Geometric Applications of Non-Abelian Hodge Theory

非阿贝尔霍奇理论的几何应用

• 批准号: 9800790
• 负责人: Tony Pantev
• 金额: $ 8.67万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800790&HistoricalAwards=false
• 关键词: Geometric; Applications; Non; Abelian; Hodge

Pantev 9800790 Pantev will use techniques from abelian and non-abelian Hodge theory to approach concrete geometric questions as well as problems in mathematical physics and string theory. Four problems will be studied. The first one is to look for a construction of a linear algebraic object that is canonically associated to a projective variety X and calculate the cohomology of all local systems on X together with their Hodge structures whenever the latter makes sense. The second concerns the non-abelian Hodge conjecture on curves and provides an intrinsic geometric characterization of the locus of motivic local systems. The third discusses a construction of a singular plane curve whose complement has a non-residually finite fundamental group. The fourth project suggests a geometric strategy for understanding the appearance of enhanced gauge symmetry in string theory and proposes a method for analyzing the exotic components of F theory moduli spaces. The understanding of these questions is essential for unifying various linearization procedures in algebraic geometry and mathematical physics. On one hand, it will bring us closer to understanding the basic structure of algebraic varieties, and on the other hand it will bring a geometric perspective to the interaction of the recently discovered M and F string theories. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one that has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.

潘特夫9800790 潘特夫将使用阿贝尔和非阿贝尔霍奇理论的技术来解决具体的几何问题以及数学物理和弦理论的问题。将研究四个问题。第一个是寻找一个线性代数对象的构造，该线性代数对象与一个射影簇X正则相关，并计算X上所有局部系统的上同调，以及它们的霍奇结构，只要后者有意义。第二个是关于曲线上的非阿贝尔Hodge猜想，并给出motivic局部系统轨迹的一个内在几何刻画。第三章讨论了其补曲线具有非剩余有限基本群的奇异平面曲线的构造。第四个项目提出了一种几何策略，用于理解弦理论中增强规范对称的出现，并提出了一种分析F理论模空间奇异分量的方法。理解这些问题对于统一代数几何和数学物理中的各种线性化过程是必不可少的。一方面，它将使我们更接近理解代数簇的基本结构，另一方面，它将带来一个几何的角度来相互作用的最近发现的M和F弦理论。 这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，这些方法在这些领域以及物理学，理论计算机科学和机器人学中也得到了应用。