For my dissertation project, I worked on an `L^{p}` version the Cuntz-Krieger-Pimsner algerbas. The goal was to construct a general class of `L^p` operator algebras motivated by Katsura’s construction for `C^{**}`- correspondences. As a motivation for all this, I started looking at representations of `C^{**}`- correspondences on pairs of Hilbert spaces which yielded natural definitions for modules over `L^p` operator algebras and more generally `L^p`-correspondences.
During my PhD, I also worked on the spectral synthesis problem for `F^p(\mathbb{Z})`, the `L^p` group algebra of the integers. It is known that spectral sythesis holds for `p=2` but fails when `p=1`. The problem is still open for `p \in (1,\infty) \setminus \{2\}`.
In 2020 I worked on a side project with Marcin Bownik. A potential goal of this project is to characterize shift-modulation invariant subspaces in the irrational case. The rational case was done by Marcin a while ago and the one-dimensional irrational case was done by Deguang Han (unpublished notes). I hope to eventually post my notes about this.