My research interests are in approximation problems in different disciplines such as complex analysis, operator theory, harmonic analysis, and summability theory.
My first set of projects is in approximation problems in Banach spaces of functions such as Reproducing Kernel Hilbert Spaces (RKHS), Reproducing Kernel Banach Spaces (RKBS), and spaces of holomorphic functions (Hardy spaces, de Branges-Rovnyak spaces, Bergman spaces, and weighted Dirichlet spaces with superharmonic weights) using summability theory. I got interested recently in problems related to optimal polynomial approximants and I am currently giving a reading course to two PhD students from the University of Hawaii at Manoa on this topic.
I am also interested in problems in geometric function theory and dynamical systems. I am currently interested in rational lemniscates and the problem of characterizing those Jordan curves that are pre-images of the unit circle under a rational map. Rational lemniscates have shown to be really useful in approximation of shapes in the complex plane and their study goes back to the 18-th problem of Hilbert on the density of polynomial lemniscates in the family of Jordan curves.
I am also interested in holomorphic dynamics and its generalization to higher dimensions using hypercomplex structures. In a joint work with N. Doyon and W. Verreault, we obtained a complete characterization of the involutions of the set of multicomplex numbers. In a future work, I intend to apply this result to multicomplex analysis and to the multicomplex version of the Mandelbrot set.