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)提供资金的。首席研究员将探索Heegaard Floer同调和Khovanov同调之间的联系，这两个理论受到物理学思想的启发，在过去十年里改变了低维拓扑的格局。该项目将集中在一个部分理解的联系上--即Khovanov纠缠理论和Heegaard Floer双分支覆盖理论之间的关系，该理论首先由Peter Ozsvath和Zoltan Szabo发现，后来由首席研究员和Stephan Wehrli使用Andras Juhasz的缝合流形的Heegaard Floer同调的相对版本重新解释。在不同的TQFT型运算下，连接的自然性提供了一条为低维拓扑中更广泛的对象开发Khovanov型不变量的途径，这反过来将产生新的应用。本项目的主要目的是提高我们对三维和四维空间拓扑的理解，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。

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.