p-adic and mod p Galois representations

“这项奖励是根据2009年美国复苏和再投资法案（公法111-5）资助的。”研究者的研究领域是数论和表征理论。特别是，研究者研究p进伽罗瓦表示和p进霍奇理论，着眼于伽罗瓦表示和朗兰兹程序的模块化应用。PI将承担围绕p进表示及其约简模p的主题组织的几个项目。他将研究任意分裂约简群和任意数域的Serre猜想中的权重，目标是给出一个显式的具有相当普遍性的Serre权重公式。研究者将为布鲁伊-梅扎德猜想的推广提供证据，并将证明这种推广的一些情况，并应用于朗兰兹程序。该项目的另一个组成部分涉及某些p进伽罗瓦表示的显式约简模p。数论是数学最古老的分支之一。从根本上说，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。近几十年来，数论在密码学（创建密码）和密码分析（破译密码）领域有着革命性的应用。例如，大多数蜂窝电话通信都是由基于椭圆曲线的密码系统保护的，椭圆曲线是研究者领域的主要研究对象之一。PI是加拿大/美国Mathcamp的负责人之一，这是一个针对有数学天赋的高中生的暑期项目，他认为积极从事研究的数学家参与这种努力是至关重要的。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The investigator's research areas are number theory and representation theory.  In particular, the investigator studies p-adic Galois representations and p-adic Hodge theory, with an eye towards applications to the modularity of Galois representations and the Langlands program.  The PI will undertake several projects organized around the theme of p-adic representations and their reduction modulo p.  He will study the weight in Serre's conjecture for arbitrary split reductive groups and arbitrary 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 rather far-removed from this goal. In recent decades, number theory has had revolutionary applications to the fields of cryptography (creating codes) and cryptanalysis (breaking codes). For instance, most cellular telephone communications are protected by a cryptosystem based on elliptic curves, one of the primary objects of study in the investigator's field. The PI is one of the directors of Canada/USA Mathcamp, a summer program for mathematically talented high school students, and believes it is vital that research-active mathematicians participate in such endeavors.