The PI proposes to investigate the existence of homotopical dualizing modules in number theory and algebraic geometry and the implications thereof. Specifically, there will be two valuable instances of such dualizing modules. One will be studied in joint work with Tomer Schlank; it will give a homotopical extension of the arithmetic duality theorems of Tate-Poitou, which state that the cohomology of certain absolute Galois groups has a twisted form of self-duality. The PI and Tomer Schlank will also investigate applications of such duality results to problems of existence of rational points on algebraic varieties. A different example is related to the spectra of topological modular forms with or without level structures. Previous work of the PI suggests that the spectrally derived moduli stack of generalized elliptic curves has a simply describable dualizing sheaf of commutative ring spectra; an objective of the proposed project is to prove that result. Topological modular forms are crucial for understanding v2 periodic homotopy in the sphere spectrum; though somewhat removed from the theme of duality, the PI will collaborate with Mark Behrens, Kyle Ormsby, and Nathaniel Stapleton to compute the cooperations in the homology based on connective topological modular forms.
Duality is a pervasive concept in mathematics; the proposed project will study different types of duality in a unified framework, thereby arriving at novel and otherwise inaccessible information. In particular, the project is motivated by the idea that introducing a homotopical viewpoint can shed new light on our understanding of geometry and arithmetic. One of the objects with such curious self-duality properties, namely topological modular forms, lends itself to vast applications, some already explored and others not, as it mirrors itself in homotopy, algebraic geometry, number theory, and even quantum field theory as the receptacle of the Witten genus.
