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

PI计划研究流形在欧氏空间中的嵌入空间。特别地，这包括各种维度的欧几里得空间中的纽结。PI计划从同伦论的角度进一步加深我们对其结构的理解。他计划通过应用函子微积分来研究纽结，他已经在最近的工作中将其与纽结理论联系起来。具体地说，函子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类型结不变量的纽结不变量的所有信息。这些不变量被发现以错综复杂的方式与拓扑学和几何学的其他领域以及物理学联系在一起。其最终目的是证明这些不变量将纽结分开，即，给定任意两个不同的纽结，存在一个有限类型不变量来区分它们。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、有希望的工具来攻击这个问题。更广泛地说，PI计划建立在他和他的合作者已经完成的关于欧氏空间中具有适当的维度限制的流形嵌入空间的完整拓扑描述的工作的基础上。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崩溃，将某些积分推广到更一般的纽结空间，以便推广有限类型不变量，并建立各种具有广泛重要性和各种用途的基础性技术结果。纽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，也因为它们对物理学家、化学家等感兴趣。关于纽结的一些基本问题，如它们的分类，或区分它们的有效方法的构造(即找到好的纽结不变量)，仍然产生了大量令人兴奋的研究。这个项目的主要目标之一是通过函子演算这一相对较新的技术来研究纽结理论，以加深对它的理解。然而，事实证明，这一理论中使用的方法是相当普遍的，并且超越了纽结，扩展到更大类的拓扑空间。因此，这个项目有望产生的拓扑学、几何学、组合学和物理学之间的新联系将潜在地提供构造各种嵌入空间的不变量的新方法，回答关于纽结理论的几个重要猜想，并在代数拓扑学中引入新的观点。

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.