I am an Associate Professor in the Mathematics
Department at the University of
Houston and a member of the Analysis
Group at UH. I have supervised a number of
graduate students, and postdocs, and published a number of papers.
A Brief Description of the Areas in Which I Work
Although I work primarily in Functional Analysis, the work I do in Analysis has
important connections with topics in Algebra. I began my mathematical
career in Functional Analysis doing research related to
C*-algebras, which are algebras consisting of operators on Hilbert space.
In recent years my interests have broadened and, intrigued by algebraic
phenomena related to analysis, I now do research that examines topics in
both Functional Analysis and Algebra, as well as interactions between topics in the
Much of my work in Functional Analysis involves C*-algebras constructed
from discrete and dynamical structures. This involves building C*-algebras
from objects (e.g., graphs, matrices, bimodules, shift spaces) and then
using the object to study properties of the associated C*-algebra. In
these investigations one often considers the following questions: When can
a known C*-algebra be modeled as the C*-algebra of an
object? Can one translate well-known properties of the object into
properties of the associated C*-algebra? Can invariants for the constructed C*-algebra
be computed in terms of the object? Can the
objects be used to classify the associated C*-algebras?
At what level of generality can
one use these methods to model well-known classes of C*-algebras?
My work in Algebra has been motivated by the recent development of Leavitt
path algebras, which are algebras constructed from graphs in a manner similar
to how graph C*-algebras are constructed. These Leavitt
path algebras are intimately related to graph C*-algebras, and both the
C*-algebraic and algebraic theories have guided, influenced, and assisted
each other. Indeed, each theory has had nontrivial applications to the
other, and together they have given a deeper understanding of
certain classes of algebras and C*-algebras. Much of my work in this area involves
describing the structure of Leavitt path algebras, classifying Leavitt path algebras in terms of K-theory,
examining the relationship between Leavitt path
algebras and graph C*-algebras, and using Functional Analysis results to
motivate the creation of new algebras from other objects besides graphs.
Operator Algebra Resources
Past Conferences and Workshops with Online Notes, Papers, and Lectures
Applications of model theory to operator algebras
, July 31--August 4, 2017
Classification of C*-algebras, flow equivalence of shift spaces,
and graph and Leavitt path algebras.
, May 11--15, 2015
- Flow equivalence of graphs, shifts, and C*-algebras, Nov 18--22, 2013.
- Masterclass on Classification
of non-simple purely infinite C*-algebras at the University of Copenhagen in August, 2013.
- Graph Algebras: Bridges between graph C*-algebras and Leavitt path algebras,
held at the Banff International Research Station from April 21--26, 2013.
- Classification of amenable C*-algebras,
held at the Banff International Research Station from September 19--24, 2010
- Lecture Notes from the
Mathematical Institute of the Polish Academy of Sciences.