CAREER: Decomposition, duality and Picard groups in chromatic homotopy theory

This CAREER award supports research in algebraic topology: an area of mathematics which studies geometric objects through the lens of algebra. Namely, it classifies geometric shapes by assigning to them various algebraic invariants, such as numbers. These tools are very powerful for studying geometric objects in spaces of larger dimensions that we can not easily picture, for example, the high dimensional analogues of spheres. The PI's work involves using and developing tools from algebraic topology for describing phenomena that arise when we consider continuous maps between large dimensional spheres. The educational component of the project is centered around creating a series of annual conferences with the goal of building a topology community in the South Central region and providing mentoring and networking opportunities for junior mathematicians from this region. The PI will also develop and run an enrichment program in mathematics for high school students and lead a research team in a Topology workshop. Chromatic homotopy theory is a conceptual and computational framework for understanding the stable homotopy category through the Lubin–Tate theory of deformations of formal group laws. It is a key tool for stable homotopy theory, both for calculations and for organizing the search for large scale phenomena. The PI will investigate several projects related to duality and invertibility in chromatic homotopy theory. More specifically, the PI will study Spanier-Whitehead and Gross-Hopkins duality in this setting, particularly as applied to the spectrum of topological modular forms and other important spectra in the K(2)-local category. The second part of the project focuses on invertibility phenomena which are closely intertwined with duality. The PI will investigate the Picard groups of K(n)-local categories of particular interest. 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.