This project aims to study endpoint maximal inequalities for families of convolution operators with smooth measures on manifolds under anisotropic dilation structures, making use of local smoothing estimates in the context of an advanced Calderon-Zygmund argument. The (properly scaled) convolution in three dimensions with the arc-length measure on the twisted cubic is the ultimate model case. In addition, LaVictoire and his collaborators hope to apply these methods to obtain new estimates for singular averaging operators with canonical folding relations in nilpotent groups, and also to complete a related project on nonstandard ergodic averages derived from sparse sets in discrete virtually nilpotent groups. Finally, LaVictoire plans to develop and test an innovative approach to supplementary mathematics instruction inspired by the website Project Euler.
Methods and techniques from the mathematics field of harmonic analysis have often led to important applications to the natural sciences and engineering. Harmonic analysis, in its broadest sense, concerns the study of functions by means of calculus-type integrals that lead to the discovery or recovery of important information which can be very hard to detect from the function's technical definition. This project looks at various processes that distort a given function (the data) in a geometrically patterned way similar e.g., to a photograph taken with a shaking camera, or to a deep-space image warped by a massive gravitational field. LaVictoire will investigate the important problem of when can the original data be satisfactorily reconstructed from the distorted data; such problems can be studied using recent advances in several areas of harmonic analysis.
