Geometrical and Physical Phenomena in Algebraic Topology

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

DMS-0305853 Igor Kriz Algebraic 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, primarily groups, 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 puzzle of modern physics, namely unification of gravity with the other forces 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 theory suggests 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 generalized cohomology theories related to D-branes and M-theory. In the process, he will work on mathematical theories which will serve as axiomatic systems for fundamental and extended objects of string theory. Many aspects of this project are joint work with Po Hu.