Higher-dimensional Heegaard Floer homology

主要研究者将使用PI与他的合作者Vincent Colin正在开发的称为“高维Heegaard Floer同源性”的理论进行3维，4维和更高维空间的研究。这些空间在局部上类似于标准的（欧几里德）n维空间，在全局上可能非常复杂，但局部观察者无法区分两者的区别，就像蚂蚁无法分辨它是坐在一个平面上还是一个非常大的球体上一样。 高维Heegaard Floer同调是通过接触和辛几何定义的，而接触和辛几何又与数学物理和弦理论密切相关。 这些在更高维度上的研究反过来应该对我们理解低维（即，3-和4维）形状，从微观水平上的DNA打结到宏观水平上的宇宙形状。PI将通过Heegaard Floer同调的高维推广来研究高维中的接触和辛几何。 由于Ozsváth和Szabó，Heegaard Floer同调是3维和4维空间以及3维空间中的结和链的不变量的包，它在区分这些空间和结和链时非常有效。 PI研究计划的主要目标是在更高的维度上发展Heegaard Floer同源性理论，即，尺寸大于4。 这一理论预计将产生重要的应用接触和辛几何在更高的维度，这反过来又涉及到代数几何和数学物理。 Khovanov同调是三维空间中纽结和链环的另一个重要不变量，至少在理论上可以看作是高维Heegaard Floer同调的一个特例。

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.