Interactions of homotopy theory and algebra

该项目有三个组成部分，都涉及代数和拓扑之间的新的相互作用。 在第一部分中，研究者将探讨正交格拉斯曼在动机同伦理论中的作用。 重点将是他们的关系与埃尔米特K-理论和他们的动机上同调，后者是相关的动机特征类的二次丛。 一个目标是几何地构造这样的类并研究它们的基本属性。 在项目的第二部分，研究者将探索（与D。Biss和D. Isaksen）空间，试图对Cayley-Dickson代数中的零因子空间进行完全分类。 这可能最终应用于Kervaireinvariant问题，这将被探讨。 最后，研究者将继续与B合作。Shipley等人研究了微分分次代数与环谱之间的关系，在过去的十年中，拓扑学和代数之间的相互作用有了许多新的、独特的发现。 从历史上看，代数一直被用来给出拓扑学的信息;但最近的发展令人惊讶的是，他们经常使用拓扑学来给出代数学的信息。 这个项目的重点是三个实例。 研究的主要部分涉及动机上同调，这是一个非常令人兴奋的领域，正在迅速发展。 在其核心，它涉及应用拓扑技术的研究系统的多项式方程。 该项目的目标是研究从拓扑学的角度很好理解的某些基本对象的动机上同调，并研究代数和拓扑学如何在这些重要的例子中相互作用。 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.