The Algebra of Spectra, Group Actions, and Classifying Spaces

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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.***