Geometrical and Physical Phenomena in Algebraic Topology

DMS-0305853Igor KrizAlgebraic拓扑学是一个检查不受连续变形影响的形状属性的区域：在一个传统的例子中，用模型粘土制成的咖啡杯可以连续地变形成甜甜圈的形状，而不是实心的球(因为手柄上有个洞)。用代数(主要是群)来描述和解释这些概念的复杂方法被开发出来。弦理论是理论物理学的一个领域，目前是解决现代物理学最大难题的最佳候选者，即重力与自然其他力的统一。弦理论的基本思想是，非常小的粒子可能不是‘点’，而是$1$维物体(‘弦’)。本课题主要研究代数拓扑学和弦理论之间的联系。虽然物理学家知道拓扑学与弦的现象学有关，但本项目探索的是一种新性质的联系。值得注意的是，需要代数拓扑学来使弦理论的概念在数学上严谨，这反过来又是可能将此类概念用于科学预测的必要步骤。另一方面，弦理论提出了令人兴奋的代数拓扑的新方法，这也将在本项目中进行探索。研究人员将继续使用弦理论来构造椭圆上同调的数学模型。他还将考虑将这些方法扩展到寻找与D-膜和M-理论相关的广义上同调理论。在这个过程中，他将致力于数学理论，这些理论将为弦理论的基本和扩展对象提供无理系统。这个项目的许多方面都是与博湖合作的。

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.