Contact structures, Floer homology and TQFT

摘要奖：DMS-0805352首席研究员：Ko Honda最近，接触结构在低维拓扑和拓扑量子场论(TQFT)中的核心作用变得越来越明显。包括PI在内的许多作者发展的三维接触结构的剪切粘贴理论，现在可以引入Heegaard FloerHomology，一个(3+1)维的TQFT。PI和他的合作者试图更好地理解接触几何、Floer同调理论、TQFT和弦理论以及双曲几何之间的关系。更具体地说，PI提出了以下建议：(1)分析Heegaard/缝合Floer同调的TQFT性质。(2)理解通过接触几何得到的某个范畴，它满足三角范畴的许多性质。(3)研究了接触流形辛化中Legendrian环之间的精确拉格朗日余弦的接触不变量。(4)计算了双曲三维流形上紧接触结构的接触同调。PI提出了一种称为接触结构的探针来研究三维和四维空间。我们研究的三维空间将局部地类似于标准的(欧几里得)三维空间。这些物体可能在全球范围内非常复杂，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一个非常大的球体上一样。接触结构是三维空间中的一个场，类似于物理中的电场。正如电场的性质，如任何一点的强度和方向，提供了一些关于电荷分布的信息，即电场的来源，接触结构提供了关于周围三维空间形状的信息。对三维接触结构的更全面的数学理解可能会大大有助于我们对宇宙形状的理解。

AbstractAward: DMS-0805352Principal Investigator: Ko HondaRecently it has become more evident that contact structures playa central role in low-dimensional topology and TopologicalQuantum Field Theories (TQFTs). The cut-and-paste theory ofcontact structures in dimension three, developed by numerousauthors including the PI, can now be imported into Heegaard Floerhomology, a (3+1)-dimensional TQFT. The PI, together with hiscollaborators, seeks to better understand the relations amongcontact geometry, Floer homology theories, TQFTs and stringtheory, and hyperbolic geometry. More specifically, the PIproposes the following: (1) Analyze the TQFT properties ofHeegaard/sutured Floer homology. (2) Understand a certaincategory obtained via contact geometry, which satisfies many ofthe properties of a triangulated category. (3) Study contacthomology invariants of exact Lagrangian cobordisms betweenLegendrian links in the symplectization of a contact manifold.(4) Calculate the contact homology of tight contact structures onhyperbolic 3-manifolds.The PI proposes a study of 3- and 4-dimensional spaces using aprobe called a contact structure. The 3-dimensional spaces westudy will locally be similar to the standard (Euclidean)3-dimensional space. These objects may be very complicatedglobally, but a local observer cannot tell the difference, justas an ant cannot tell whether it is sitting on a flat plane or avery large sphere. A contact structure is a field in3-dimensional space, analogous to an electric field in physics.Just as the nature of the electric field, such as the strengthand the direction at any point, gives some information about thedistribution of the electric charge, i.e., the source of theelectric field, a contact structure gives information about theshape of the ambient 3-dimensional space. A more thoroughmathematical understanding of contact structures in dimensionthree may contribute significantly towards our understanding ofthe shape of the universe.