Calculus of the embedding functor

嵌入函子的微积分

• 批准号: 0504390
• 负责人: Ismar Volic
• 金额: $ 6.21万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-15 至 2007-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504390&HistoricalAwards=false
• 关键词: Calculus; embedding; functor

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.

这个项目的主要目标是通过一个相对较新的嵌入函子微积分理论来研究经典的节点和更一般的嵌入空间，这个理论是由T。Goodwillie和M.韦斯。 Volic最近的研究结果表明，两者之间存在着很强的联系。 事实上，从函子演算中产生的某个空间塔可以作为有限型结不变量的分类对象，这是一类迷人的不变量，它以复杂的方式与拓扑学和几何学以及物理学的其他领域联系在一起。 该项目的其他目标涉及更一般的空间节点，特别是崩溃的某些频谱序列与这些空间。 这应该给新的信息同源性和同伦空间的结，以及新的见解出现有趣的组合在他们的研究。最近由P.Lambrechts和Volic关于位形空间的一个结果将作为本项目这一部分的起点。 由于构造模拟节点空间的微积分塔是相当通用的，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计划使用嵌入函子的两个版本的微积分的相互作用，流形和正交，以表明这些理论产生的谱序列崩溃。 这也将导致对这两种版本的微积分如何相互作用的新的认识。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建（即找到好的结不变量），仍然产生了丰富的令人兴奋的研究。 这个项目的主要目标之一是通过函子演算的新技术来研究纽结理论，以进一步理解纽结理论。 然而，事实证明，所使用的方法是相当普遍的，并扩展到更大的拓扑空间类以外的结。 因此，拓扑学，几何学，组合学和物理学之间的新联系，预计将从这个项目中产生广泛的影响，并以意想不到的方式将拓扑学中的各种思想汇集在一起。