Collaborative Research: The Geometry of Duality in Mathematics and Physics

合作研究：数学和物理中的对偶几何

• 批准号: 0073657
• 负责人: Sheldon Katz
• 金额: $ 33万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-09-01 至 2002-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0073657&HistoricalAwards=false
• 关键词: Collaborative; Research; Geometry; Duality; Mathematics

AbstractAward: DMS-0073657Principal Investigator: Sheldon KatzThis research project assembles a team at two campuses to work onstring theory and related geometry. The project is led by twomathematicians and a physicist, and the award supportspostdoctoral research fellows, graduate students, and thecollaborations and communications of a broadly based researchnetwork. The research program supported by this awardconcentrates on dualities in string theory and M theory, a topicin which a broad range of geometric and physical concepts cometogether. The best known of these dualities is mirror symmetry,which has had dramatic consequences for algebraic and symplecticgeometry, and this project intends to explore new dualities andconstraints along with mirror symmetry.String theory is a promising candidate for a unifying theory ofthe universe at its most fundamental levels. The basic idea issimple - elementary particles should be modeled as mathematicalloops of string rather than as points - but working out thedetails of this theory has involved and inspired somesophisticated mathematical tools and ideas. Constraints andobservations from physics, sometimes posed as a claim that twodistinct geometries must generate the same physical theory, canhave large numbers of consequences for geometry since quantitiesof physical interest are often expressed as the average value ofan observable quantity over space, or as a way of counting thenumber of times two objects meet. The program in Algebra, NumberTheory, and Combinatorics is cofunding this award.

摘要奖项：DMS-0073657 首席研究员：Sheldon Katz 该研究项目在两个校区组建了一个团队，致力于弦理论和相关几何学的研究。 该项目由两名数学家和一名物理学家领导，该奖项支持博士后研究员、研究生以及基础广泛的研究网络的合作和交流。 该奖项支持的研究项目集中于弦理论和 M 理论的对偶性，该主题融合了广泛的几何和物理概念。 这些对偶性中最著名的是镜像对称性，它对代数和辛几何产生了巨大的影响，该项目旨在探索新的对偶性和约束以及镜像对称性。弦理论是宇宙最基本的统一理论的有希望的候选者水平。 基本思想很简单——基本粒子应该被建模为数学弦环而不是点——但研究这个理论的细节涉及并启发了一些复杂的数学工具和思想。 物理学的约束和观察有时被认为是两个不同的几何图形必须产生相同的物理理论，可能对几何学产生大量影响，因为物理兴趣量通常表示为空间上可观测量的平均值，或者作为一种计数方式两个物体相遇的次数。 代数、数论和组合学项目共同资助了该奖项。