I study operator algebras, a sub-field of functional analysis. Mostly I look at C*-algebras affiliated to directed graphs, groupoids, and related objects.
The unifying theme of my research is that interesting analytic and algebraic properties of such a C*-algebra are often controlled by relatively simple properties of the parent graph or groupoid.
I am interested in studying the state and tracial state structure of C*-algebras; I have also studied the representation theory of graph C*-algebras,
and their behavior when combined via tensor products. With Gabriel Nagy I've obtained effective simplicity criteria for groupoid C*-algebras as well as descriptions of the tracial state space of graph and groupoid C*-algebras.
Recently I've been interested in a set of ``largeness'' conditions for inclusions of C*-algebras arising from étale groupoids and groupoid crossed products, including the pure extension property and unique conditional expectation property.
I've adapted some ideas of Vrej Zarikian to the groupoid setting.
Recently I have studied problems involving pursuit games on undirected graphs, which have no connection to functional analysis. Natasha Komarov, John Mackey, and I
obtained upper bounds on the number of "cops" needed to contain a "robber" in a variant pursuit game.
Here is my research statement.
Accepted Papers
Simplicity of etale groupoid C*-algebras (with Gabriel Nagy). Journal of Operator Theory , Vol 83, Number 1 (2020). 95–138
Continuous-trace graph algebras. (Rocky Mountain Journal of Mathematics, Vol 48, Number 5.)
Traces arising from regular inclusions (with Gabriel Nagy. Journal of the Australian Mathematical Society , Vol 103, No 2)
Containment: A Variation of Cops and Robber (with Natasha Komarov and John Mackey). Accepted at Graphs and Combinatorics .
In preparation
Stability of C*-algebras associated to graphs, k-graphs, and groupoids
Large subalgebras arising from etale groupoid crossed products.
Slides from selected presentations
Montana Colloquium Slides, February 2020
JMM 2019 Slides
GPOTS 2018 slides ( handout )
Slides on Total Positivity and Determinants (SLU Math Open House)
GPOTS 2017 slides
Iowa Noncommutative Analysis Workshop 2016 slides
GPOTS 2016 slides
Miscellaneous notes/expository writing
Notes on enveloping algebras of Lie algebras
Undergraduate research by my students
REU writeup by A. Isvik, N. Kolo, M. Ross
REU Beamer slides (prepared by SuMaR REU Participants Amy Isvik , Nate Kolo , Maria Ross )