Calculus of the embedding functor

该项目的主要目标是通过T. Goodwillie和M. Weiss提出的相对较新的嵌入函子演算理论来研究经典结和更一般的嵌入空间。Volic最近的研究结果表明两者之间有很强的联系。事实上，由函子演算产生的某个空间塔作为有限型结不变量的分类对象，这是一类迷人的不变量，已被发现以复杂的方式与拓扑和几何以及物理的其他领域联系在一起。该项目的其他目标涉及更一般的结空间，特别是与这些空间相关的某些谱序列的崩溃。这将提供关于结空间的同调和同伦的新信息，以及在他们的研究中有趣组合的出现的新见解。P. Lambrechts和Volic最近关于配置空间的研究结果将作为这部分项目的起点。由于对结点空间建模的微积分塔的构建是非常通用的，另一个目标是提取关于其他嵌入空间的信息。特别是，G. Arone和Volic计划使用两种版本的嵌入函子演算的相互作用，流形和正交，来证明由这些理论产生的谱序列崩溃。这也将导致对这两个版本的微积分如何相互作用的新见解。结是拓扑学中最有趣的研究对象之一，因为它们很容易定义和可视化，而且物理学家、化学家等都对它们感兴趣。一些关于结的基本问题，比如它们的分类，或者区分它们的有效方法的构建（即找到好的结不变量），仍然产生了大量令人兴奋的研究。本课题的主要目标之一是利用函子演算的新技术来研究结理论，进一步加深对结理论的理解。然而，事实证明，所使用的方法是相当普遍的，并且可以扩展到更大的拓扑空间类。因此，拓扑学、几何学、组合学和物理学之间的新联系有望从这个项目中产生，它可能具有广泛的含义，并以意想不到的方式将拓扑学的各种思想流派结合在一起。

Project Abstract for Ismar VolicThe main goal of this project is the study of classical knots and more general spaces of embeddings through a relatively new theory of calculus of the embedding functor, developed by T. Goodwillie and M. Weiss. Recent results of Volic show that there is a strong connection between the two. In fact, a certain tower of spaces arising from calculus of functors serves as a classifying object for finite type knot invariants, a fascinating class of invariants which has been found to connect in intricate ways to other areas of topology and geometry, as well as physics. Other goals of the project concern more general spaces of knots and in particular the collapse of certain spectral sequences associated to these spaces. This should give new information about homology and homotopy of spaces of knots, as well as new insight into appearance of interesting combinatorics in their study. A recent result due to P. Lambrechts and Volic concerning configuration spaces will serve as the starting point for this part of the project. Since the construction of the calculus tower modeling the space of knots is quite general, another goal is to extract information about other spaces of embeddings. In particular, G. Arone and Volic plan to use the interplay of two versions of calculus of the embedding functor, manifold and orthogonal, to show that the spectral sequences arising from these theories collapse. This should also result in new insight into how these two versions of calculus interact.Knots are some of the most interesting objects of study in topology both because they are easy to define and visualize and because they are of interest to physicists, chemists, etc. Some fundamental questions about knots, such as their classification, or construction of efficient ways of telling them apart (i.e. finding good knot invariants), still generate a wealth of exciting research. One of the main objectives of this project is to further the understanding of knot theory by studying it through the new technique of calculus of functors. It turns out, however, that the methods used are quite general and extend beyond knots to larger classes of topological spaces. Thus the new connections between topology, geometry, combinatorics, and physics which are expected to arise from this project could have broad implications as well as bring together various schools of thought in topology in unexpected ways.