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CAREER: Structure and Interpolation in Number Theory and Beyond

职业：数论及其他领域的结构和插值

基本信息

批准号：
1751281
负责人：
Ellen Eischen
金额：
$ 40万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1751281&HistoricalAwards=false
关键词：
CAREER Structure Interpolation Number Theory

项目摘要

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.

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的许多领域有着深厚联系的数学领域。特别是，PI将研究与整数，分数和相关集合（称为“数域”）相关的结构。PI旨在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，几何学和其他领域的神秘现象。在项目过程中，PI将组织研讨会，向研究生介绍最新的研究进展，并促进各种合作。为了让本科生参与激发她研究的主题，PI将开发一个创新的，互动的课程，整合艺术的方法。通过与博物馆馆长的合作，PI还将为更广泛的社区组织公开展览。该项目的研究重点是L-函数，自守形式和p-adic方法，作为理解特定结构及其在家庭中如何变化的工具。预期的结果包括朝着格林伯格-岩泽主图（将伽罗瓦群的结构与p进L-函数联系起来）和布洛赫-加藤图（将某些群的秩与相关L-函数在中心点的消失阶等同起来，类似于伯奇和斯温纳顿-戴尔猜想）的实例取得进展。关键组成部分，这是相互联系的，使一个进步的其他，包括调查之间的相互作用的p-adic和阿基米德性质的L-函数和自守形式，研究p-adic方面的酉志村品种，并桥接不同的方法，以p-adic自守形式，以深入了解行为的相关数据，其意义延伸到数论，同伦理论，该奖项反映了NSF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。