Interactions of homotopy theory and algebra

这个项目有三个组成部分，都是关于代数和拓扑学之间的新的相互作用。在第一部分中，研究者将探索正交Grassmannians在动机同伦理论中的作用。重点将集中在它们与厄米特K-理论的关系和它们的动机上同调，后者与二次丛的动机特征类有关。OneGoal将以几何形式构造此类类并研究它们的基本性质。在项目的第二部分，研究人员将探索(与D.Biss和D.Isaksen一起)Cayley-Dickson代数中的零因子空间，试图对它们进行完全分类。这最终可能会应用到Kervaire不变性问题中，这一点将被探索。最后，研究人员将继续与B.Shipley合作，研究微分分次代数与环谱之间的关系。在过去的十年中，发现了许多新的、令人惊讶的拓扑学和代数相互作用的方式。在历史上，代数总是用来提供关于拓扑的信息；但令人惊讶的是，最近的发展是他们经常使用拓扑学来提供关于代数的信息。这个项目关注于三个这样的例子。这项研究的主要内容涉及动机上同源性，这是一个非常令人兴奋的领域，正在迅速增长。其核心是将拓扑学技术应用于多项式方程组的研究。该项目的目的是研究从拓扑观点理解的某些基本对象的基元上同调，并在这些重要的例子中调查代数和拓扑是如何相互作用的。研究结果对该领域的进一步发展具有重要意义。

The project has three components, all concerning new interactionsbetween algebra and topology. In the first component, theinvestigator will explore the role of orthogonal Grassmannians inmotivic homotopy theory. The focus will be on their relationship withHermitian K-theory and on their motivic cohomology, the latter beingrelated to motivic characteristic classes for quadratic bundles. Onegoal will be to geometrically construct such classes and toinvestigate their basic properties. In the second part of the projectthe investigator will explore (with D. Biss and D. Isaksen) spaces ofzero-divisors in Cayley-Dickson algebras, in an attempt to completelyclassify these. This may eventually have applications to the Kervaireinvariant one problem, and this will be explored. Finally, theresearcher will continue joint work with B. Shipley which studies therelationship between differential graded algebras and ring spectra.The last ten years have seen the discovery of many new, surprisingways in which topology and algebra interact. Historically, algebrahas always been used to give information about topology; but what isamazing about recent developments is that they often use topology togive information about algebra. This project focuses on threeinstances of this. The main component of the research involvesmotivic cohomology, which is a very exciting area that is growingquickly. At its core, it involves the application of topologicaltechniques to the study of systems of polynomial equations. Theobjective of the project is to study the motivic cohomology of certainbasic objects which are well-understood from a topological viewpoint,and to investigate how the algebra and topology interact in theseimportant examples. The results should be very important for thefurther development of the field.