Moduli of Galois Representations

数论是数学中最古老的分支之一。数论最基本的内容是研究多项式方程的整数解，特别是几何方程的整数解。例如，直角三角形的三条边的长度由毕达哥拉斯定理联系起来。虽然找到边长为有理数的所有直角三角形很简单，但可能令人惊讶的是，确定哪些整数可以是边长为有理数的直角三角形的面积仍然是一个悬而未决的问题。数论有许多重要的实际应用。例如，大多数蜂窝电话是由基于椭圆曲线的代码保护的，椭圆曲线是朗兰兹计划的主要研究对象之一，也是主要研究人员的研究领域。近几十年来，在理解朗兰兹猜想自同构形式和伽罗瓦表示之间的互易性方面取得了重大进展。证明互易定律的技术通常需要考虑伽罗瓦表示的p-进数族。将开展三个项目，以调查伽罗瓦代表的P-adary家庭。第一个项目将产生某些伽罗瓦表示的模堆栈，其中剩余表示被允许变化。这将打开在朗兰兹计划中证明结果的可能性，通过在这些空间上一般地建立它们。第二个项目，关于布鲁尔关于p-adic朗兰兹程序中类型的局部-全局兼容性猜想，将是这一技术的一个演示。第三个项目是研究非一般情形下，对于n大于2的GL(N)的Serre猜想的权重部分。

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.