Invariants in Low Dimensional Topology

DMS-0305012 James Conant首席研究员，一位低维拓扑学家，正在利用他与Peter Teichner共同开发的Tool(称为`‘摸索共边法’)来寻找一类称为Vassiliev不变量的重要变量的拓扑应用和解释。特别是，他和Teichner正在探索Vassiliev不变量是否可以检测到结的问题。主要研究人员还在寻找Vassiliev不变量理论的更强大的版本，希望能产生四维(和谐)信息。其中一些是与雅各布·莫斯托沃伊和泰德·斯坦福合作的。与Matt Horak和KarenVogtmann合作，首席研究员正在通过Heller，Penner开发并由Kontsevich(图同调)阐明的工具来研究曲面的映射类群。图同调与Vassiliev不变量上下文中出现的代数非常相似。作者还通过图同调研究了自由群的外自同构群。最后，首席研究员瑞安·布德尼、凯文·斯坎内尔和德夫·辛哈正在进行一个将经典同伦理论和微分拓扑学中的Vassiliev不变量扎根的程序。这个程序已经在最简单的Vassiliev不变量的几何解释方面取得了成果。拓扑不变量理论背后的瑜伽是它们提供了有用的、可计算的拓扑信息。当维数较大时，通常可以使用代数不变量来找到拓扑问题的完整答案。在低维情况下，如3维和4维，情况更为复杂。不变量仍然存在，但它们要弱得多，而且通常不会给出完整的答案。另一方面，在低维空间中存在着大量的不变量，它们没有高维的相似之处，在拓扑学上也很难理解。例如，有一个多项式可以与任何打结的环路(如一段DNA或时空中的奇点)联系在一起，即使环路被推拉成新的形状，它也不会改变。这种多项式被称为琼斯多项式，它的定义是任何拓扑学上都无法解释的，其拓扑学应用也很有限。首席研究人员正在努力准确地理解这个多项式(和其他类似的对象)与拓扑的联系，这将导致更有效的方法来回答低维拓扑和更高的问题。

DMS-0305012James ConantThe principal investigator, a low dimensional topologist, is utilizing atool he developed with Peter Teichner (called ``grope cobordism") to findtopological applications and interpretations for an important class ofinvariants called Vassiliev invariants. In particular he and Teichner arepursuing the question of whether Vassiliev invariants detect knottedness.The principal investigator is also looking for more powerful versions ofthe theory of Vassiliev invariants, hopefully giving rise to fourdimensional (concordance) information. Some of this is joint with JacobMostovoy and Ted Stanford. Working jointly with Matt Horak and KarenVogtmann, the principal investigator is studying the mapping class groupof surfaces via a tool developed by Harer, Penner, and elucidated byKontsevich (graph homology). Graph homology is very similar to the algebrathat appears in the context of Vassiliev invariants. The principalinvestigator is also studying the group of outer automorphisms of a freegroup via graph homology. Finally, the principal investigator, workingjointly with Ryan Budney, Kevin Scannell, and Dev Sinha, is pursuing aprogram for grounding Vassiliev invariants in classical homotopy theoryand differential topology. This program has already borne fruit in termsof geometric interpretations of the simplest Vassiliev invariant.The yoga behind the theory of topological invariants is that they provideuseful, computable topological information. When the number of dimensionsis large, complete answers to topological questions can often be foundusing algebraic invariants. In low dimensions, such as 3 and 4, thesituation is more complicated. The invariants still exist, but they aremuch weaker, and usually do not give complete answers. On the other hand,there are lots of invariants in low dimensions which don't have highdimensional analogues, and which are poorly understood topologically. Forexample, there is a polynomial you can associate to any knotted loop (suchas a piece of DNA or a singularity in spacetime) which will not changeeven if the loop is pushed and pulled into a new shape. This polynomial,called the Jones polynomial, has a definition which eludes any topologicalinterpretation and whose topological applications have been modest. Theprincipal investigator is working to understand precisely the connectionof this polynomial (and other similar objects) with topology, and thiswill lead to more effective methods for answering questions in lowdimensional topology and beyond.