Arithmetic and Quantum Intersection Theory on Homogeneous Spaces

齐次空间的算术与量子相交理论

• 批准号: 0098551
• 负责人: Harry Tamvakis
• 金额: --
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2001-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0098551&HistoricalAwards=false
• 关键词: Arithmetic; Quantum; Intersection; Theory; Homogeneous

This research is in different aspects of intersection theory in algebraic geometry: higher dimensional Arakelov theory, a part of arithmetic algebraic geometry, and quantum cohomology, a theory on the border between mathematics and quantum physics. The investigator studies the arithmetic and quantum intersection rings of homogeneous spaces, such as flag manifolds, as part of a general program of extending results of classical algebraic geometry to the new settings. The following are some of the problems considered: a) Understanding the arithmetic and quantum Schubert calculus for homogeneous spaces of classical Lie groups; b) finding determinantal formulas for Lagrangian and orthogonal degeneracy loci and developing further the investigator's theory of double Schubert polynomials; c) obtaining arithmetic analogues of Fulton's results on degeneracy loci and numerically positive polynomials; d) computing arithmetic intersections and heights on Shimura varieties.During the 19th century, mathematicians and physicists were fascinated by the symmetries observed in geometric objects. The many experiments and computations made at that time eventually led to the cohomology theories of the 20th century, which were applied to solve long-standing open problems in both fields. Today, we are in a similar situation in modern number theory and quantum field theory, and it is important to have calculations of specific examples to support and guide our intuition. The investigator examines two new theories, one motivated by number theory and the other by quantum physics, in many specific examples which are prototypes for this purpose. The potential applications are a better understanding of Diophantine equations and approximation, used in coding theory and theoretical computer science, and enumerative geometry of curves, related to string theory and symmetry in physics.

这项研究涉及代数几何中交集理论的不同方面：高维阿拉克洛夫理论（算术代数几何的一部分）和量子上同调（数学和量子物理之间的边界理论）。 调查研究算术和量子相交环的齐次空间，如旗流形，作为一个通用程序的一部分，扩展结果的经典代数几何的新的设置。 本文讨论了以下几个问题：（a）理解经典李群齐性空间的算术和量子Schubert演算;（B）找到Lagrange和正交退化轨迹的行列式公式，进一步发展研究者的双Schubert多项式理论;（c）获得富尔顿关于退化轨迹和数值正多项式的结果的算术类比; d）计算Shimura簇上的算术交和高度。在世纪，数学家和物理学家对在几何物体中观察到的对称性着迷。 当时的许多实验和计算最终导致了世纪的上同调理论，该理论被应用于解决这两个领域长期存在的开放问题。 今天，我们在现代数论和量子场论中处于类似的情况，重要的是要有具体例子的计算来支持和指导我们的直觉。 调查员检查两个新的理论，一个由数论和其他量子物理学的动机，在许多具体的例子，这是原型为这一目的。 潜在的应用是更好地理解丢番图方程和近似，用于编码理论和理论计算机科学，以及曲线的枚举几何，与弦理论和物理学中的对称性有关。