HOMOTOPICAL ALGEBRA: COALGEBRAS, DGAS, AND RATIONAL EQUIVARIANT SPECTRA

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The 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 with symmetries that allow complete calculations. The PI continues to train graduate students and disseminate research results. In addition, the PI is involved with several organizations that promote the participation of women and underrepresented minorities in math and science. 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. More specifically, the PI and Greenlees plan to extend their model for tori to provide an algebraic model for rational G-equivariant commutative and associative ring spectra. The PI and Kathryn Hess continue to develop the homotopical setting for coalgebras with motivations coming from Rognes' Hopf-Galois extensions of ring spectra and Hess' homotopical framework for descent. The PI and Birgit Richter are developing a simple algebraic model for homotopically commutative differential graded algebras.