Probabilistic approaches to Brauer groups and rationality problems

布劳尔群和理性问题的概率方法

• 批准号: 2302231
• 负责人: Anthony Varilly-Alvarado
• 金额: $ 38万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302231&HistoricalAwards=false
• 关键词: Probabilistic; approaches; Brauer; groups; rationality

Diophantine geometry seeks to understand the set of integer or rational solutions to systems of polynomial equations in several variables. Every collection of polynomial equations with integer coefficients has a geometric avatar, called a variety. The geometry of this variety often governs the set of rational solutions to the original system of equations. This idea is summarized by the mantra "Geometry determines Arithmetic". The project focuses on developing theoretical tools to understand large classes of diphantine equations. Applications of understanding these equations, and their related shadows over finite number systems, abound, e.g., in cryptography and coding theory. The PI has experience working on the latter applications. The project will also fund mentoring and outreach efforts, with particular attention to increasing retention of students and researchers from underrepresented backgrounds in Mathematics, including leading small research project groups outside his primary institution, and organizing research conferences that specifically foster a sense of community and belonging. The PI currently advises five PhD students and two postdoctoral scholars, and expects to maintain a vigorous research and training group. He will also embark on a book project on the arithmetic of algebraic surfaces, to fill a gap in the literature, to help educate future generations of diophantine geometers. This project addresses foundational questions in the arithmetic of Diophantine equations whose geometric avatars are surfaces. The PI will use Bayesian inference to devise probabilistic algorithms that take as input a set of equations defining, e.g., a low-degree del Pezzo or K3 surface, and determine, with a prescribed degree of confidence, if these systems of equations have rational solutions. It is expected that these ideas will have wide application in other problems of arithmetic geometry around Galois groups. In a related direction, the PI will systematically study the behavior of rationality over number-theoretic bases, by leveraging the connection between certain kinds of fourfolds and surfaces twisted by Brauer classes. The project naturally leads to considering new cases of the Tate conjecture for divisors on surfaces over finitely generated fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

丢番图几何旨在了解一组整数或合理的解决方案，以系统的多项式方程在几个变量。 每一个整数系数多项式方程的集合都有一个几何化身，称为簇。 这类几何通常支配着原始方程组的有理解集。 这一思想可以概括为“几何决定算术”。 该项目的重点是开发理论工具，以了解大量的diphantine方程。 理解这些方程的应用，以及它们在有限数系统上的相关阴影，比比皆是，例如，在密码学和编码理论方面。 主要研究者具有处理后一种应用程序的经验。 该项目还将资助辅导和外联工作，特别注意增加来自数学代表性不足背景的学生和研究人员的保留，包括在他的主要机构之外领导小型研究项目组，并组织专门培养社区和归属感的研究会议。 PI目前为五名博士生和两名博士后学者提供咨询，并希望保持一个充满活力的研究和培训团队。 他还将着手一本书的项目上的算术代数曲面，以填补空白的文学，以帮助教育后代的丢番图geometers。本计画探讨以曲面为几何化身的丢番图方程演算的基础问题。 PI将使用贝叶斯推理来设计概率算法，该算法将一组方程作为输入，一个低程度的del Pezzo或K3表面，并确定，与规定的置信度，如果这些方程组有合理的解决方案。 预计这些想法将有广泛的应用在其他问题的算术几何周围伽罗瓦群。 在一个相关的方向，PI将系统地研究数论基上的理性行为，通过利用某些类型的四重和布劳尔类扭曲的曲面之间的联系。该项目自然会导致考虑新的情况下，泰特猜想的除数表面上产生的领域。这一奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。