Group proposal in topology

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DMS-0204615J. Peter MayThis project deals with research in a wide range of topics in topology, geometry, and related areas of mathematics. May studies a variety of categorical and homotopical structures in stable homotopy theory, equivariant algebraic topology, and other fields. Although his current work is primarily foundational, his students and collaborators, among others, make extensive calculational applications of it. He and his collaborators have recently unified the foundations of stable homotopy theory by proving that all of the new highly structured categories of spectra are Quillen equivalent model categories, via structure-preserving functors. The analogous, and deeper, unification of the foundations of equivariant stable homotopy has also been carried out. Within algebraic topology, May has been working on various other projects in equivariant and non-equivariant homotopy theory. He has also been working on various topics that have roots in algebraic topology but are of a more general nature. In particular, he has recently obtained a new definition of enriched weak $n$-categories and has spearheaded a very large scale unification project in higher category theory. Weinberger studies various areas of geometric topology and differential geometry. He has a longstanding interest in transformation groups, especially surgery theory on manifolds with group actions. In particular, he studies problems concerned with removing the ``gap hypothesis'' that obstructs the direct generalization of the nonequivariant theory. Weinberger also has a longstanding interest in the Novikov, Borel, and Baum-Connes conjectures, and he has made recent progress on them. In a new direction, Weinberger has been engaged in a large scale collaborationwith Nabutovsky that concerns applications of logic to Riemannian variational problems and to the large scale geometry of moduli spaces. Some of his recent results defy easy characterization. For one example, he has applied an old theorem of Browder about finite H-spaces to obtain a theorem about the "social choice problem". For another, in recent work with Farb he has shown that "hidden symmetries"on a locally symmetric manifold force it to be arithmetic.Besides May, the algebraic topology group at Chicago includes four nontenured faculty and eight graduate students. The geometry group includes Weinberger and four other tenured faculty, six nontenured faculty, and fifteen graduate students. There is considerable interactionbetween these groups, and between them and other groups at Chicago, such as the geometric Langlands group and the algebraic geometry group. The research funded by this grant is part of a web of research projects in progress at the University of Chicago.