CAREER: Connections between algebraic and geometric invariants in low-dimensional topology

主要研究人员将探索低维拓扑中对象的两类“同调型”不变量之间的联系：Khovanov同调，其(代数)构造源于量子群的更高表示理论；Heegaard Floer同调，其(几何/解析)构造源于辛几何和规范理论的思想。这个项目建立在Ozsvath-Szabo的基础上，并试图从第一原理重新解释他的基础工作，他构造了一个环的(简化的)Khovanov同调的变形，从而将它连接到它的双分支覆盖的Heegaard Floer同调。通过理解这两个理论的“开放”版本之间的关系，P.I.将获得关于辫子和缠结的问题的应用。本项目的广泛目标是提高我们对三维和四维空间的拓扑的理解，即这些空间在拉伸和收缩(但不在撕裂和粘合)下保持不变的性质。拓扑思想为高效计算机芯片和信息网络的发展奠定了基础。分子和蛋白质的形状决定了它们的电学性质和生物功能。将量子计算算法建立在量子系统的大规模特征上，可以最大限度地减少它们对随机错误的敏感度。此外，纽结理论，即研究嵌入在三维空间中的环，在我们理解DNA在细胞中的行为方式方面已经变得越来越重要。

The principal investigator will probe the connection between two classes of "homology-type" invariants of objects in low-dimensional topology: Khovanov homology, whose (algebraic) construction originates in the higher representation theory of quantum groups and Heegaard Floer homology, whose (geometric/analytic) construction arises from ideas in symplectic geometry and gauge theory. This project builds on, and seeks to re-explain from first principles, foundational work of Ozsvath-Szabo, who constructed a deformation of the (reduced) Khovanov homology of a link, thereby connecting it to the Heegaard Floer homology of its double-branched cover. By understanding the relationship between "open" versions of the two theories, the P.I. will obtain applications to questions about braids and tangles.The broad aim of the present project is to improve our understanding of the topology of 3- and 4-dimensional spaces, i.e., the properties of these spaces that remain unchanged under stretching and contracting (but not under tearing and gluing). Topological ideas underpin the development of efficient computer chips and information networks. The shapes of molecules and proteins determine their electrical properties and biological functions. Basing quantum computing algorithms on large-scale features of a quantum system minimizes their susceptibility to random error. Moreover, knot theory, the study of loops imbedded in 3-dimensional space, has become increasingly important in our understanding of how DNA behaves in cells.