Higher-dimensional Heegaard Floer homology

首席研究员将使用一种叫做“高维Heegaard flower同源”的理论对3维、4维和高维空间进行研究，该理论是PI与他的合作者Vincent Colin共同开发的。这些空间局部与标准（欧几里得）n维空间相似，全局可能非常复杂，但局部观察者无法分辨其中的区别，就像蚂蚁无法分辨自己是坐在一个平面上还是一个非常大的球体上一样。高维Heegaard flower同调是通过接触几何和辛几何来定义的，而接触几何和辛几何又与数学物理和弦理论密切相关。这些高维的研究反过来又会对我们对低维（即3维和4维）形状的理解做出重大贡献，从微观层面的DNA结到宏观层面的宇宙形状。PI将通过Heegaard flower同调的高维推广来研究高维的接触几何和辛几何。由于Ozsváth和Szabó， Heegaard flower同源性是三维和四维空间以及三维空间中的结和链接的不变量的集合，它在区分这些空间、结和链接方面非常有效。PI研究计划的主要目标是在高维（即大于4维）中发展Heegaard flower同调理论。这一理论有望在高维的接触几何和辛几何中产生重要的应用，而接触几何和辛几何又与代数几何和数学物理有关。Khovanov同调是三维空间中结点和连杆的另一个重要不变量，至少可以推测为高维Heegaard flower同调的一种特殊情况。

The Principal Investigator will undertake a study of 3-, 4-, and higher-dimensional spaces using a theory called "higher-dimensional Heegaard Floer homology'' which the PI is developing with his collaborator Vincent Colin. These spaces will locally be similar to the standard (Euclidean) n-dimensional spaces and may be very complicated globally, but a local observer cannot tell the difference, just as an ant cannot tell whether it is sitting on a flat plane or a very large sphere. Higher-dimensional Heegaard Floer homology is defined through contact and symplectic geometry, which in turn are closely related to mathematical physics and string theory. These studies in higher dimensions should in turn contribute significantly towards our understanding of low-dimensional (i.e., 3- and 4-dimensional) shapes, from the knotting of DNA at the microscopic level to the shape of the universe at the macroscopic level. The PI will study contact and symplectic geometry in higher dimensions through a higher-dimensional generalization of Heegaard Floer homology. Heegaard Floer homology, due to Ozsváth and Szabó, is a package of invariants of 3- and 4-dimensional spaces as well as knots and links in 3-space, which is extremely effective at distinguishing these spaces and knots and links from one another. The main goal of the PI's research program is to develop the theory of Heegaard Floer homology in higher dimensions, i.e., in dimensions greater than 4. This theory is expected to yield important applications to contact and symplectic geometry in higher dimensions, which in turn are related to algebraic geometry and mathematical physics. Khovanov homology, another important invariant of knots and links in 3-space, can at least conjecturally be viewed as a special case of higher-dimensional Heegaard Floer homology.