Low Dimensional and Semi-infinite Dimensional Topology

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

DMS-0206485 Tomasz Mrowka Mrowka is engaged in number of projects centering around Floer homology. The first is a book project with Kronheimer giving a detailed and general account of the foundational aspects of Seiberg-Witten Floer homology using the Morse complex. It is hoped that this will provide both a route for graduate students into the field as well as be a jumping off point for more ambitious projects. The second project also joint with Kronheimer is to use some of the tools developed in the previous project to relate the Seiberg-Witten and Instanton Floer homologies. A consequence of this would be the Property P conjecture. The third project is to give a definition of Floer homology directly using infinite dimensional cycles. Mrowka will continue investigations into the study of mathematical models for space-time. These investigations center around understanding properties of solutions and spaces of solutions to the various equations of high energy physics, primarily the Yang-Mills and Sieberg-Witten equations. The beauty of these equations is that gross properties of the spaces of solutions reflect subtle properties the space-time that they live on. One application of Mrowka's previous work is to theoretical biology in particular the knotted properties of DNA. The mathematical question of estimating the unknotting number give estimates on the number of times topoisomerase needs to act on a given loop of DNA.