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.

这是代数几何领域的研究-一个研究多项式方程组的解决方案的经典课题。目前的项目解决了四个问题，为复杂几何学、弦理论和量子物理学提供了令人兴奋的接口。 第一个问题的目的是建立一个工具箱的霍奇理论技术计算不变量的复杂的射影簇有关的高同伦。第二部分探讨了复杂几何中经典Shafarevich一致化猜想的局限性。第三个问题提出了一种在由K3曲面纤维化的gerby Calabi-Yauman流形上的层的显式构造，并研究了它对场的非对易形变物理概念的影响。第四个项目探讨了Calabi-Yau流形的退化族的极值转移、可积系统的极限行为和非交换代数几何之间的关系，理解这些问题对于统一代数几何、辛拓扑、理论物理和数学物理中的各种线性化过程是必不可少的。该项目为理解代数簇的基本结构奠定了基础，以便在广泛的应用中实用。这项工作的提出将直接关系到深层次的问题incategory理论，组合群论，理论的积分系统，弦理论，量子引力和宇宙学。该项目提出了具体的跨学科应用矩阵量子力学，弦对偶和粒子物理模型的建立。