Equivariant motivic homotopy theory

主要部分调查员提出的研究是应用等变方法在两个密切相关的领域，即拓扑和代数几何。调查员正在进行的主要项目之一（与I. Kriz和K. Ormsby）是应用机器的稳定同伦理论，如亚当斯和亚当斯-诺维科夫谱序列，莫雷尔和Voevodsky的motivic同伦理论。 研究者的另一个项目是动机同伦理论，也是与I。Kriz和K.奥姆斯比，是研究等变motivic稳定同伦理论，这反过来也导致了新的信息，在世界上的等变拓扑。和我一起。Kriz，研究者有一个项目，研究由motivic谱的实现产生的拓扑中的等变谱，例如所谓的拓扑埃尔米特协边谱。作用于这种光谱的基团包含Z/2，并且作用包含真实的（或复共轭）作用。正如希尔，霍普金斯和拉文埃尔在解决Kervaire不变量1问题的最新工作所示，这些对象是非常有趣的新同伦理论信息的来源。此外，研究者也有一个项目，在理解弦拓扑，以及操作行动和变形理论在代数和拓扑。这方面的调查员的工作密切相关的J.卢里最近的概念非交换庞加莱对偶流形。总的主题调查员的研究是在相互作用的两个重要领域的数学，即代数拓扑和代数几何。 代数拓扑可以被认为是“纯粹定性”的几何，其中允许变形形状或拓扑空间，并将它们与某些代数和数值不变量相关联。另一方面，代数几何可以被认为是研究某些更严格的数学对象，基本上是从代数方程的解集建立的。莫雷尔和Voevodsky构建了一种方法的应用方法的代数拓扑领域的代数几何，引起了一个新的领域的数学，即motivic同伦理论。 调查员的一个项目（与I. Kriz和K.奥姆斯比）是应用某些成熟的机器代数拓扑这个世界，这有可能回答长期存在的问题。另一个项目是获得对等变动机同伦理论的理解，其目标是通过在故事中加入群体的行动来阐明动机同伦理论的结构。反过来，获得对等变动机同伦理论的理解也将导致关于拓扑本身的对象的新信息。

The main part the investigator's proposed research is the application of equivariant methods in two closely related areas, that of topology and of algebraic geometry. One of the investigator's main ongoing projects (joint with I. Kriz and K. Ormsby) is to apply the machinery of stable homotopy theory, such as the Adams and Adams-Novikov spectral sequences, to Morel and Voevodsky's motivic homotopy theory. Another of the investigator's projects in motivic homotopy theory, also joint with I. Kriz and K. Ormsby, is the study of equivariant motivic stable homotopy theory, which in turn also leads to new information in the world of equivariant topology. Together with I. Kriz, the investigator has a project studying equivariant spectra in topology arising from the realizations of motivic spectra, such as the so-called topological hermitian cobordism spectrum. The groups acting on such spectra contain Z/2, and the actions incorporate a Real (or complex conjugation) action. As shown by the recent work of Hill, Hopkins and Ravenel in solving the Kervaire invariant 1 problem, these objects are highly interesting sources of new homotopy theoretical information. In addition, the investigator also has a project in understanding string topology, as well as operad actions and deformation theory in both algebra and topology. This aspect of the investigator's work closely related to J. Lurie's recent notion of non-abelian Poincare duality on manifolds.The overall theme of the investigator's research is in the interaction between two important areas of mathematics, that of algebraic topology and algebraic geometry. Algebraic topology can be thought of as "purely qualitative" geometry, where one is allowed to deform shapes or topological spaces, and associate to them certain algebraic and numerical invariants. On the other hand, algebraic geometry can be thought of as the study of certain much more rigid mathematical objects, built essentially from the solution sets of algebraic equations. Morel and Voevodsky have constructed a way of applying the methods of algebraic topology to the area of algebraic geometry, giving rise to a new field of mathematics, that of motivic homotopy theory. One of the investigator's projects (joint with I. Kriz and K. Ormsby) is to apply certain well-established machinery of algebraic topology to this world, which has the potential to answer long-standing questions. Another project is to gain an understanding of equivariant motivic homotopy theory, the goal of which is to shed light on structures in motivic homotopy theory by adding in the actions of groups to the story. In its turn, gaining an understanding of equivariant motivic homotopy theory will also lead to new information about objects in topology itself.