CAREER: p-adic and mod p Galois representations

PI的研究方向是数论和表示论；广义上，它的目标是理解与自同构形式和自同构表示相关的伽罗瓦表示(有时是猜想的)。PI将承担几个与Serre猜想、Fontaine-Mazur猜想和Breuil-Mezard猜想(及其推广)以及新兴的p-进和mod p-朗兰兹对应有关的项目。PI将研究数域上约化群的Serre猜想的权重部分，目的是给出一个相当普遍的明确的Serre权重公式。研究人员将为Breuil-Mezard猜想的推广提供证据，并将通过对朗兰兹计划的应用来证明这种推广的一些情况。该项目的另一个组成部分涉及某些p-进Galois表示的模p的显式约简。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与这一目标相去甚远的印象。数论有许多重要的实际应用；例如，大多数移动电话呼叫都受到基于椭圆曲线的代码的保护，这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，该奖项还将支持几项教育倡议，包括亚利桑那大学针对少数族裔微积分学生的过渡计划，开发本科问题解决课程，以及招募代表不足的少数族裔学生参加高中数学暑期计划。PI是加拿大/美国数学营的主任之一，这是一个为数学有天赋的高中生提供的暑期项目。

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.