2025 Iowa State REU Projects
2025 Iowa State REU Projects
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.
Project mentor: Zhijun Wu
This project is to use the evolutionary game theory to study a critical language dynamic problem emerging worldwide in recent decades where multiple languages compete in societies with many of them becoming extinct as a result. In this project, the students will learn basic theory and methods for game dynamic analysis, and use these knowledges to build a game theoretic model for language competition. The basic idea here is to consider language competition as a population game where every individual decides how to use the available languages for his/her best social interests and collectively, the whole population reaches an equilibrium state with a “balanced” use of multiple languages. The project will use the game theoretic model to investigate the population threshold for language growth or decline; the effect of society intervention for language shift; and the importance of maintaining a language group for keeping a language. A Matlab code is to be implemented to simulate the language games in small-world social networks as well.
Prerequisites: Knowledge of linear algebra, differential equations, and Matlab.