Collaborative Research: The Geometry of Duality in Mathematics and Physics

奖项：dms -0073657首席研究员：Sheldon katz本研究项目在两个校区组建了一个团队，研究弦理论和相关几何。该项目由两位数学家和一位物理学家领导，该奖项支持博士后研究员、研究生以及广泛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，这是一个将广泛的几何和物理概念结合在一起的主题。这些对偶性中最著名的是镜像对称，它对代数和辛几何产生了戏剧性的影响，这个项目打算探索新的对偶性和镜像对称的约束。在最基本的层面上，弦理论很有希望成为宇宙统一理论的候选者。基本思想很简单——基本粒子应该被建模为弦的数学环，而不是点——但研究这一理论的细节涉及并激发了一些复杂的数学工具和思想。来自物理学的约束和观察，有时被认为是两种不同的几何必须产生相同的物理理论，可以对几何产生大量的结果，因为物理上感兴趣的量通常被表示为空间上可观察量的平均值，或者作为计算两个物体相遇次数的一种方式。代数、数论和组合学项目共同资助了这个奖项。

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.