Equivariant motivic homotopy theory

研究人员提出的主要研究内容是等变方法在拓扑学和代数几何这两个密切相关的领域中的应用。研究人员正在进行的主要项目之一(与I.Kriz和K.Ormsby合作)是将稳定同伦理论的机制，如Adams和Adams-Novikov谱序列，应用于Morel和Voevodsky的动机同伦理论。这位研究者与I.Kriz和K.Ormsby在动机同伦理论方面的另一个合作项目是等变动机稳定同伦理论的研究，这反过来也带来了等变拓扑界的新信息。与I.Kriz一起，这位研究者有一个项目，研究由Motivic谱的实现产生的拓扑学中的等变谱，例如所谓的拓扑厄米特余边谱。作用在这些光谱上的基团包含Z/2，并且这些作用结合了实数(或复共轭)作用。正如Hill、Hopkins和Ravenel最近在解决Kervaire不变量1问题时所做的工作所表明的那样，这些天体是新的同伦理论信息的非常有趣的来源。此外，研究人员还研究了弦拓扑学，以及代数和拓扑学中的算符作用和形变理论。研究人员的这方面工作与J.Lurie最近提出的流形上的非阿贝尔Poincare对偶概念密切相关。研究人员研究的总主题是两个重要数学领域--代数拓扑学和代数几何之间的相互作用。代数拓扑学可以被认为是“纯粹定性的”几何，它允许人们变形形状或拓扑空间，并将某些代数和数值不变量与它们联系在一起。另一方面，代数几何可以被认为是对某些更严格的数学对象的研究，这些对象基本上是从代数方程的解集建立起来的。Morel和Voevodsky构造了一种将代数拓扑的方法应用于代数几何领域的方法，从而产生了一个新的数学领域--动机同伦理论。研究人员的项目之一(与I.Kriz和K.Ormsby合作)是将某些公认的代数拓扑学机制应用于这个世界，这有可能回答长期存在的问题。另一个项目是了解等变动机同伦理论，其目的是通过在故事中加入群体的行动来阐明动机同伦理论中的结构。反过来，对等变基元同伦理论的理解也将导致关于拓扑学本身中对象的新信息。

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.