CAREER: p-adic and mod p Galois representations

职业生涯：p-adic 和 mod p Galois 表示

• 批准号: 1564367
• 负责人: David Savitt
• 金额: $ 16.49万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1564367&HistoricalAwards=false
• 关键词: CAREER; adic; mod; Galois; representations

The PI's research is in number theory and representation theory; its goal, broadly, is to understand the Galois representations associated (sometimes conjecturally) to automorphic forms and automorphic representations. The PI will undertake several projects related to the conjectures of Serre, Fontaine-Mazur, and Breuil-Mezard (and their generalizations) as well as to the emerging p-adic and mod p Langlands correspondences. The PI will study the weight part of Serre's conjecture for reductive groups over number fields, with the goal of giving an explicit Serre weight recipe in considerable generality. The investigator will produce evidence for generalizations of the Breuil-Mezard conjecture, and will prove some cases of such a generalization, with applications to the Langlands program. Another component of the project involves the explicit reduction modulo p of certain p-adic Galois representations.Number theory is one of the oldest branches of mathematics. At its most fundamental, number theory is the study of whole number solutions to equations, although sophisticated modern techniques can sometimes give the appearance of being far-removed from this goal. Number theory has many important practical applications; for instance, most cellular telephone calls are protected by a code based on elliptic curves, one of the primary objects of study in the investigator's field. In addition to these broader impacts, this award will support several educational initiatives, including a transition program for minority calculus students at the University of Arizona, the development of an undergraduate problem-solving curriculum, and the recruitment of underrepresented minority students to high school mathematics summer programs. The PI is one of the directors of Canada/USA Mathcamp, a summer program for mathematically talented high school students.

PI的研究是在数论和表示论;其目标，广泛地说，是理解伽罗瓦表示相关（有时是抽象的）自守形式和自守表示。 PI将进行几个与Serre，Fontaine-Mazur和Breuil-Mezard（及其推广）以及新兴的p-adic和mod p Langlands对应关系相关的项目。 PI将研究Serre猜想的权重部分，用于数域上的还原群，目标是给出一个显式的Serre权重配方。研究者将为Breuil-Mezard猜想的推广提供证据，并将证明这种推广的一些情况，并应用于朗兰兹计划。 该项目的另一个组成部分涉及某些p-adic伽罗瓦表示的显式约化模p。数论是数学中最古老的分支之一。 数论最基本的是研究方程的整数解，尽管复杂的现代技术有时会给人一种远离这一目标的印象。 数论有许多重要的实际应用;例如，大多数蜂窝电话呼叫都受到基于椭圆曲线的代码的保护，椭圆曲线是研究者领域的主要研究对象之一。 除了这些更广泛的影响，该奖项将支持几项教育举措，包括亚利桑那大学少数民族微积分学生的过渡计划，本科生解决问题课程的开发，以及招募代表性不足的少数民族学生参加高中数学暑期课程。 PI是加拿大/美国数学夏令营的董事之一，这是一个为数学天才高中生开设的暑期项目。