Low Dimensional Topology and Gauge Theory

PI将继续他对其中三个流形和纽结的Floer同调不变量的研究。其中一个特别的焦点将是与彼得·克朗海默共同研究Floer同调理论，该理论建立在三个流形中的一个链接上与指定奇点的联系上。当专用于三个球面中的链环时，这个意志不变量似乎与链环的霍万诺夫同调密切相关。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。在学生Maksim Lypyanskiy的帮助下，PI打算进一步探索基于半无限维循环理论的Floer同调的新基础。这似乎导致了理论的极大简化。与另一名学生Ben Mares一起，PI希望开始实施N=4个超对称杨-米尔斯方程的数学理论。最后，和第三个学生Timothy Nguyen一起，他希望了解双曲型Yang-Mills方程，特别是3+1维的。物理学家为理解高能物理中的粒子而构建的模型，如Yang-Mills和Seiberg-Witten方程，也被证明是许多令人兴奋的数学发展的来源。最值得注意的是，这些模型在理解三维和四维空间的现象方面发挥了至关重要的作用，而这些现象似乎仍然无法用其他方法来实现。PI在这些模型的数学发展方面一直处于领先地位，并将继续在这些模型的若干不同方向上进行研究。彼得·克朗海默的一个项目试图将两个看起来完全不同的模型联系起来。另一种是寻求为探索这些模型开发新的数学基础，并有望极大地简化这些模型的严格数学结构。最后，莫罗卡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.