Arithmetic of Rationally Simply Connected Varieties

多项式方程在科学和工程中普遍存在。特别重要的是出现在密码学和计算机科学等领域的多项式方程，其中多项式的输入和输出是分数或它们的近亲(“全局域”中的元素)。对于这些方程，重要的是不仅要知道分数形式的解，而且要知道有很多解，并且我们可以在有限的运行时间内高效地找到这样的解(即，某些分数解的分子和分母的大小存在有效的界限)。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，特别是物理学的思想，给出了全局函数域上许多特殊的多项式方程的积分解的存在性的证明。这个项目的主要目的是利用这一进展，证明这些特殊的多项式方程(“有理单连通”方程组)在全局函数域上有理解的某些猜测界的准确性，从而给出一个寻找有理解的有效算法。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，帮助高中生参加数学夏令营，每年为数学学生举办两次为期一天的培训研讨会，同时举办一次重要的周末代数几何研讨会系列。从技术上讲，主要目的是研究特殊的“有理单连通”簇的Batyrev-Manin猜想：这类簇包括，例如，在表示理论中无处不在的射影齐次簇。最近的工作将整体函数域上的有理点与复数上有理曲线的模空间的几何性质联系起来。通过利用这一点，有望给出由Batyrev-Manin猜想所隐含的有理点的有效高度界，并有望在重要的特殊情况下解决该猜想。次要目标是了解这些模空间的Picard群(粗略地说，不同可能的“高度函数”)，并将农隆模型令人惊叹的成功故事扩展到比阿贝尔变种更一般的变种。更广泛的影响包括继续和延长由国际数学协会开始的数学写作和专业发展研讨会，继续为国际数学协会在石溪大学的数学夏令营提供部分支持，以及为研究生开设一系列新的为期一天的培训讲习班，时间安排在Agnes一年两次的代数几何周末讲习班的时间段。

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.