The Algebra of Spectra, Group Actions, and Classifying Spaces

谱代数、群作用和空间分类

• 批准号: 9971953
• 负责人: Clarence Wilkerson
• 金额: $ 34.3万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971953&HistoricalAwards=false
• 关键词: Algebra; Spectra; Group; Actions; Classifying

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.***

小行星9971953 威尔克森将继续他的工作与德怀尔的扩展Steenrod问题的理解空间的性质在一个总理类似的分类空间的连接紧凑李群。 他还将与杰夫史密斯同伦自同构群的finitenilpotent复杂和行动的有限团体等复杂。 此外，他将与吉姆特纳和阿夫拉莫夫的类似物的塞尔定理，建立了非平凡的简单连接finitecomplex有非平凡的同伦在无限数量的层面。麦克卢尔将与杰夫史密斯在更高的中心的关联环谱。 他还将研究一大类光谱序列，这些序列应该允许链复合模型，以便于计算。 在这个项目的一个步骤，应该是有趣的，在其本身的权利是表明，某些同伦类别的模谱可以模拟链复合物。 希普利将开发一个新的模型等变稳定同伦理论，这将提供代数模型的rationalequivariant稳定同伦理论在紧凑的李群。 她还将研究（与德怀尔）cobar谱序列及其收敛性质。 与Rezk和Schwede，她将研究一种方法，通过单纯模型类别来描述earbitrary模型类别。 史密斯将继续他的工作，利用模型范畴理论作为研究环谱和交换环谱的工具。 特别是，他将研究同伦理论的环谱在一个固定的环R，同伦理论的余代数的R-双模，和thebar建设。 他将改进他的组合模型分类理论，使其成为更有效的工具。 在联合工作与阿德姆，他将研究同伦理论的群体行动的同伦产品的领域。 从这个总结中可以看出，这个项目将涉及同伦理论中许多不同方向的研究。 几个theprojects要调查一直是重要的很长一段时间（例如，Steenrod问题和联合工作的史密斯和Adem gobback到20世纪60年代），而其他人涉及最近的发展（特别是理论的严格结合和交换环谱;史密斯和希普利是先驱在这一领域，和麦克卢尔的工作使大量使用它）。 麦克卢尔和史密斯的联合工作可应用于数学物理，因为它与孔采维奇和塔马金最近关于变形量子化的工作密切相关。*