RUI: Embedding spaces via calculus of functors and generalizations of finite type invariants

PI计划研究欧几里德空间中流形嵌入的空间。特别地，这包括不同维度欧几里得空间中的结点。PI计划从同伦理论的角度进一步了解它们的结构。他计划应用函子演算来研究结，他在最近的工作中已经将其与结理论联系起来。特别地，函子的演算产生了一个特定的空间序列，它捕获了一类相对较新的称为有限型结不变量的结不变量的所有信息。人们发现，这些不变量以复杂的方式与拓扑学、几何以及物理学的其他领域联系在一起。最终的目标是证明这些不变量可以区分结点，也就是说，给定任意两个不同的结点，存在一个有限型不变量可以将它们区分开来。PI相信他可以在这个猜想上取得一些进展，因为函子演算提供了一个新的、强大的、有前途的工具，可以用来解决这个问题。更一般地说，PI计划以他和他的合作者已经完成的工作为基础，对欧几里得空间中流形的嵌入空间进行完整的拓扑描述，并对维度进行适当的限制。特别是，他希望能够证明计算这些嵌入空间的同伦的某个谱序列的坍缩，将某些积分推广到更一般的结点空间，以推广有限型不变量，并建立各种具有广泛重要性和各种用途的基础技术成果。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。该项目的主要目标之一是通过相对较新的函子演算技术来研究结理论，从而进一步了解结理论。然而，事实证明，在这个理论中使用的方法是相当普遍的，并且超越了结点扩展到更大的拓扑空间类。由此产生的拓扑学、几何学、组合学和物理学之间的新联系，将有可能为构造各种嵌入空间的不变量提供新的方法，回答关于结理论的几个重要猜想，并引入代数拓扑学的新观点。

The PI plans to study spaces of embeddings of manifolds in Euclidean spaces. In particular, this includes knots in Euclidean spaces of various dimensions. The PI plans to further our understanding of their structure from the homotopy-theoretic point of view. He plans to study knots by applying the calculus of functors, which he has already connected to knot theory in recent work. In particular, calculus of functors produces a certain sequence of spaces which captures all the information about a relatively new class of knot invariants called finite type knot invariants. These invariants have been found to connect in intricate ways to other areas of topology and geometry, as well as physics. The ultimate goal is to show that these invariants separate knots, i.e. that, given any two knots which are different, there exists a finite type invariant which can tell them apart. The PI believes he can make some progress on this conjecture since the calculus of functors provides a new, powerful, promising tool with which this problem can be attacked. More generally, the PI plans to build on the work he and his collaborators have already accomplished concerning a complete topological description of spaces of embeddings of a manifold in a Euclidean space with suitable restrictions on dimensions. In particular, he expects to be able to show the collapse of a certain spectral sequence which computes the homotopy of these embedding spaces, to extend certain integrals to more general spaces of knots in order to generalize finite type invariants, and establish a variety of foundational technical results of wide importance and variety of uses. 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 relatively new technique of calculus of functors. It turns out, however, that the methods used in this theory 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 will potentially provide new ways of constructing invariants of various spaces of embeddings, answer several important conjectures about knot theory, and to introduce new points of view in algebraic topology.