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。每一个整数系数多项式方程的集合都有一个几何化身，称为变种。这种几何形式常常支配着原方程组的一组有理解。这个想法可以用“几何决定算术”来概括。该项目的重点是开发理论工具来理解大类别的狄芬汀方程。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，在密码学和编码理论中比比皆是。PI具有开发后一种应用程序的经验。该项目还将资助指导和推广工作，特别关注增加来自代表性不足的数学背景的学生和研究人员的留存，包括在他的主要机构之外领导小型研究项目小组，以及组织专门培养社区意识和归属感的研究会议。PI目前指导5名博士生和2名博士后，并希望保持一个充满活力的研究和培训小组。他还将着手写一本关于代数曲面的算术的书，以填补文学上的空白，帮助教育未来几代的丢番图几何学家。该项目解决了丢番图方程的算术基础问题，其几何化身是曲面。PI将使用贝叶斯推理来设计概率算法，该算法将一组定义（例如，低次del Pezzo或K3曲面）的方程作为输入，并以规定的置信度确定这些方程系统是否具有合理解。期望这些思想能在伽罗瓦群周围的其他算术几何问题中得到广泛应用。在一个相关的方向上，PI将系统地研究理性在数论基础上的行为，通过利用某些类型的四折和由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表面上的因子的Tate猜想的新情况。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.