Connections between Khovanov - and Heegaard Floer - type Homology Theories

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。首席研究员将探讨Heegaard Floer同源性和Khovanov同源性之间的联系，这两个理论受到物理学思想的启发，在过去十年中改变了低维拓扑的景观。 该项目将专注于一个部分理解的连接-即Khovanov缠结理论和Heegaard Floer双分支覆盖理论之间的关系，首先由Peter Ozsvath和Zoltan Szabo发现，后来重新解释，使用Andras Juhasz的相对版本的Heegaard Floer同调缝合流形，由首席研究员和Stephan Wehrli。 各种TQFT型操作下的连接的自然性表明了一条发展Khovanov型不变量的路径，用于低维拓扑中更广泛的一类对象，这反过来又会产生新的应用。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。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The principal investigator will probe the connection between Heegaard Floer homology and Khovanov homology, two theories, inspired by ideas in physics, that have transformed the landscape of low-dimensional topology during the past decade. The project will focus on one partially-understood connection--namely, the relationship between Khovanov theories of tangles and Heegaard Floer theories of their double-branched covers, first discovered by Peter Ozsvath and Zoltan Szabo and later reinterpreted, using Andras Juhasz's relative version of Heegaard Floer homology for sutured manifolds, by the principal investigator and Stephan Wehrli. The naturality of the connection under various TQFT-type operations suggests a path for developing Khovanov-type invariants for a wider class of objects in low-dimensional topology which should, in turn, yield new applications.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.