Homotopy Theory, Geometry, and Arithmetic

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Algebraic topology arose out of development of algebraic methods to answer concrete questions about geometric shapes. One of the most successful tools originally developed is homology theory, which gives qualitative information about shapes that remains unchanged if the shapes are moved, twisted, or stretched. Several decades were dedicated to understanding homology and producing a sequence of new, more general tools that shared similar properties but detected different types of information. Many new and powerful tools were developed; in addition, a surprising connection was discovered with number theory. This research focuses on expanding on our understanding of this connection by lifting recent developments from number theory to topology, and applying this new machinery of algebraic topology to concrete problems about geometric shapes such as knots and links. This research project extends work of the investigator and collaborators in constructing structured ring spectra realizing connections to specific moduli of abelian varieties. These phenomena and their impact on the chromatic filtration in stable homotopy theory have been examined up through chromatic height two. The project aims to produce new information at chromatic height three through the study of Picard modular surfaces and to apply these results to detecting higher periodic phenomena in the stable homotopy category. Further, the project will study positivity phenomena in the theories of modular and automorphic forms and their connection to the additive locus in chromatic homotopy theory. The work also aims to extend recent results in constructing Khovanov spectra associated to knots and to show Khovanov's work on complexes associated to tangles lifts to a theory of stable homotopy types associated to tangles. Finally, the project will apply work in stable equivariant homotopy theory, in particular the equivariant Tate diagonal, to realize homology-level constructions in Heegaard-Floer homology.