Character Varieties

本课题的主要研究内容是光滑射影代数曲线到约化代数群G的基本群表示的模空间。这个模空间是非交换霍奇理论中上同调群的贝蒂版本。它的其他版本是，Dolbeault：模空间的半稳定希格斯G-丛的表面和，德拉姆：模空间的平坦G-连接它。在Dolbeault和德拉姆版本这个空间一直是中央最近的重要工作在朗兰兹计划，无论是算术的工作Ngo和Laumon和几何的工作Kapustin和维滕。 该项目的基本思想是使用数论，组合学和李群表示论的工具来计算有限域上Betti空间上的点。于是，魏尔几何给出了关于它的上同调和几何信息（因此也给出了关于它的另外两种味道的信息）。从长远来看，数学总是有办法在它自己的学科之外发挥作用。比如说，20年前，谁会想到像“椭圆曲线”这样明显与日常生活无关的概念最终会成为在线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通过几何方法“相加”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做“几何”（工作与点，直线，曲线等）。 在纯有限的上下文中，其中真实的或复数被它们的有限对应物有限域所取代。这提供了一个最有用的方法来应用抽象的数学概念，从几何和数论到真实的生活情况。该研究项目以不同的方式使用有限域：在通常意义上探索数学和物理都非常感兴趣的某些空间的几何。PI发现在这个项目中离散（有限域上的计数点）和连续（复数上的几何）之间的相互作用，以及涉及相当遥远的数学领域（数论，组合数学，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.