Character Varieties

本文主要研究光滑射影代数曲线的基本群到一个约化代数群G的表示的模空间。这个模空间是非阿贝尔Hodge理论中上同调群的Betti形式。它的其他版本是：Dolbeault：曲面上半稳定Higgs G-丛的模空间，de Rham：曲面上的平坦G-连通模空间。在Dolbeault和De Rham的版本中，这个空间一直是朗兰兹计划最近重要工作的核心，既有Ngo和Laumon的算术，也有Kapustin和Witten的几何。这个项目的基本思想是利用数论、组合学和李型有限群的表示理论来计算有限域上Betti空间上的点。然后，韦尔猜想产生关于它的上同调和几何信息(因此也产生关于它的另外两种风格的信息)。从长远来看，数学总是有一种方法使自己在自己的学科之外有用。比方说，20年前，谁会想到，像“椭圆曲线”这样明显从日常生活中消失的东西，最终会成为网上购物安全的关键工具？椭圆曲线是一种具有非常丰富结构的几何构造。曲线上的两个点可以用几何方法“相加”，从而产生第三个点。这一事实可以追溯到300多年前。在更现代的时代，我们已经学会了如何“几何”(处理点、直线、曲线等)。在纯有限的上下文中，其中实数或复数被其有限对应的有限域所代替。这提供了将几何和数论中的抽象数学概念应用到现实生活中的最有用的方法之一。该研究项目以一种不同的方式使用有限域：探索数学和物理都非常感兴趣的某些空间的几何，在通常的单词意义上。PI发现这个项目中离散的(有限域上的点计数)和连续的(复数上的几何)之间的相互作用，以及在很大程度上涉及到相当遥远的数学领域(数论、组合学、群论和微分几何)的事实，令人着迷。

The main focus of the research project concerns the moduli space of representations of the fundamental group of a smooth projective algebraic curve to a reductive algebraic group G. This moduli space is the Betti version of a cohomology group in non-abelian Hodge theory. Its other versions are, Dolbeault: the moduli space of semistable Higgs G-bundles on the surface and, de Rham: the moduli space of flat G-connections on it. In the Dolbeault and de Rham versions this space has been central to recent important work in the Langlands program, both the arithmetic by work of Ngo and Laumon and the geometric by work of Kapustin and Witten. The basic idea of the project is to use tools of Number Theory, Combinatorics and the Representation Theory of finite groups of Lie type to count points on the Betti space over finite fields. The Weil conjectures then yield cohomological and geometrical information about it (and hence also about its other two flavors).In the long run, Mathematics always has a way of making itself useful outside its own discipline. Who would have thought 20 years ago, say, that something as apparently removed from everyday life as the concept of an "elliptic curve" could end up as a crucial tool for the security of online shopping? An elliptic curve is a geometric construct with a very rich structure. Two points on the curve can be "added" by geometric means to produce a third point. This fact goes back more than 300 years. In more modern times we have learned how do to "geometry" (work with points, lines, curves, etc.) in a purely finite context, where the real or complex numbers are replaced by their finite counterparts, finite fields. This provides one of the most useful ways to apply abstract mathematical concepts from Geometry and Number Theory to real life situations. The research project uses finite fields in a different way: to probe the geometry, in the usual sense of the word, of certain spaces of great interest to both Mathematics and Physics. The PI finds the interplay between the discrete (counting points over finite fields) and the continuous (geometry over the complex numbers) in this project, as well as the fact that involves in a substantial way fairly distant areas of Mathematics (Number Theory, Combinatorics, Group Theory and Differential Geometry), fascinating.