Character and Quiver Varieties

性格和箭袋品种

• 批准号: 1101484
• 负责人: Jose Voloch
• 金额: $ 18万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101484&HistoricalAwards=false
• 关键词: Character; Quiver; Varieties

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.

该研究项目的主旨是继续 PI 与 T. Hausel 和 E. Letellier 在某些特征簇的几何、黎曼曲面基本群到还原代数群的表示的模空间方面的合作。这些空间具有丰富的几何形状。在其他化身中，他们对表面上的某些希格斯束进行参数化，这是一个与物理学理论相关的空间。 PI和他的合作者的主要猜想描述了字符簇的混合霍奇结构，并涉及组合学的麦克唐纳多项式。在一个单独的项目中，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其学科之外的生活。但总体而言，它迟早会发生。在数字通信和数据处理日益增长的世界中，算术几何尤其变得至关重要。例如，有限域上的几何曾经是专家的课题，但现在许多广泛使用的加密和信息电子传输算法都需要它。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一些相关空间的几何形状的探针。