Investigating connections between homotopy theory and algebra

本项目主要集中在三个方面：动机稳定同伦群，等同伦理论的新基础，以及微分分次代数的拓扑等价。在第一个领域，PI将继续他与Isaksen在Motivic Adams谱序列方面的工作，并将研究源自Cayley-Dickson代数的Motivic Hopf元素之间的特定关系。Etale同伦理论项目包括使用拟范畴来创建空间的同伦上滤图的Artin-Mazur风格的同伦理论。最后，在与Shipley的合作中，PI将继续研究微分分次代数可以产生弱等价Eilenberg-MacLane谱的情况。同伦论是数学中通过代数技术研究高维空间中几何对象的部分。信息的流动可以是双向的：通常情况下，一个人从代数知识开始，然后使用同伦理论来推导几何信息，但特别是在最近几年，从几何知识开始并推导出一些复杂的代数信息已经成为可能。这个研究项目将试图增加我们关于代数和几何以神秘方式相互作用的几个主题的知识。通过更深入地理解这些联系，我们将更好地理解某些精细的几何图案，并可能攻击一些困难的猜想。

The project focuses on three areas: motivic stable homotopy groups, new foundations for etale homotopy theory, and topological equivalences of differential graded algebras. In the first area the PI will continue his work with Isaksen on the motivic Adams spectral sequence, and will investigate specific relations amongst the motivic Hopf elements that arise from Cayley-Dickson algebras. The etale homotopy theory project involves using quasi-categories to create an Artin-Mazur style homotopy theory of homotopically cofiltered diagrams of spaces. Finally, in joint work with Shipley the PI will continue the investigation of situations in which differential graded algebras can give rise to weakly equivalence Eilenberg-MacLane spectra.Homotopy theory is the part of mathematics that studies geometric objects in higher-dimensional space via algebraic techniques. The flow of information can go in both directions: typically one starts with algebraic knowledge and uses homotopy theory to deduce geometric information, but especially in recent years it has become possible to start with geometric knowledge and deduce some sophisticated algebraic information. This research project will try to add to our knowledge about several topics in which algebra and geometry interact in mysterious ways. By coming to a deeper understanding of these connections, we will be in a better position to understand certain elaborate geometric patterns and possibly to attack some difficult conjectures.