Low Dimensional Topology and Gauge Theory

PI将继续他的调查弗洛尔同源不变量的三个流形和结在他们。 一个特别的重点将是联合工作与彼得Kronheimer关于弗洛尔同源理论建立从连接与一个规定的奇点沿着一个链接在三个流形。 当专门的链接在三个领域，这将不变量似乎是密切相关的Khovanov同源的链接。 Kronheimer和PI打算进一步探索这个理论，并更清楚地了解它与Khovanov同调的关系。 与学生马克西姆Lypyanskiy，PI打算进一步探索基于半无限维循环理论的Floer同源性的新基础。 这似乎导致了理论的急剧简化。 PI希望与另一名学生Ben Mares一起开始将N=4超对称杨-米尔斯方程的数学理论付诸实践。 最后，与第三个学生，蒂莫西·阮，他希望了解双曲杨米尔斯方程，特别是在3+ 1维。模型物理学家已经建立了理解高能物理中的粒子，如杨米尔斯和塞伯格-威滕方程，已被证明是一个来源，许多令人兴奋的发展，以及数学。 最值得注意的是，这些模型在理解三维和四维现象方面发挥了至关重要的作用，这些现象似乎仍然无法用其他方法来理解。 PI一直是这些模型的数学发展的领导者，并将继续在这些模型的一些不同方向的研究。 彼得·克朗海默（Peter Kronheimer）的一个项目试图将两种完全不同的表面模型联系起来。 另一个目的是开发新的数学基础，探索这些模型，并有望导致一个很大的简化，在严格的数学构造这些模型。 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.