Probabilistic approaches to Brauer groups and rationality problems

丢番图几何试图理解多变量多项式方程组的整数或有理解的集合。每一组具有整数系数的多项式方程都有一个几何化身，称为变种。这种几何形式通常支配着原始方程组的有理解集。这一思想被概括为一句口头禅：几何决定算术。该项目的重点是开发理论工具，以理解大类的敌我方程。理解这些方程及其在有限数系统上的相关阴影的应用程序比比皆是，例如在密码学和编码理论中。PI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，特别注意留住数学专业背景不足的学生和研究人员，包括领导其主要机构以外的小型研究项目小组，以及组织专门培养社区意识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，并预计将保持一支充满活力的研究和培训团队。他还将着手一个关于代数曲面算术的图书项目，以填补文献中的一个空白，帮助教育未来几代丢番图几何学家。这个项目解决了丢番图方程算法中的基本问题，该方程的几何头像是曲面。PI将使用贝叶斯推理来设计概率算法，该算法将定义例如低次Del Pezzo或K3曲面的一组方程作为输入，并以规定的置信度来确定这些方程系统是否具有有理解。这些思想可望在伽罗华群周围的其他算术几何问题中得到广泛的应用。在一个相关的方向上，PI将系统地研究数论基础上的理性行为，通过利用某些类型的四重折叠和被Brauer类扭曲的曲面之间的联系。该项目自然导致考虑有限生成场上表面上的约数的泰特猜想的新案例。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.