Asymptotic Hodge Theory, Fibered Motives, and Algebraic Cycles

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The focus of this project is on polynomial equations and their solution sets, which have been studied specifically and intensively by algebraic geometers. Deep conjectures, including those of Bloch, Beilinson and Hodge, seek to relate the behavior of analytic objects (like integrals and differential equations), which are a priori non-algebraic, to the underlying algebraic structure and topological shape of such solution sets. Even as these conjectures remain intractable in general, solutions in individual cases both continue to bear out their validity and produce new algebraic structures (for example "cycles" and "motives") which facilitate the solutions of problems in apparently remote areas of mathematics and other sciences, for example, at the interface of number theory and physics (such as string theory and quantum field theory). Recent technical innovations, based on considering polynomial equations in families, have begun to provide access to new cases of these conjectures. Their further development and application is the subject of this project, whose results will be disseminated through conferences, summer schools, journal articles and websites. The project consultants brought to Washington University by the grant will contribute to its research atmosphere, and specialized problems embedded in the project will provide training for the PI's graduate students.Hodge-theoretic invariants such as period and regulator maps provide the basic interface between the algebraic and transcendental worlds in modern geometry. The goal of this project is to better understand the asymptotic properties of these invariants, and apply the results to closely intertwined problems of current interest in arithmetic geometry, physics, and algebraic geometry. Specifically, the PI plans to: (I) exhibit the Apery constants of Fano varieties as limits of higher normal functions (hence periods), and construct motives related to motivic Gamma functions to verify specific instances of conjectures of Beilinson and Green-Griffiths-Kerr; (II) compute the Feynman amplitudes associated to a family of two-loop graphs, and relate the spectra of quantum curves to zeroes and limits of normal functions (thereby confirming two consequences of a conjecture of Marino); and (III) use the mixed Hodge theory of miniversal deformations of singularities to interpret fibrations of geometric boundary components in moduli, and use Lie-theoretic methods to study local and global aspects of Hodge-theoretic compactifications of period maps.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.