Character and Quiver Varieties

该研究项目的主旨是继续PI与T的联合工作。Hausel和E. Letellier的几何形状的某些字符品种，模空间的代表性的基本群体的黎曼曲面的一个还原代数群。这些空间具有丰富的几何形状。在其他化身中，他们将表面上的某些希格斯束参数化，这是一个与物理学理论有关的空间。PI及其合作者的主要猜想描述了字符簇的混合霍奇结构，并涉及组合数学的麦克唐纳多项式。在一个独立的项目中，PI将继续研究与超几何动机相关的L函数的理论和计算方面。很难预测当前的纯数学将如何影响学科之外的生活。但总的来说，这在很大程度上迟早会发生。特别是算术几何，在数字通信和数据处理不断增加的世界中变得至关重要。例如，有限域上的几何，曾经是专家的一个课题，现在需要许多广泛使用的加密算法和信息的电子传输。在这个项目中，PI在另一个方向上使用有限域：作为研究数学和物理中相关空间几何的探针。

The main thrust of the research project is the continuation of the joint work of the PI with T. Hausel and E. Letellier on the geometry of certain character varieties, the moduli space of representations of the fundamental group of a Riemann surface to a reductive algebraic group. These spaces have a rich geometry. In other incarnations they parametrize certain Higgs bundles on the surface, a space which arose in connection to theories in Physics. The main conjecture of the PI and his collaborators describes the mixed Hodge structure of the character varieties and involves the Macdonald polynomials of combinatorics. In a separate project the PI will continue to investigate the theoretical and computational aspects of the L-functions attached to hypergeometric motives.It is hard to predict how current pure Mathematics will impact life outside its discipline. But overall, it largely does sooner or later. Arithmetic geometry in particular, has become crucial in a world of ever increasing digital communication and data manipulation. Geometry over finite fields, for example, once a subject of specialists, is now needed for many widely used algorithms for encryption and electronic transmission of information. In this project the PI uses finite fields in the other direction: as a probe to study the geometry of some spaces of relevance in Mathematics and Physics.