p-adic and mod p Galois representations

“该奖项是根据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示论。 特别是，研究者研究p-adic伽罗瓦表示和p-adic霍奇理论，着眼于伽罗瓦表示的模块化和朗兰兹程序的应用。 PI将进行几个项目组织围绕主题的p-adic表示和他们的减少模p.他将研究的重量在塞尔的猜想为任意分裂还原群和任意数域，目标是给一个明确的塞尔重量食谱相当普遍。 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。近几十年来，数论在密码学（创建代码）和密码分析（破解代码）领域有着革命性的应用。例如，大多数蜂窝电话通信受到基于椭圆曲线的密码系统的保护，椭圆曲线是研究者领域的主要研究对象之一。PI是加拿大/美国数学夏令营的负责人之一，这是一个为数学天才高中生开设的暑期项目，PI认为研究活跃的数学家参与这些活动至关重要。

"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.