Research



I am an Associate Professor in the Mathematics Department at the University of Houston and a member of the Analysis Research Group at UH. I have supervised a number of undergraduate students, 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 two subjects.

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.



Survey Articles



Operator Algebra Resources



Past Conferences and Workshops with Online Notes, Papers, and Lectures





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