CAREER: Arithmetic of Surfaces

职业：曲面算术

• 批准号: 1352291
• 负责人: Anthony Varilly-Alvarado
• 金额: $ 40.51万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1352291&HistoricalAwards=false
• 关键词: CAREER; Arithmetic; Surfaces

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.

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项目。第一个重点是K3曲面：建立在PI和他的合作者的早期工作，并着眼于算术应用，PI将追求系统的，概念性的和实用的方法来显式地构建非分歧的Azumaya代数，表示K3曲面上的超越Brauer类。第二个项目侧重于del Pezzo曲面：证明了Colliot-Thélène和Sanjiang猜想的新案例，即Brauer-Manin障碍足以解释del Pezzo曲面上局部到全局现象的失败，低次曲面上这些障碍的有效计算，以及Hasse原理失败的统计。算术几何中的一个首要主题是研究多变量多项式方程组，约束条件是解的坐标是有理数或整数（例如，11/17或-9）。当不存在这样的解决方案时，人们试图理解缺乏解决方案背后的现象。与多项式系统相关的几何关系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精确。虽然这个项目从理论的角度研究基本问题，但某些类型的多项式方程的解的结构有着很好的应用（例如，在建立从未见过的双方之间传输信息的安全协议方面）。除了研究活动外，PI还将在赠款期间每年夏天开展为期两周的项目。该计划的目标是促进大学一级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的8年级和9年级学生，包括来自STEM领域代表性不足群体的学生。该程序的内容是研究多项式方程有理解的基础材料。