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

职业：低维拓扑中代数和几何不变量之间的联系

• 批准号: 1151671
• 负责人: Julia Grigsby
• 金额: $ 41.07万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1151671&HistoricalAwards=false
• 关键词: CAREER; Connections; between; algebraic; geometric

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.

主要研究者将探讨低维拓扑中两类“同调型”不变量之间的联系：Khovanov同调，其（代数）构造起源于量子群的高级表示理论和Heegaard Floer同调，其（几何/分析）构造源于辛几何和规范理论中的思想。该项目建立在Ozsvath-Szabo的基础工作之上，并试图从第一原理重新解释，Ozsvath-Szabo构建了链接的（约化）Khovanov同调的变形，从而将其连接到其双分支覆盖的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.将获得有关辫子和缠结问题的应用。本项目的广泛目标是提高我们对三维和四维空间拓扑结构的理解，即，这些空间的性质在拉伸和收缩下保持不变（但在撕裂和粘合下不保持不变）。 拓扑思想支撑着高效计算机芯片和信息网络的发展。 分子和蛋白质的形状决定了它们的电学性质和生物学功能。 基于量子系统的大规模特征的量子计算算法最大限度地减少了它们对随机错误的敏感性。 此外，纽结理论（knot theory）--研究嵌入三维空间中的环的理论--在我们理解DNA在细胞中的行为方面变得越来越重要。