Contact structures and Floer homology theories

接触结构在三维和四维拓扑学中起着核心作用，这在很大程度上是由于它与各种Floer同调理论的关系-接触同调，Heegaard Floer同调，嵌入接触同调和Seiberg-Witten Floer同调-所有这些都在过去十年中有着惊人的应用。PI研究计划的主要目标是了解当我们将接触三维流形与边界粘合时，这些Floer同调理论会发生什么。三维接触结构的剪切粘贴理论-由包括PI在内的许多作者开发-预计将发挥重要作用。 作为计划的一部分，PI计划研究一个特定的范畴，称为接触范畴，并从接触结构中构造出来，这是粘合接触流形与边界的“范畴胶水”。 PI与Colin和Ghiggini共同工作，也计划建立两个Floer同调理论- Heegaard Floer同调和嵌入接触同调-的等价性，使用Giroux的开卷分解框架;这可以解释为相对嵌入接触同调的胶合结果。PI提出使用称为接触结构的探针研究3维和4维空间。我们研究的3维和4维空间在局部上类似于标准的（欧几里德）3维和4维空间。这些物体在全局上可能非常复杂，但局部观察者无法区分，就像蚂蚁无法分辨它是坐在一个平面上还是一个非常大的球体上一样。对接触结构的更透彻的数学理解应该会大大有助于我们对三维和四维形状的理解，从微观水平上的DNA打结到宏观水平上的宇宙形状。

Contact structures play a central role in 3- and 4-dimensional topology, in no small part due to its relationship with the various Floer homology theories - contact homology, Heegaard Floer homology, embedded contact homology, and Seiberg-Witten Floer homology - all of which have had spectacular applications over the last decade. The main goal of the PI's research program is to understand what happens to these Floer homology theories when we glue contact 3-manifolds with boundary. The cut-and-paste theory of contact structures in dimension three - developed by numerous authors including the PI - is expected to play an important role. As part of the program, the PI plans to study a certain category, called the contact category and constructed from contact structures, which is the "categorical glue" for gluing contact manifolds with boundary. The PI, in joint work with Colin and Ghiggini, also plans to establish the equivalence of two of the Floer homology theories - Heegaard Floer homology and embedded contact homology - using the framework of open book decompositions due to Giroux; this can be interpreted as a gluing result for relative embedded contact homology.The PI proposes a study of 3- and 4-dimensional spaces using a probe called a contact structure. The 3- and 4-dimensional spaces we study will locally be similar to the standard (Euclidean) 3- and 4-dimensional spaces. These objects 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. A more thorough mathematical understanding of contact structures should contribute significantly towards our understanding of 3- and 4-dimensional shapes, from the knotting of DNA at the microscopic level to the shape of the universe at the macroscopic level.