Homotopical Coalgebras, Algebraic Models, and Realizing Derived Equivalences

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

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, with co-authors, the PI will develop algebraic models for several different topological settings in spectra. 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 mathematics and science. In the first broad project the PI will continue her work with various coauthors on developing homotopical settings for comodules and coalgebras. The motivations for this project include applications to algebraic K-theory, connections with chromatic homotopy theory and string theory, and development of computational tools. In a second broad project, the PI continues to study when derived equivalences can be realized by underlying richly structured equivalences. This includes recognizing some homotopy categories as rigid, or having a unique (up to equivalence) underlying homotopy theory (as has been shown, for example, for rational circle-equivariant homotopy theory) and also recognizing some homotopy categories as being modeled by non-equivalent homotopy theories. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.