Ideas for Undergraduate Research Projects

INFORMAL DEFINITIONS


Directed Graph: A directed graph is a set of dots (called vertices) together with a set of arrows (called edges) each of which points from one vertex to another. We impose no restrictions on our directed graphs -- in particular, there may be multiple edges between vertices, and the set of vertices or set of edges may be infinite.

C*-algebra: If H is a (possibly infinite-dimensional) Hilbert space, then a C*-algebra is an algebra of continuous linear operators on H that is norm closed and closed under taking adjoints. This definition is sufficiently general to contain n x n matrices, as well as C(X) for any compact Hausdorff space X, and many other familiar objects.

GRAPH C*-ALGEBRAS


If G is a directed graph it is possible to create a C*-algebra C*(G) which is generated by a collection of elements that satisfy certain relations determined by G. These graph C*-algebras include many common C*-algebras, and consequently there is a great deal of interest in them. Despite the fact that graph C*-algebras include such a wide class of C*-algebras, their basic structure is fairly well understood and their invariants are readily computable. As mentioned before, the graph provides a convenient presentation of the relations that the generators satisfy. More surprisingly, however, the graph also reflects many of the properties of the associated C*-algebra. Thus C*-algebraic properties of a graph algebra may be translated into properties of the corresponding graph -- for example, the C*-algebra associated to a graph is an AF-algebra (i.e. close to a finite-dimensional algebra) if and only if the graph contains no cycles. This correspondence between C*-algebraic properties and graph theoretic properties results in a beautiful and elegant theory in which the graph provides a tool for visualizing characteristics of the associated C*-algebra.

Much of my work involves taking a C*-algebraic question for C*(G) and translating it into an equivalent question about the graph G. Often, the resulting graph question is easy to answer, thereby giving a solution to the C*-algebra question. Sometimes, however, the graph question is not so easily answered, or there are questions as to what kind of behavior a directed graph can exhibit.

(If you'd like, you can read a more technical description of my work -- but keep in mind that you don't have to understand the details in order to get started on a project with me.)

PROJECTS FOR UNDERGRADUATES


I have a number of questions about directed graphs that I am interested in answering. Depending on the background and interest of the student, there are a few possibilities of what to do:
  1. I have some purely graph theoretic problems suitable for students. Since these problems are motivated by C*-algebra considerations, many of them involve properties of directed graphs that others have not looked at before. Consequently, many of them are open problems. In addition, since these projects involve directed graphs, there is very little background required -- the problems can often be explained fairly quickly and students can begin working on them right away. (In particular, although solutions to the problems may have application to C*-algebras, absolutely no knowledge of C*-algebras is necessary to work on them.)

  2. For students interested in learning about Functional Analysis and C*-algebras, the topic of graph algebras can serve as a tool for introducing these topics. The correspondence described above allows one to use directed graphs to "visualize" abstract properties of certain C*-algebras, and therefore they can serve as motivating examples for more general study. In addition, a student could begin to work on some of the problems involved with translating C*-algebraic properties into graph properties.


If you are interested in working on any of the above projects, or if you would like to talk about other projects on which you would like to work, please contact me so that we can set up a meeting to discuss possibilities.