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

首席研究员将探索低维拓扑中物体的两类“同调型”不变量之间的联系：Khovanov同调，其（代数）构造源于量子群的高级表示理论；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的基础工作的基础上，并试图从第一原理重新解释，他构建了一个链接的（简化的）Khovanov同调的变形，从而将其连接到其双分支盖的Heegaard flower同调。通过理解这两种理论的“开放”版本之间的关系，P.I.将应用于关于辫子和缠结的问题。本项目的总体目标是提高我们对三维和四维空间拓扑结构的理解，即这些空间在拉伸和收缩（但不包括撕裂和粘合）下保持不变的性质。拓扑学思想是高效计算机芯片和信息网络发展的基础。分子和蛋白质的形状决定了它们的电学特性和生物学功能。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误差的敏感性。此外，研究嵌入三维空间的环的结理论在我们理解DNA在细胞中的行为方面变得越来越重要。

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.