The Algebra of Spectra, Group Actions, and Classifying Spaces

Wilkerson Wilkerson将继续他与Dwyer在理解具有类似于连通紧李群的分类空间性质的素数的空间的扩展Steenrod问题上的工作。他还将与Jeff Smith一起研究有限幂零复形的同伦自同构群以及有限群在此类复形上的作用。此外，他将与Jim Turner和Avramov合作研究Serre定理的类似，该定理建立了非平凡的单连通有限复形在无限维上具有非平凡同伦。McClure将与Jeff Smith合作研究结合环谱的更高中心。他还将研究一大类谱序列，这些序列应该允许使用便于计算的链式复杂模型。ASTEP在这个项目中，本身应该很有趣的是证明某些同伦范畴的模块光谱可以由链复合体建模。Shipley将为等变稳定同伦理论发展一个新的模型，这将为紧Lie群上的有理等变稳定同伦理论提供代数模型。她还将(与德怀尔一起)研究Cobar谱序列及其收敛性质。她将与Rezk和Schwede一起研究一种用单纯模型范畴取代二元模型范畴的方法。Smith将继续他的工作，将模型范畴理论作为研究环谱和交换环谱的工具。特别是，他将研究固定环R上的环谱的同伦理论，R-双模的余代数的同伦理论，以及杆结构。他将改进他的组合模型范畴理论，使其成为更有效的工具。在与Adem的合作中，他将研究球面同伦乘积上的群作用的同伦理论。从这篇综述中可以看出，这个项目将涉及同伦理论中许多不同方向的研究。要研究的问题中有几个长期以来一直是重要的(例如，Steenrod问题和Smith和Adem GoBack的联合工作，直到1960年的S)，而另一些问题涉及非常新的发展(特别是严格结合环谱和交换环谱理论；Smith和Shipley是这一领域的先驱，McClure的工作大量使用了它)。麦克卢尔和史密斯的联合工作在数学物理中有应用，因为它与康采维奇和塔马尔金最近在形变量子化方面的工作密切相关。

9971953Wilkerson Wilkerson will continue his work with Dwyer on the extended Steenrodproblem of understanding spaces with properties at a prime similar tothose of the classifying spaces of connected compact Lie groups. He willalso work with Jeff Smith on homotopy automorphism groups of finitenilpotent complexes and actions of finite groups on such complexes. Inaddition, he will work with Jim Turner and Avramov on analogues of theSerre theorem that establishes that non-trivial simply connected finitecomplexes have nontrivial homotopy in an infinite number of dimensions.McClure will work with Jeff Smith on higher centers of associative ringspectra. He will also investigate a large class of spectral sequencesthat should admit chain-complex models that facilitate calculation. Astep in this project that should be interesting in its own right is toshow that certain homotopy categories of module spectra can be modeled bychain complexes. Shipley will develop a new model for equivariant stablehomotopy theory; this will provide algebraic models for rationalequivariant stable homotopy theory over compact Lie groups. She willalso study (with Dwyer) a cobar spectral sequence and its convergenceproperties. With Rezk and Schwede she will study a way to replacearbitrary model categories by simplicial model categories. Smith willcontinue his work on using the theory of model categories as a toolfor the study of ring spectra and of commutative ring spectra. Inparticular, he will study the homotopy theory of ring spectra over afixed ring R, the homotopy theory of coalgebras of R-bimodules, and thebar construction. He will improve his theory of combinatorial modelcategories to make it a more effective tool. In joint work with Adem hewill study the homotopy theory of group actions on homotopy products ofspheres. As can be seen from this summary, this project will involve researchin many different directions within homotopy theory. Several of theproblems to be investigated have been important for a long time (forexample, the Steenrod problem and the joint work of Smith and Adem goback to the 1960's), while others involve very recent developments(especially the theory of strictly associative and commutative ringspectra; Smith and Shipley were pioneers in this area, and McClure'swork makes heavy use of it). The joint work of McClure and Smith hasapplications to mathematical physics, since it is closely related torecent work of Kontsevich and Tamarkin on deformation quantization.***