Arithmetic of Rationally Simply Connected Varieties

有理单连通簇的算术

• 批准号: 1405709
• 负责人: Jason Starr
• 金额: $ 16.8万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405709&HistoricalAwards=false
• 关键词: Arithmetic; Rationally; Simply; Connected; Varieties

Polynomial equations are ubiquitous in science and engineering. Particularly important are the kinds of polynomial equations that arise in fields like cryptography and computer science, where the inputs and the outputs of the polynomials are fractions or their near-cousins (elements in a "global field"). For these equations, it is important not only to know that there are solutions in fractions, but also to know that there are many solutions, and that we can find such solutions efficiently in a finite amount of runtime (i.e., there are efficient bounds on the size of the numerators and denominators of some fraction solution). Investigating these questions is a major goal of algebraic geometry. Remarkably, ideas from geometry, particularly ideas suggested by physics, give a proof of existence of integral solutions for many special polynomial equations over global function fields. The main goal of this project is to exploit this advance and prove the veracity of certain conjectured bounds on rational solutions of these special polynomial equations ("rationally simply connected" systems of equations) over global function fields, thus giving an efficient algorithm for finding rational solutions. In addition, there are several educational and training goals: improving the writing of math students through weekly meetings, helping with a summer math camp for high school students, and holding twice-annual one-day training workshops for math students coinciding with one an important weekend workshop series in algebraic geometry. Technically, the main objective is to study the Batyrev-Manin conjecture on asymptotics of rational points of bounded height over global function fields for the special class of "rationally simply connected" varieties: a class that includes, for instance, the projective homogeneous varieties so ubiquitous in representation theory. Recent work relates rational points over global function fields to geometric properties of moduli spaces of rational curves on lifts of the varieties over the complex numbers. By exploiting this, there is hope to give efficient height bounds on rational points implied by the Batyrev-Manin Conjecture, and hopefully to settle the conjecture in important special cases. Secondary goals are to understand the Picard groups of these moduli spaces (roughly, the different possible "height functions"), and to extend the amazingly successful story of Neron models to a more general class of varieties than Abelian varieties. Broader impacts include continuing and extending a mathematical writing and professional development seminar begun by the PI, to continue the partial support of the PI for the Mathematics Summer Camp at Stony Brook University, and to institute a new series of one-day training workshops for graduate students timed to coincide with the AGNES series of twice-annual weekend workshops in algebraic geometry.

多项式方程在科学和工程中无处不在。特别重要的是密码学和计算机科学等领域中出现的多项式方程，其中多项式的输入和输出是分数或其近亲（“全局域”中的元素）。对于这些方程，重要的是不仅要知道分数有解，而且要知道有很多解，并且我们可以在有限的运行时间内有效地找到这些解（即，存在有效界限某些分数解的分子和分母的大小）。研究这些问题是代数几何的一个主要目标。值得注意的是，几何学的思想，特别是物理学提出的思想，证明了全局函数域上许多特殊多项式方程的积分解的存在性。该项目的主要目标是利用这一进步并证明这些特殊多项式方程（“有理简单连接”方程组）在全局函数域上的有理解的某些猜想界限的准确性，从而给出一种有效的算法来寻找有理数解决方案。此外，还有几个教育和培训目标：通过每周会议提高数学学生的写作能力，帮助高中生数学夏令营，以及每年两次为数学学生举办为期一天的培训研讨会，恰逢重要的一年代数几何周末研讨会系列。从技术上讲，主要目标是研究“有理简单连通”簇这一特殊类的全局函数域上有界高度有理点渐近的巴特列夫-马宁猜想：该类包括例如射影同质簇，在表示论中无处不在。最近的工作将全局函数域上的有理点与有理曲线在复数上的簇升力上的模空间的几何性质联系起来。通过利用这一点，我们有望在巴特列夫-马宁猜想所隐含的有理点上给出有效的高度界限，并有望在重要的特殊情况下解决该猜想。次要目标是理解这些模空间的皮卡德群（粗略地说，不同的可能的“高度函数”），并将 Neron 模型的惊人成功故事扩展到比阿贝尔簇更通用的簇。更广泛的影响包括继续和扩展由 PI 发起的数学写作和专业发展研讨会，继续 PI 对石溪大学数学夏令营的部分支持，以及为研究生举办一系列新的为期一天的培训研讨会学生们的时间与 AGNES 系列每年两次的代数几何周末研讨会相一致。