I work in functional analysis and operator theory and like to joke that I do glorified linear algebra. (But don't we all?)
Recently, I've been interested in various incarnations of the shift and backward shift operators. I wrote a blog post about the classical setting a while back, but the idea is that in one variable, multiplying by z "shifts" the coefficients of a power series, and you can "shift them back" but this lops off your constant term.
I've done work with polynomial approximation that was originally motivated by questions about cyclicity under the shift operator; these are the optimal approximants problems, and I've been exploring them in several variable contexts, both commutative and noncommutative. These also make for nice undergraduate research projects because the basic idea can be explained to students with a basic understanding of Taylor series and orthogonal projection, and there are lots of computational projects that can be done.
I've also been looking at shift (and backward shift) invariant subspaces which get strange in several variables since you have different shift operators for each independent variable. You also have different reasonable choices of domain (the ball vs the bidisk, for example), and there's also a whole world of noncommutative questions, so I'm sure I'll be busy for a long time!
Work in Progress
Noncommutative optimal polynomial approximants with Palak Arora, Meric Augat, and Michael Jury
Compressions of two variable shift operators with Kelly Bickel, Katie Quetermous, and Matina Trachana
Dirichlet series optimal approximants with Alan Sola
Noncommutative de Branges-Rovnyak models with Robert T. W. Martin