ALGEBRAIC MODELS OF HOMOTOPY THEORIES AND HOMOTOPICAL MODELS OF ALGEBRA

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The PI continues to study when derived equivalences can be realized by underlying richly structured equivalences. Examples of this arise in the PI's long term project with John Greenlees of constructing algebraic models for rational G-equivariant spectra for compact Lie groups G. The PI and Greenlees plan to extend their model for connected groups in the free case to non-connected groups. The PI and Greenlees also plan to extend their model for tori to provide an algebraic model for rational G-equivariant commutative and associative ring spectra. As a step towards the general case, the specific case of SO(3) is also being considered. The PI and Kathryn Hess propose to develop model categories for coalgebras over various comonads in various settings. The main motivation here is to provide a homotopical setting for studying Hopf-Galois extensions of ring spectra. The proposed research projects involve the interplay between the study of algebraic structures and topology, the study of shapes or spaces. Algebraic topologists use algebraic structures to describe and simplify topological phenomena. Spectra, which represent cohomology theories, are algebraic structures built out of topological spaces and hence are useful for translating from one field to the other. In one project, the PI and Greenlees develop algebraic models for certain types of spectra which allow complete calculations. The PI also continues to train graduate students and disseminate research results. In addition, the PI is involved with several organizations which promote the participation of women and underrepresented minorities in science.