。虽然这个项目从理论的角度研究基本问题，但某些多项式方程的解的结构有很好的应用记录（例如，在从未见过面的双方之间建立信息传输的安全协议）。除了研究活动外，项目负责人还将在资助期间每年夏天开展一个为期两周的项目。该计划的目标是提高大学水平STEM专业的入学率和坚持率。目标人群包括来自休斯顿独立学区（Houston Independent School District）即将升学的八年级和九年级学生，包括来自STEM领域代表性不足群体的学生。该计划的内容是研究多项式方程的有理解的基础材料。

The PI will study the existence and distribution of rational points on algebraic surfaces defined over global fields. The proposed research has two mayor component projects. The first focuses on K3 surfaces: building on earlier work of the PI and his collaborators, and with a view towards arithmetic applications, the PI will pursue systematic, conceptual and practical methods to explicitly construct unramified Azumaya algebras representing transcendental Brauer classes on K3 surfaces. The second project focuses on del Pezzo surfaces: proving new cases of a conjecture of Colliot-Thélène and Sansuc that Brauer-Manin obstructions suffice to explain failures on local-to-global phenomena for del Pezzo surfaces, efficient computation of these obstructions on low-degree surfaces, and statistics on failures of the Hasse principle.An overarching theme in arithmetic geometry is the study of systems of polynomial equations in many variables, with the constraint that the coordinates of the solutions be rational numbers or integers (for example, 11/17, or -9). When no such solutions exist, one tries to understand the phenomena behind the absence of solutions. The geometry associated to a system of polynomials bears on the possible obstructions to the existence of solutions, and this project seeks to make such an intuition precise in some cases when a system of polynomial equations defines a surface. Although this project studies fundamental questions from a theoretical point of view, the structure of solutions to certain kinds of polynomial equations has well-documented applications (for example, in establishing secure protocols for the transmission of information between two parties that have never met). In addition to research activities, the PI will run a two-week program each summer for the duration of the grant. The goal of the program is to foster enrollment and persistence rates in STEM majors at the college level. The target demographic consists of rising 8th and 9th graders from the Houston Independent School District, including students from underrepresented groups in STEM fields. The content of the program is foundational material in the study of rational solutions to polynomial equations.