Knots in Washington XXI: Skein Modules, Khovanov Homology and Hochschild Homology

华盛顿结 XXI：绞纱模块、Khovanov 同源性和 Hochschild 同源性

• 批准号: 0555648
• 负责人: Jozef Przytycki
• 金额: --
• 依托单位: George Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-01-01 至 2006-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555648&HistoricalAwards=false
• 关键词: Knots; Washington; XXI; Skein; Modules

AbstractAward: DMS-0555648Principal Investigator: Jozef H. PrzytyckiThis award provides partial support for participants of the 21stconference in the very successful ``Knots in Washington''series. This meeting is devoted to Skein Modules, KhovanovHomology and its relation to Hochschild Homology. ``Khovanovhomology'' is a new development in quantum topology that emergedin the last several years. It is a generalization of the Jonesand Homflypt polynomials of links to homologies of certain chaincomplexes. These homologies turn out to be significantlystronger than the original invariants. Khovanov's ideas were alsoapplied to the polynomial invariants of graphs, as well asKauffman bracket skein module of some 3-manifolds. One of themost recent developments is Przytycki's observation that Khovanovhomology (or its comultiplication-free variant developed byHelme-Guizon and Rong) can be interpreted as Hochschild homologyof underlying algebras. This shows how seemingly distant branchesof mathematics can be put together.Low dimensional topology studies shapes of three and fourdimensional spaces. These dimensions are of particular interestto us because they are the dimensions of our space and ourspace-time. Knot theory is a subfield of low dimensionaltopology, which studies the knottedness in our space. One of themain goals of knot theory is to distinguish knotted objects. Thisis often done by means of the so called "knot invariants,"functions that replace geometric objects, such as knots, withthose that are easier to compare. Over the past two decades,there have been a flourish of new invariants in low dimensionaltopology, boosted by ideas from gauge theory, quantum algebras,and mathematical physics. In particular, a new invariant forknots, developed by Khovanov using ideas from homologicalalgebra, has sparked a great deal of interest recently. Thepurpose of the Knots in Washington Conferences is to bringtogether the researchers of the field, including establishedmathematicians as well as graduate students and recent PhDs, todiscuss the state of the art of the subject.

摘要奖：DMS-0555648首席研究员：Jozef H.Przytyck这个奖项为第21届会议的与会者提供了部分支持，这是非常成功的“在华盛顿的结”系列。这次会议致力于Skein模块，KhovanovHomology及其与Hochschild同调的关系。“KhovanovHomology”是过去几年量子拓扑学中出现的新发展。它是某些链复形同调的链环的Jones和Homflypt多项式的推广。事实证明，这些同调比原来的不变量强得多。Khovanov的思想也被应用于图的多项式不变量，以及一些三维流形的Kauffman括号斜线模。最新的发展之一是Przytycki观察到Khovanov同调(或由Helme-Guizon和Rong开发的无余乘变体)可以解释为基础代数的Hochschild同调。这说明了数学中看似遥远的分支是如何组合在一起的。低维拓扑学研究三维和四维空间的形状。我们对这些维度特别感兴趣，因为它们是我们的空间和我们时空的维度。结点理论是低维拓扑学的一个子域，它研究我们空间中的结点。纽结理论的主要目的之一是区分纽结对象。这通常是通过所谓的“纽结不变量”来完成的，这些函数用更容易比较的几何对象来代替几何对象，如纽结。在过去的二十年里，在规范理论、量子代数和数学物理的思想的推动下，低维拓扑中的新不变量蓬勃发展。特别是，Khovanov利用同调代数的思想提出了一种新的不变量ForNots，最近引起了人们的极大兴趣。华盛顿会议的目的是将该领域的研究人员聚集在一起，包括知名数学家以及研究生和新近的博士，讨论该学科的最新进展。