Algebraic Structures in Equivariant Homotopy Theory

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Algebraic topology is an area of pure mathematics that approaches the study of complicated and usually high dimensional spaces via the use of algebraic invariants. It has connections to fields ranging from data analysis to physics. The PI's research specifically focuses on the algebra used to study spaces that have interesting symmetries. This type of structure has become increasingly important in recent developments in algebraic topology but is still comparatively poorly understood. The PI's work will advance state-of-the-art knowledge in this area of pure mathematics. Her research program is centered on developing the rich algebraic structures that arise in this context and using these to construct and analyze both classical and new algebro-topological deformation invariants of spaces with symmetries. Additionally, the funds from this grant will assist the PI in her outreach activities designed to promote women and underrepresented minorities in mathematics, including her work with the newly formed Vanderbilt Women in Math group.The PI plans to conduct research in equivariant stable homotopy theory. This area is primarily concerned with extracting features of spaces that are invariant under deformations and symmetries. Recent developments in homotopy theory have highlighted the importance of equivariance, as well as the many ways in which equivariance is remains poorly understood. Equivariant homotopy theory is key to modern computations in algebraic K-theory and in results in nonequivariant homotopy theory, such as the recent solution to the Kervaire invariant one problem. The PI's research program will develop new tools for extending these kinds of calculations as well as deepening our overall picture of the surprising ways in which symmetry manifests itself in homotopical considerations. A primary component of her work will be establishing methods of constructing cohomology theories with group actions from algebraic data in order to provide new and improved tools for further research in the field. Her work will also focus on developing algebraic structures inherent in equivariant cohomology theories with commutative ring structures. This work will provide calculational tools for use in a variety questions in homotopy theory. Additionally, the PI will investigate a new cohomology theory for coalgebras, including its connections to equivariant cohomology theories arising from algebras. The overall goals of these avenues of research are to advance state-of-the-art knowledge in homotopy theory and more concretely in group actions in a homotopy theoretic context.