Hodge Theory, Dualities and Non-Commutative Geometry

霍奇理论、对偶性和非交换几何

• 批准号: 0403884
• 负责人: Tony Pantev
• 金额: $ 10.5万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0403884&HistoricalAwards=false
• 关键词: Hodge; Theory; Dualities; Non; Commutative

This is a research in the field of algebraic geometry - a classicalsubject studying the solutions to systems of polynomial equations. Thecurrent project addresses four problems providing exciting interfacesbetween complex geometry and string theory and quantum physics. Thefirst problem aims to build a toolbox of Hodge theoretic techniques forcomputing invariants of complex projective varieties related to higherhomotopy. The second explores the limits of the classical Shafarevichuniformization conjecture in complex geometry. The third problemproposes an explicit construction of sheaves on a gerby Calabi-Yaumanifold which is fibered by K3 surfaces, and studies its impact on thephysical notion of non-commutative deformation of fields. The fourthproject probes the relationship between the extremal transitions indegenerating families of Calabi-Yau manifolds, the limiting behavior ofintegrable systems, and non-commutative algebraic geometry.The understanding of these questions is essential for unifying variouslinearization procedures in algebraic geometry, symplectic topology,theoretical and mathematical physics. The project sets the stage forunderstanding the basic structure of algebraic varieties in a waysuitable for pragmatic use in a broad spectrum of applications. Thework proposed will be immediately relevant to deep questions incategory theory, combinatorial group theory, the theory of integrablesystems, string theory, quantum gravity and cosmology. The projectproposes concrete interdisciplinary applications to matrix quantummechanics, string dualities and particle physics model building.

这是代数几何学领域的一项研究，这是一种研究多项式方程系统解决方案的经典主体。 Thecurrent项目解决了四个问题，这些问题在复杂的几何学与弦理论与量子物理学之间提供了令人兴奋的互动。 第一个问题旨在构建Hodge理论技术的工具箱，从而导致复杂的投射品种与高敏感性相关的复杂投射品种的不变性。第二个探讨了复杂几何形状中经典的Shafarevichifiction猜想的限制。第三个问题将带有K3表面纤维的Gerby Calabi-Yaumanifold上的滑轮的显式结构，并研究了其对田间非交通变形的物理概念的影响。第四个项目探究了calabi-yau歧管的极端转变，可融合系统的限制行为与非交通性代数几何形状之间的关系。这些问题的理解对于统一多种测量过程的理解对于在代数阶段的各种核算程序中至关重要。该项目以适合在广泛的应用程序中务实使用的方式来理解代数品种的基本结构的阶段。提出的工作将立即与深度问题理论，组合群体理论，综合系统理论，弦理论，量子引力和宇宙学有关。 ProjectPropose对矩阵量定机电，字符串二元性和粒子物理模型构建的具体跨学科应用。