Higher-dimensional contact topology

接触结构在现代几何学和拓扑学中占有重要地位，与物理学有着密切的联系。 它们在数学物理中自然地作为时空的边界出现，在三维和四维空间的数学研究以及DNA的打结中发挥着重要作用。 在三维中，由于在过去几十年中的巨大进步，接触结构相对较好地理解。 该研究项目的目标是进一步发展高维接触几何，尽管近年来取得了重大进展，但仍然非常神秘，处于起步阶段。 作为该项目的一部分，首席研究员还将促进未来数学家的培训。这个关于高维接触几何及其Floer理论不变量的研究项目包括两个部分：首先是继续系统地研究凸超曲面理论--一种将接触流形分解为可分析的块的技术--由主要研究者和他的合作者黄博士发起的任意维度的研究，并进一步发展所使用的技术。第二是发展Heegaard Floer同调的高维推广。 目标是获得有效区分接触和辛流形及其子同态的不变量，以及三流形和四流形不变量。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Contact structures occupy a central role in modern geometry and topology and have close connections with physics. They appear naturally as the boundary of space-time in mathematical physics and play an essential role in the mathematical study of three- and four-dimensional spaces and the knotting of DNA. In dimension three, contact structures are comparatively well-understood due to dramatic advances in the previous decades. The goal of this research project is to further develop higher-dimensional contact geometry, which is still quite mysterious and in its infancy despite significant advances in recent years. As part of this project, the Principal Investigator will also promote the training of future mathematicians.This research project on higher-dimensional contact geometry and its Floer-theoretic invariants has two parts: The first is to continue the systematic study of convex hypersurface theory - a technique to decompose a contact manifold into easier-to-analyze pieces - in arbitrary dimensions initiated by the Principal Investigator and his collaborator Dr. Huang and further develop the techniques that were used. The second is to develop a higher-dimensional generalization of Heegaard Floer homology. The goal is to obtain invariants that are effective at distinguishing contact and symplectic manifolds and their diffeomorphisms, as well as three- and four-manifold invariants.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.