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

摘要奖：DMS-0555648主要研究者：Jozef H.该奖项为非常成功的“华盛顿之结”系列中的第21届会议的与会者提供了部分支持。这次会议专门讨论Skein模，Khovanov同调及其与Hochschild同调的关系。“Khovanov同源性”是近几年来量子拓扑学的一个新发展。这是一个推广的琼斯和Homflypt多项式的链接到同调的某些chaincomplex。 这些同源性比原来的不变量要强得多。Khovanov的思想也被应用于图的多项式不变量，以及一些3-流形的Kauffman括号skein模。其中最新的发展是Przytycki的观察，Khovanov同调（或其余乘法自由变种开发的Helme-Guizon和荣）可以被解释为Hochschild homologyof的基础代数。这表明了数学中看似遥远的分支是如何被联系在一起的。低维拓扑学研究三维和四维空间的形状。这些维度对我们特别有意义，因为它们是我们的空间和时空的维度。纽结理论是低维拓扑学的一个分支，它研究我们空间中的纽结性。纽结理论的主要目标之一是区分打结的物体。这通常是通过所谓的“结不变量“来完成的，这些函数用更容易比较的对象来替换几何对象，如结。在过去的二十年里，在规范理论、量子代数和数学物理的推动下，低维拓扑中出现了大量新的不变量。特别是，一个新的不变量为结，开发的Khovanov使用的思想从homologicalalgebra，最近引发了极大的兴趣。华盛顿会议的目的是把这个领域的研究人员聚集在一起，包括已经建立的数学家以及研究生和最近的博士生，讨论这个问题的最新进展。

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.