Higher-dimensional Heegaard Floer homology

首席调查员将使用名为“高维Heegaard Floer同调”的理论对3维、4维和更高维空间进行研究，该理论是PI与他的合作者文森特·科林共同开发的。这些空间将局部地类似于标准的(欧几里得)n维空间，可能在全局上非常复杂，但局部观察者无法区分，就像蚂蚁无法分辨它是坐在平面上还是坐在非常大的球体上一样。高维Heegaard Floer同调是通过接触几何和辛几何来定义的，而辛几何又与数学物理和弦理论密切相关。从微观层面的DNA打结到宏观层面的宇宙形状，这些在更高维度上的研究反过来应大大有助于我们对低维(即，3维和4维)形状的理解。PI将通过Heegaard Floer同调的高维推广来研究高维上的接触和辛几何。由于Ozsváth和Szabó的Heegaard Floer同调，Heegaard Floer同调是三维和四维空间以及三维空间中的纽结和链环的不变量的集合，它对于区分这些空间和纽结和链环是非常有效的。PI的研究计划的主要目标是发展高维的Heegaard Floer同调理论，即大于4维的Heegaard Floer同调理论。该理论有望在高维接触几何和辛几何中产生重要的应用，而高维辛几何又与代数几何和数学物理有关。Khovanov同调是三维空间中纽结和链环的另一个重要不变量，至少可以猜想地认为是高维Heegaard Floer同调的特例。

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.