p-adic and mod p Galois representations

“这个奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。”调查者的研究领域是数论和代表论。特别是，研究人员研究了p-进Galois表示和p-进Hodge理论，着眼于在Galois表示的模块化和朗兰兹计划中的应用。他将研究Serre猜想中任意分裂约化群和任意数域的权重，目的是给出一个相当普遍的显式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案例。该项目的另一个组成部分涉及到某些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。近几十年来，数论在密码学(创码)和密码分析(破译密码)领域有了革命性的应用。例如，大多数蜂窝电话通信都受到基于椭圆曲线的密码系统的保护，这是研究人员领域的主要研究对象之一。PI是加拿大/美国数学营的主任之一，这是一个针对数学有天赋的高中生的暑期项目，他认为活跃于研究的数学家参与这样的努力是至关重要的。

"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.