Here is where you can find slides and/or recordings from some of my most recent talks.
PIMS Emergent Research: Shift operators and their adjoints in the context of optimal approximants
Abstract: I will give a very broad overview discussing various uses and generalizations of the shift operator (and its adjoint). In the classical case we consider the Hardy space of analytic functions on the complex disk with square summable Taylor coefficients. The shift operator is simply multiplication by z and this "shifts" the coefficients of the function. The backward shift does the opposite, and in the case of the Hardy space, it's actually the adjoint of the shift. (This doesn't happen in every function space!) There are many classical results about subspaces that are invariant under the shift or its adjoint and connecting these to functions and operators. I'll discuss some of the generalizations of the shift operators and some of my recent and current projects and how they connect to the classical theory.
OTTER: A beginner's view of noncommutative realizations
OTTER (Operatory Theory Talks for Early Researchers) is a seminar I help organize!
Abstract: Noncommutative analysis is a topic of interest for lots of us in the operator theory community, and realizations are an important and useful tool. However, many of the references are quite abstract, and not geared towards beginners. In this talk, I hope to give an introduction to the topic of noncommutative rational functions and realizations not in full generality, but with some detailed computations and examples.
FUNctional analysis! Optimal approximants and orthogonal polynomials
Talk Math With Your Friends (TMWYF) is a virtual mathematics colloquium. This talk was geared to be accessible to undergraduates and fun for everybody!
Abstract: Optimal polynomial approximants are the "best" way to approximate functions that might not be "nice." I think they're a good introduction to some of the ideas in functional analysis, including what I mean when I say "best" and "nice." I think they also provide a fertile ground for undergraduate research projects!
The geogebra applet I use in the talk: https://www.geogebra.org/m/mqajpefa
Optimal approximants and orthogonal polynomials in several variables (OTWIA 2020)
Abstract: In recent years, optimal polynomial approximants have been used to study cyclicity of functions in Dirichlet-type spaces on the complex unit disk, with particular interest being paid to the connection between the location of the zero sets of the optimal approximants and cyclicity, as well as a correspondence between optimal approximants and orthogonal polynomials. In this talk we discuss generalizing the concept of optimal approximants to several variables, including to the cases of Dirichlet-type spaces on the bidisk, and to a scale of Drury-Arveson-type spaces on the ball, as well as some of the inherent complications.