Moduli of Galois Representations

数论是数学最古老的分支之一。从根本上说，数论是对多项式方程的整数解的研究，特别是对几何方程的研究。例如，直角三角形的三条边的长度是由勾股定理联系起来的。虽然很容易找到所有边长为有理数的直角三角形，但令人惊讶的是，确定哪些整数是有理数边的直角三角形的面积仍然是一个未解决的问题。数论有许多重要的实际应用。例如，大多数移动电话通话都受到基于椭圆曲线的代码的保护，椭圆曲线是朗兰兹计划（Langlands program）的主要研究对象之一，也是首席研究员的研究领域。近几十年来，在理解朗兰兹自同构形式和伽罗瓦表示之间的推测互易性方面取得了重大进展。证明互易律的技术通常需要考虑伽罗瓦表示的p进族。将进行三个项目来调查伽罗瓦代表的p进族。第一个项目将产生某些伽罗瓦表示的模堆栈，其中残差表示允许变化。这将通过在这些空间上一般地建立朗兰兹方案的结果，从而开辟证明结果的可能性。第二个项目是关于p进Langlands程序中类型的Breuil局部-全局兼容性猜想，它将演示这种技术。第三个项目是研究非泛型情况下GL(n)的Serre猜想的权重部分，当n大于2时。

Number theory is one of the oldest branches of mathematics. At its most fundamental, number theory is the study of whole number solutions to polynomial equations, and especially to equations motivated by geometry. For example, the lengths of the three sides of a right triangle are related by the Pythagorean theorem. While it is straightforward to find all right triangles whose side lengths are rational numbers, it perhaps surprisingly remains an unsolved problem to determine which whole numbers can be the area of a right triangle with rational sides. 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 Langlands program, the principal investigator's area of research.Significant progress has been made in recent decades towards understanding Langlands's conjectural reciprocity between automorphic forms and Galois representations. Techniques for proving reciprocity laws generally require the consideration of p-adic families of Galois representations. Three projects to investigate p-adic families of Galois representations will be undertaken. The first project will produce moduli stacks of certain Galois representations in which the residual representation is allowed to vary. This will open up the possibility of proving results in the Langlands program by establishing them generically on these spaces. The second project, on Breuil's local-global compatibility conjecture for types in the p-adic Langlands program, will be a demonstration of this technique. A third project is to study the weight part of Serre's conjecture for GL(n), for n larger than 2, in non-generic cases.