IMPA - O Instituto de Matemática Pura e Aplicada

The Weil conjectures predict that the zeta function of a non-singular variety over a finite field can be written as a rational function in one variable. To find such a rational function is a very hard problem. In this talk, I will explain how to find it when the variety is a Fermat variety (and more generally a diagonal variety). This will be done via the classical argument using Gauss and Jacobi sums.

The classical Lie functor identifies Lie algebras as the infinitesimal objects underlying Lie groups, and although it has been naturally extended to Lie groupoids and Lie algebroids, a differentiation functor in higher Lie theory has remained elusive. In recent joint work with A. Cabrera, we develop a geometry-driven solution to this problem which admits a clean algebraic formulation in terms of cosimplicial and differential graded algebras. In this talk, I will review the classical Dold–Kan correspondence, discuss explicit formulas for the monoidal denormalization functor, and describe its left adjoint, which can be interpreted as an algebraic differentiation functor. Finally, I will relate this construction to Beilinson’s small algebras, and explain how it resolves the representability problem proposed by Ševera.

Warping-based geoacoustic inversion techniques based on time-frequency dispersion analysis is discussed. The dispersion of acoustic waves in a shallow-water environment with a two-layer bottom (gradient layer over a uniform basement) is investigated. This model leads to an analytical solution for eigenfunctions, and its versatility allows many different sound velocity profiles to be considered. Ordinary differential equations describing the dependence of horizontal wavenumbers of normal modes on the geoacoustic parameters of both layers are derived. It is shown that integration of these equations allows one to model dispersion curves for every point of a gridded multi-dimensional parameter space for a shallow-water waveguide model at very low computational cost. This advancement has many potential applications, including new ways to perform Bayesian single-hydrophone geoacoustic inversion by highly-efficient maximization of the posterior probability density over the corresponding space of geoacoustic parameters. This approach is demonstrated by performing geoacoustic inversion of measured modal-dispersion data collected on the New England Mud Patch.