CAREER: p-adic and mod p Galois representations

PI的研究方向为数论和表示论；从广义上讲，其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（有时是推测性的）。 PI 将承担多个与 Serre、Fontaine-Mazur 和 Breuil-Mezard（及其概括）猜想以及新兴的 p-adic 和 mod p Langlands 对应相关的项目。 PI 将研究数域上还原群的塞尔猜想的权重部分，目标是给出具有相当普遍性的明确的塞尔权重配方。研究者将提供布勒伊-梅扎德猜想推广的证据，并将证明这种推广的一些案例，并将其应用于朗兰兹纲领。 该项目的另一个组成部分涉及某些 p 进伽罗瓦表示的模 p 的显式约简。数论是数学最古老的分支之一。 从本质上讲，数论是对方程的整数解的研究，尽管复杂的现代技术有时看起来与这个目标相去甚远。 数论有许多重要的实际应用；例如，大多数蜂窝电话都受到基于椭圆曲线的代码的保护，椭圆曲线是研究人员领域的主要研究对象之一。 除了这些更广泛的影响之外，该奖项还将支持多项教育举措，包括亚利桑那大学少数族裔微积分学生的过渡计划、开发本科问题解决课程，以及招募代表性不足的少数族裔学生参加高中数学暑期课程。 PI 是加拿大/美国数学营的主任之一，这是一个针对数学天才高中生的暑期项目。

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.