Low Dimensional and Semi-infinite Dimensional Topology

DMS-0206485 Tomasz Mrowka Mrowka参与了许多以Floer同源性为中心的项目。 第一个是一本书的项目与克朗海默给予详细和一般性的基本方面的塞伯格-维滕弗洛尔同源性使用的莫尔斯复杂。希望这将为研究生进入该领域提供一条途径，并为更雄心勃勃的项目提供一个出发点。第二个项目也与Kronheimer联合使用在前一个项目中开发的一些工具来关联Seiberg-Witten和InstantonFloer同源性。一个后果，这将是财产P猜想。 第三个项目是直接用无限维圈给出Floer同调的定义。Mrowka将继续研究时空的数学模型。 这些研究的中心是理解高能物理学中各种方程的解和解空间的性质，主要是杨-米尔斯方程和Sieberg-Witten方程。这些方程的美妙之处在于，解空间的总体性质反映了它们所处的时空的微妙性质。 Mrowka以前的工作的一个应用是理论生物学，特别是DNA的打结特性。估计解结数的数学问题给出了拓扑异构体作用于给定DNA环所需的次数的估计。

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.