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.

首席研究者（PI）将研究数字理论中似乎不同的数据之间的联系，这是一个数学领域，与科学和工程领域的许多领域有着深厚的联系。特别是，PI将研究与总数，分数和相关集相关的结构，称为“数字字段”。 PI旨在阐明这些结构在某些无限的数字字段集合中如何变化，此外，它们的行为如何解释了数字理论，几何形状及以后的当前神秘现象。在该项目的过程中，PI将组织研讨会，以教育研究生有关最近的研究发展并促进多样化的合作。为了让本科生参与激励她的研究的主题，PI将开发一种创新的，互动的课程，以整合艺术的方法。 PI部分通过与博物馆策展人的合作，还将为更广泛的社区组织公众展览。该项目的研究侧重于L功能，自动形式和P-ADIC方法，作为了解特定结构及其在家庭中如何变化的工具。预期的后果包括朝着格林伯格 - iwasawa主要猜想的实例（将Galois组的结构与P-ADIC L功能连接起来）和Bloch-Kato猜想（某些群体的等级等同于与Birch and Swinnerton-Dyerertouter-DyeriuteRuter-DyeriuteRuter类似）。与之相互联系的关键组成部分可以进步，包括调查L功能和自增生形式的P-ADIC和ARCHIMEDEAN特性之间的相互作用，研究单位shimura品种的p-辅助方面算术几何形状。该奖项反映了NSF的法定任务，并被认为是使用基金会的知识分子优点和更广泛影响的评论标准的评估值得支持的。