Geometrical and Physical Phenomena in Algebraic Topology

代数拓扑中的几何和物理现象

• 批准号: 0305853
• 负责人: Igor Kriz
• 金额: $ 14.57万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305853&HistoricalAwards=false
• 关键词: Geometrical; Physical; Phenomena; Algebraic; Topology

DMS-0305853Igor KrizAlgebraic topology is an area which examines properties of shapes not affected by continuous deformation: in a traditional example, a coffee cup made out of modelling clay can be continuously, without gluing or tearing,deformed into the shape of a donut, but not a solid ball (because of the `hole' in the handle). Sophisticated methods using algebra, primarilygroups, were developed to describe and explain such concepts.String theory is an area of theoretical physics which is currently the best candidate for solving the greatest puzzleof modern physics, namely unification of gravity with the otherforces of nature. The fundamental idea of string theory is that very small particles may not be `dots', but $1$-dimensional objects (`strings'). The present project focuses on connections between algebraic topology and string theory. While physicists know about the fact that topology is connected to the phenomenology of strings, the present project explores connections of a new nature. Notably, algebraic topology is needed in making the concepts of string theory mathematically rigorous, which in turn is a necessary step toward possibly using such concepts for scientific prediction. On the other hand, string theorysuggests exciting new methods for algebraic topology, which will also be explored in this project.The investigator will continue to work on using string theory to construct mathematical models for elliptic cohomology. He will also consider extending these methods to finding generalizedcohomology theories related to D-branes and M-theory. In the process,he will work on mathematical theories which will serve asaxiomatic systems for fundamental and extended objects of stringtheory. Many aspects of this project are joint work with Po Hu.

代数拓扑学是一个研究不受连续变形影响的形状属性的领域：在一个传统的例子中，一个由建模粘土制成的咖啡杯可以连续地变形成甜甜圈的形状，而不需要粘合或撕裂，但不是一个实心球（因为手柄上的“洞”）。使用代数（主要是群）的复杂方法被用来描述和解释这些概念。弦理论是理论物理学的一个领域，它目前是解决现代物理学最大难题的最佳候选，即引力与自然界其他力的统一。弦理论的基本思想是非常小的粒子可能不是“点”，而是一元维的物体（“弦”）。本课题的重点是代数拓扑和弦理论之间的联系。虽然物理学家知道拓扑结构与弦的现象学有关，但本项目探索的是一种新性质的联系。值得注意的是，要使弦理论的概念在数学上变得严格，代数拓扑是必要的，而这反过来又可能是将这些概念用于科学预测的必要步骤。另一方面，弦理论为代数拓扑提供了令人兴奋的新方法，这也将在本项目中进行探索。研究者将继续致力于利用弦理论构建椭圆上同调的数学模型。他还将考虑扩展这些方法来寻找与d膜和m理论相关的广义上同调理论。在这个过程中，他将研究数学理论，这些理论将为弦理论的基本和扩展对象提供公理系统。这个项目的许多方面都是与胡珀共同完成的。