Project mentor: Steve Butler

There will be two potential projects that will run (based on interest and availability of co-mentors). They are as follows:

• Spectral graph theory. Graphs look at how objects (called "vertices") connect together (via "edges"). Graphs are ubiquitous structures and have found extensive applications in both theoretical and practical setting. One way to understand graphs, particularly large graphs, is through the use of linear algebraic tools, primarily the eigenvalues of matrices associated with the graph. This project will start by looking at the "halfian" matrix which is a modification of the well studied Laplacian matrix and try to determine the spectrum of various families as well as properties that the matrix might possess.

• Parking functions. Consider the problem of parking cars on a one-way street where there are as many spots to park as their are cars and each car first drives to a preferred spot and starts looking for the first available open spot to park on or after that location. If all cars can park, then that set of preferences is known as a parking function. This project will look at counting various objects associated with parking functions, e.g. lucky spots.

Prerequisites: Proof-based math course; basic linear algebra course; mathematical curiosity; good communication skills and ability to work well in a group. Freshman and sophomores are encouraged to apply.
Students with advanced coursework (e.g. graduate level courses) are likely not to get as much benefit from this program; but may still apply.