Ring Spectra, DGAs and Derived Equivalences

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

Abstract Award: DMS-0706877 Principal Investigator: Brooke E. Shipley The PI proposes to study algebraic derived equivalences using methods from homotopy theory. Specifically, the main project proposed here is to consider when an equivalence between derived categories can be realized by an underlying richly structured equivalence. The PI and Dugger have shown that certain derived equivalences of differential graded algebras (DGAs) can only be realized by "topological equivalences" (involving tilting spectra) which arise by considering DGAs in the broader context of ring spectra. One goal of this project is to produce simple invariants for detecting topological equivalences of DGAs. Another goal is to extend the study of topological equivalences to DG-categories. In on-going joint work with Dugger, the PI is also investigating an exotic example of two derived equivalent DGAs with no underlying structured equivalence. The proposed research projects involve the interplay beween the study of algebraic structures and topology, the study of shapes or spaces. Algebraic topologists use algebraic structures to describe and simplify topological phenomena. In a project on realizing derived equivalences, the investigator hopes to use techniques developed in algebraic topology to attack questions which originate in algebra. The PI also plans to organize conferences which will help to train graduate students as well as disseminate results. In addition, the PI will continue to be involved with several organizations which promote the participation of women in science.