CAREER: Symmetry in Geometry, Topology, and Algebra

Symmetry is a fundamental property of physical and mathematical objects. This project will investigate two forms of symmetry in mathematics: reflectional symmetry (e.g., flipping a circle across a diameter) and rotational symmetry (e.g., rotating a circle around its center). The PI will use modern techniques from algebraic topology to classify the rotational symmetries of certain high-dimensional geometric objects and understand their behavior under reflectional symmetries. In another direction, the PI will study how certain algebraic and geometric objects with reflectional symmetry appear in different areas of mathematics. The PI will also develop and study the effects of community-engaged pedagogy in undergraduate math courses, with an emphasis on K-12 education and outreach, continue co-organizing the Electronic Computational Homotopy Theory Online Research Community, and organize two regional workshops and a regional conference in the Mid-Atlantic. Specific research projects include the study of the stable homotopy groups of spheres using topological modular forms and applications of these results to problems in geometric topology and Riemannian geometry; namely, the classification of exotic spheres, the detection of their rotational symmetries, and the study of lower curvature bounds on their Ricci curvature. In another direction, the PI will continue developing the theory of homological stability for sequences of topological spaces equipped with group actions, with the aim of developing a general framework for passing results from the non-equivariant to the equivariant setting. Finally, the PI will continue investigating real algebraic K-theory, the K-theory of rings and ring spectra equipped with anti-involutions, using trace methods, with applications in stable homotopy theory and arithmetic geometry. 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.