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.

首席研究员将探讨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）构造起源于量子群的更高表示理论和 Heegaard Floer 同调，其（几何） /解析）构造源于辛几何和规范理论的思想。该项目以 Ozsvath-Szabo 的基础工作为基础，并试图从第一原理重新解释，他构建了链接的（简化的）Khovanov 同源性的变形，从而将其与其双分支的 Heegaard Floer 同源性连接起来覆盖。 通过了解这两种理论的“开放”版本之间的关系，P.I.将应用于有关辫子和缠结的问题。本项目的广泛目标是提高我们对 3 维和 4 维空间拓扑的理解，即这些空间的属性在拉伸和收缩下保持不变（但不在撕裂和粘合下）。 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特性和生物功能。 基于量子系统的大规模特征的量子计算算法可以最大限度地降低其对随机误差的敏感性。 此外，结理论（对嵌入 3 维空间中的环的研究）对于我们理解 DNA 在细胞中的行为方式变得越来越重要。