Collaborative Research: The Geometry of Duality in Mathematics and Physics

摘要奖：DMS-0074329首席研究员：Cumrun Vafa这个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在弦理论和M理论中的对偶性，这是一个广泛的几何和物理概念汇聚在一起的主题。这些对偶中最著名的是镜像对称性，它对代数和辛几何产生了戏剧性的影响，这个项目旨在探索新的对偶性和约束以及镜像对称性。弦理论是在最基本的层次上建立宇宙统一理论的一个有前途的候选者。基本思想很简单--基本粒子应该被建模为数学上的弦循环，而不是点--但解决这一理论的细节涉及并启发了一些体系化的数学工具和想法。来自物理学的约束和观测有时被认为是两个不同的几何必须产生相同的物理理论的主张，可能会对几何产生大量的结果，因为物理兴趣的量通常表示为空间中可观察到的量的平均值，或者作为计算两个物体相遇的次数的一种方式。

AbstractAward: DMS-0074329Principal Investigator: Cumrun VafaThis 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.