Title: Fourier analysis on the infinite-dimensional torus

Abstract:  We will describe the setup for a harmonic analysis of functions defined on the infinite-dimensional torus  T^w, which is the topological compact group consisting of the Cartesian product of countably infinite many copies of the one-dimensional torus, with its corresponding Haar measure. Several results will be presented:

1.  Absolutely divergent series, Calderón-Zygmund decomposition and maximal operators associated with differentiation of integrals.
2.  Mixed norm L^p(T^w) spaces, p=(p₁,p₂,...).
3. M. Riesz Theorems on T^w, rectangular partial sums and L^p convergence.

If time permits, open problems and other questions will be considered.

About the Speaker: Luz Roncal received her PhD in 2009 from University of  La Rioja under the direction of Óscar Ciaurri.  She had several positions including as a High School teacher from 2008-2010. She joined the Basque Center for Applied Mathematics in 2016 as BCAM researcher, then as a Research Affiliate and currently as an  Ikerbasque Research and Ramón y Cajal Fellow (a very prestigious award). Her  research concerns problems from Harmonic Analysis and Partial Differential Equations. Her current interests run along several directions: - Classical Harmonic Analysis dealing with singular integrals, square funtions, spherical averages, directional operators, and weighted versions of the latter. - Functional inequalities in several settings, such as the Heisenberg group or more general H-type groups and their applications, as well as their connections with the extension problem. - Discrete Harmonic Analysis and semidiscrete fractional equations, specially Carleman estimates and unique continuation principles for discrete and non-local equations. - Topics related to Fourier Analysis in the infinite torus. Very recently she got interested into the connections between Harmonic Analysis, PDEs, and Machine Learning in the broadest sense. Roncal has over 50 publications in excellent journals,  by herself and a number of coauthors including Carlos Pérez and Sundaram Thangavelu both of whom have been visitors at UNM in the past. She has supervized three PhD students and four postdoctoral researchers.