I am a pure mathematician working in the area of
Functional Analysis,
particularly the study of
C*-algebras (pronounced
"C-star algebras"), which are algebras of
bounded
operators on Hilbert space.
C*-algebras were first introduced in 1943 to to provide algebraic
formulations of quantum
mechanics, and although they are still used by
theoretical physicists in
quantum field theory,
C*-algebras have also found important applications to many areas of mathematics
outside of Functional Analysis,
including such diverse fields as
Differential Geometry,
Knot Theory,
Algebra,
and the study of Dynamical
Systems. Indeed, in the past decades the
Fields
Medal --- often referred to as the "Nobel Prize" of mathematics --- has twice been
awarded to mathematicians for work involving C*-algebras: once in 1982 to
Alain Connes, and later in 1990 to
Vaughan Jones.
A large portion of my research has involved the study of
graph C*-algebras
and Cuntz-Pimsner algebras. Graph C*-algebras are C*-algebras constructed
from directed graphs
by a method that generalizes the construction of the
Cuntz
algebras and Cuntz-Krieger algebras. The graph C*-algebras
are especially nice for a number of reasons: (1) the class of graph C*-algebras
contains many interesting examples, including
several other widely studied classes of C*-algebras, (2) graph C*-algebras are more
manageable than general C*-algebras and the graph
provides a visual tool aiding in their study, and (3) well-known C*-algebraic invariants
of graph C*-algebra can be readily computed in terms of
information coming from the graph.
Much of my work involves the study of C*-algebras constructed from discrete and
dynamical structures. This typically involves building a C*-algebra from some
object (e.g., a
graph,
matrix,
bimodule, or
shift space), and
then using that object to
study properties of the associated C*-algebra. In these investigations
the following kinds of questions arise naturally:
Despite
the advantages provided by the graph, there do exist many examples of interesting
C*-algebras that are not graph C*-algebras. As a result, efforts have been made
to investigate larger classes of C*-algebras and determine whether
graph C*-algebras techniques can be generalized to these settings.
The Cuntz-Pimsner algebras are one such class. A Cuntz-Pimsner algebra is constructed
from a certain kind of bimodule
known as a C*-correspondence. The class of
Cuntz-Pimsner algebras is vast --- in fact, every C*-algebra may be realized as
a Cuntz-Pimsner algebra. However, despite the enormity of the class of
Cuntz-Pimsner algebras, the special case of graph C*-algebras provides important and
useful insights into the structure of general Cuntz-Pimsner algebras. In fact, it has
been discovered that a number of graph C*-algebra results can be extended to
Cuntz-Pimsner algebras, thereby allowing one to study Cuntz-Pimsner algebras in
terms of the bimodules from which they are constructed.
Although my work is rooted in Functional Analysis, the Analysis questions I investigate have
important connections with topics in Algebra. Consequently, I am intrigued by algebraic
phenomena related to analysis, and I have also done research that solely involves
Algebra, as well as research that examines the interactions between topics in
Algebra and Analysis.
More specifically, 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 graph C*-algebra. 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.
For more details on my work, you may wish to view my research publications.