Contact topology and automorphisms of surfaces

Giroux的基础工作建立了封闭三流形上的接触结构与曲面自同构之间的一一对应，通过相容的开卷分解达到稳定。这是两个经典的研究之间的对应关系，从根本上说，重要的，对象，PI建议研究。 由于正稳定的表面自同构产生一个几乎难以处理的等价关系，重要的是要辨别接触结构的性质，从一个单一的代表自同构。 这样一个结果的一个例子是这样一个事实，即自同构是组成的积极德恩扭曲诱导接触结构是斯坦填充，即出现自然边界的斯坦流形。我们正在努力理解这样的问题：什么性质的自同构保证辛可填充性，什么性质意味着吉鲁挠的存在。这些问题的研究将应用于有界和无界情形下Heegaard-Floer同调理论中接触不变量的研究。接触拓扑或几何及其偶维对应物辛拓扑或几何，都诞生于对经典力学和热力学中出现的问题的研究。三维流形是以我们生活的空间为模型的数学对象。 在研究流体流动时，这种空间上的接触结构作为垂直于流动的平面族自然出现。接触结构的一个熟悉的例子出现在DLP前投影电视的设计中，在那里他们规定使用数百万个微小的镜子，而不是一个大的曲面镜。 在过去的几十年里，三维接触拓扑和四维辛拓扑都取得了很大的进展。 最近的进展使研究人员能够将二维技术应用于接触拓扑的固有三维研究。

Fundamental work of Giroux established a one-to-one correspondence between contact structures on closed three-manifolds and automorphisms of surfaces up to stabilization via compatible open book decompositions. It is this correspondence between two classically studied, fundamentally important, objects that the PIs propose to study. Since positive stabilization of surface automorphisms generates an almost intractable equivalence relation, it is important to discern the properties of a contact structure from just a single representative automorphism. An example of such a result is the fact that automorphisms which are compositions of positive Dehn twists induce contact structures that are Stein fillable, i.e. that arise as natural boundaries of Stein manifolds. We are working to understand questions like: what property of an automorphism guarantees symplectic fillability, what property implies existence of Giroux torsion. Investigating these questions will have applications to the study of contact invariants in Heegaard-Floer homology theory in both the bounded and unbounded cases.Contact topology or geometry and its even dimensional counterparts, symplectic topology or geometry, were born out of the study of questions arising in classical mechanics and thermodynamics. Three-dimensional manifolds are mathematical objects modeled on the space in which we live. Contact structures on such spaces arise naturally in the study of fluid flows as the family of planes perpendicular to the flow. A familiar example of a contact structure occurs in the design of DLP front projection televisions where they dictate the use of literally millions of tiny mirrors rather than one large curved mirror. Considerable progress has been made in the last several decades on the three-dimensional contact topology and four-dimensional symplectic topology. Recent progress has allowed researchers to apply two-dimensional techniques to the inherently three-dimensional study of contact topology.