Chromatic Derived Algebraic Geometry and Equivariant Homotopy Theory

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Topology is about the qualitative study of (often) high-dimensional spaces. Algebraic topology uses algebraic invariants to this purposes. Classically these invariants were usually numbers. But more modern refinements use algebraic-analytic objects that contain more information about the spaces that we study. The refinements I am concerned with use modular and automorphic forms. An important early motivation for the use of modular forms in topology was Witen's work on analysis on loop spaces (i.e. spaces of closed loops or strings in spaces) as relevant in quantum field theory or string theory. My project investigates symmetries and dualities in topological modular forms and topological automorphic forms. This is, in particular, useful for understanding their relationship to other variants of them that are easier to compute.