Low Dimensional and Semi-infinite Dimensional Topology

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DMS-0206485Tomasz MrowkaMrowka is engaged in number of projects centeringaround Floer homology. The first is a book project with Kronheimer giving a detailed and general account of the foundational aspects of Seiberg-WittenFloer 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 isto use some of the tools developed in the previousproject to relate the Seiberg-Witten and InstantonFloer homologies. A consequence of this would bethe Property P conjecture. The third projectis to give a definition of Floer homology directlyusing infinite dimensional cycles.Mrowka will continue investigations into the study ofmathematical models for space-time. These investigationscenter 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 thatgross properties of the spaces of solutions reflectsubtle properties the space-time that they liveon. One application of Mrowka's previous workis to theoretical biology in particularthe knotted properties of DNA. The mathematical questionof estimating the unknotting number giveestimates on the number of times topoisomeraseneeds to act on a given loop of DNA.