The purpose of this REU was to introduce the participants to a variety of interrelated topics in analysis, topology, probability, and physics. The basic object of study was a "metrized graph," which is a finite graph G equipped with parametrizations of its edges. Such graphs serve as an accessible place to model general theories in higher dimensions, but also have many unique and interesting features in their own right.

In the first week we covered the theory of resistive circuits, and worked on an extended example involving the Peterson graph (which was not planned, it came out of a problem that some of the students posed to each other). This example elicited a nice discussion of symmetry. We also discussed some advanced topics in linear algebra. One or two days were spent talking about the connection between random walks and circuit theory. We introduced harmonic functions in this and also in other contexts. At the end of the first week, the students finally learned what a metrized graph is and began to learn about the Laplacian on such a thing.

In the second week, we got more into research-level topics: inverting the Laplacian, integral operators, eigenvalues of the Laplacian, the canonical measure, and the Tau constant. By the end of the second week, there were at least 5 open problems that the students could understand and begin to work on.

The students began actively researching these problems at the end of the second week and beginning of the third week, and continued working on problems through the end of the REU. The research activity included MAPLE programming, reading books and research papers, and working in groups on theoretical problems. One of the things accomplished during the REU was the formulation of the correct discrete analogue of the graph eigenvalue problem for an arbitrary measure. One of the main problems the students wound up investigating was the nature of the convergence of eigenvalues of in the discrete case to those in the continuous case. Another focus of research was finding a lower bound for the Tau constant. The students made a serious attack on this during the REU and found examples of graphs with Tau much smaller than any previously known. On the last day, the students made presentations about their work to one another (and some invited guests).

It was clear that each of the students had accomplished something nontrivial, and overall it seems that the program was a great success.