CAREER: Structure and Interpolation in Number Theory and Beyond

首席研究员(PI)将调查数论中出现的看似不同的数据之间的联系，数论是一个与科学和工程中的许多领域有着深刻联系的数学领域。特别是，PI将研究与整数、分数和被称为“数字域”的相关集合相关联的结构。PI的目的是阐明这些结构如何在某些无限的数域集合中变化，以及它们的行为如何解释目前数论、几何学和其他领域中的神秘现象。在项目过程中，国际和平研究所将组织讲习班，教育研究生了解最近的研究进展，并促进不同的合作。为了让本科生参与激发她的研究的主题，PI将开发一门融合了艺术方法的创新、互动课程。在一定程度上，通过与博物馆馆长的合作，PI还将为更广泛的社区组织公共展览。这个项目的研究重点是L-功能、自同构形式和p-adic方法，作为了解特定结构以及它们如何在家庭中变化的工具。预期的结果包括格林伯格-岩泽主要猜想(将伽罗华群的结构与p元L函数联系起来)和布洛赫-加藤猜想(将某些群的秩等同于相关L函数在中心点的消失顺序，类似于Birch和Swinnerton-Dyer猜想)的例子。关键部分相互联系，使一个方面的进展相互促进，包括调查L函数和自同构型的p-进和阿基米德性质之间的相互作用，研究酉下村变种的p-进方面，以及桥接p-进自同构形的不同方法，以深入了解相关数据的行为，其意义扩展到数论、同伦理论和算术几何。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。

The principal investigator (PI) will investigate connections between seemingly disparate data arising in number theory, a field of mathematics with deep ties to many areas in sciences and engineering. In particular, the PI will study structures associated to the whole numbers, fractions, and related sets called "number fields." The PI aims to elucidate how these structures vary across certain infinite collections of number fields, and furthermore, how their behavior explains currently mysterious phenomena in number theory, geometry, and beyond. In the course of the project the PI will organize workshops to educate graduate students about recent research developments and to promote diverse collaborations. To engage undergraduate students in topics motivating her research, the PI will develop an innovative, interactive course integrating approaches from the arts. Partly through a collaboration with museum curators, the PI will also organize public exhibits for the broader community.The research in this project focuses on L-functions, automorphic forms, and p-adic methods as tools to understand particular structures and how they vary in families. Anticipated consequences include progress toward instances of the Greenberg-Iwasawa main conjectures (connecting the structures of Galois groups with p-adic L-functions) and the Bloch-Kato conjectures (equating ranks of certain groups with the order of vanishing of associated L-functions at the central point, in analogy with the Birch and Swinnerton-Dyer conjecture). Key components, which are interconnected so that progress on one advances the others, include investigating the interplay between p-adic and archimedean properties of L-functions and automorphic forms, studying p-adic aspects of unitary Shimura varieties, and bridging different approaches to p-adic automorphic forms to gain insight into the behavior of associated data whose significance extends into number theory, homotopy theory, and arithmetic geometry.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.