Methods of algebraic geometry in algebraic topology

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This project aims to further the study of stable homotopy theory by methods and machinery developed in algebraic geometry. There are four main goals. The first goal is the study of specific examples of spectra of topological automorphic forms, focusing on computations for the Shimura curve of discriminant 15 and Picard modular surfaces at chromatic level 3. Second, joint work of Michael Hill and the PI aims to extend the theory of topological modular forms to the log-etale site of the moduli of elliptic curves, and in particular to functorially produce topological modular forms with level structure. Third, the PI hopes to further develop machinery employing Zink's theory of displays to produce highly structured multiplications on various spectra of importants in chromatic homotopy theory. Finally, joint work with David Gepner will study Picard and Brauer groups in the derived setting using recently-developed machinery in higher category theory.The main focus of this research field is to systematically study qualitative properties of shape by methods from algebra. In the past, these methods have led to surprising connections between many disparate fields of mathematics, and developments in these fields have translated into genuine information about geometric structures which is difficult to obtain otherwise. The goal of this research project is to take some of these most recent advances, taking place in subjects like higher category theory and algebraic geometry, and develop them into concrete tools that can advance our knowledge further.