Low Dimensional Topology and Gauge Theory

PI将继续研究三个流形及其结的花同调不变量。一个特别的重点将是与Peter Kronheimer的合作工作，关于一个Floer同调理论，该理论是由三个流形中的一个给定奇点的连接建立起来的。当专门化到三球面上的连杆时，这个will不变量似乎与连杆的Khovanov同调密切相关。Kronheimer和PI打算进一步探索这一理论，并更清楚地了解它与Khovanov同调的关系。与学生Maksim lyypyansky一起，PI打算在半无限维循环理论的基础上进一步探索Floer同调的新基础。这似乎导致了理论的急剧简化。PI和另一名学生本·马雷斯希望开始落实N=4超对称杨-米尔斯方程的数学理论。最后，与第三个学生Timothy Nguyen一起，他希望理解双曲杨-米尔斯方程，特别是在3+1维度。物理学家为理解高能物理中的粒子而建立的模型，比如Yang-Mills方程和Seiberg-Witten方程，也被证明是数学中许多令人兴奋的发展的源泉。最值得注意的是，这些模型在理解第三和第四维度的现象方面发挥了至关重要的作用，这些现象似乎是其他方法无法达到的。PI一直是这些模型数学发展的领导者，并将继续在这些模型的许多不同方向上进行研究。Peter Kronheimer的一个项目试图将两种完全不同的模型联系起来。另一种方法是为探索这些模型开发新的数学基础，并希望在这些模型的严格数学构造中得到极大的简化。最后，Mrowka将与一些学生一起探索新的模型，希望它们也能产生有趣的数学结果。

The PI will continue his investigations into Floer homology invariants for three manifolds and knots in them. One particular focus will be joint work with Peter Kronheimer concerning a Floer homology theory built from connections with a prescribed singularity along a link in a three manifold. When specialized to links in the three sphere this will invariant appears to be closely related to the Khovanov homology of the link. Kronheimer and the PI intend to further explore this theory and understand more clearly its relation to Khovanov homology. With a student, Maksim Lypyanskiy, the PI intends to further explore a new foundation for Floer homology based on a theory of semi-infinite dimensional cycles. This appears to lead to a drastic simplification of theory. With another student, Ben Mares, the PI hopes to begin to put into place the mathematical theory of the N=4 supersymmetric Yang-Mills equations. Finally with third student, Timothy Nguyen, he hopes to understand the hyperbolic Yang-Mills equations especially in dimensions 3+1.The models physicists have constructed for understanding particles in high energy physics, like that Yang-Mills and Seiberg-Witten equations, have proved to be a source for many exciting developments in mathematics as well. Most notably these models figure crucially in understanding phenomena in dimensions three and four that still seem out of reach by other methods. The PI has been a leader in the mathematical developments of these models and will continue research in a number of different directions on these models. One project with Peter Kronheimer seeks to relate two quite different seeming models. Another seeks to development new mathematical foundations for exploration of these models and will hopefully lead to a great simplification in the rigorous mathematical construction of these models. Finally with some students Mrowka will explore new models in hopes that they too will have interesting mathematical consequences.