# Undergraduate Research Opportunities

Many of our mathematics faculty frequently work with undergraduate students in research projects either at Linfield during the summer (often funded by a Linfield Student-Faculty Collaborative Research Grant) or at an REU. Here we highlight some of these ongoing research projects.

## Competitive Graph Coloring

Competitive Graph Coloring is an area of graph theory concerned with the following two-person game. Two players, Alice and Bob, alternate coloring the vertices a finite graph from a fixed finite set of r colors. On each turn, players must color such that adjacent vertices receive different colors. Alice, the first player, wins the game if the entire graph is eventually colored. Otherwise, Bob wins when there comes a time in the game when there is an uncolored vertex for which no legal color exists. The least number of colors such that Alice has a winning strategy is called the game chromatic number of the graph. The goal of this work is to explore this parameter on different classes of graphs and to examine how this parameter relates to other properties and parameters defined on graphs. This particular area of research has been quite productive and has included a great number of undergraduate collaborators. In particular, between the two of them, Professor Dunn and Professor Nordstrom have mentored over twenty-five students, both from Linfield and from other institutions through an NSF-funded REU (Research Experience for Undergraduates) and with funding from the Linfield Student-Faculty Research Grant program. At present, these projects have resulted in three published papers, and an additional book chapter soon to appear. These publications include fourteen undergraduate co-authors.

## Mathematical Modeling via Differential Equations

Over the past twenty years, a number of Linfield students have worked with our mathematics faculty on mathematical modeling problems. Some of the most recent projects (summer 2013 and summer 2014) in this area have been mentored by Professor Bricher, where he has worked with Linfield students on a variety of problems including: using nonlinear partial differential equations to model physical systems such as the spatial spread of a favored gene in a population and using nonlinear differential equations to describe the population dynamics of a terrorist organization. Both of the projects mentioned above were funded by Linfield Student-Faculty Collaborative Research Grants. In addition, the students working on these projects have presented their work at both regional and national mathematics conferences.

## The Mathematics of Tiling

Since 2008 Professor Hitchman, who first encountered the charm of tiling questions in his own capstone undergraduate project, has mentored eight undergrads in four student-inspired research projects in the mathematics of tiling. This work, some of which has been funded by Linfield College Student-Faculty Collaborative Research Grants, and some of which has been funded by the National Science Foundation, has resulted in at least six regional and national presentations by students, and led directly to four publications. Our research has focused on tile invariants, linear combinations among the number of copies of each tile that must persist in any tiling of a region. A tiler’s toolbox draws on content from combinatorics, number theory, group theory, and topology. Given their hands-on nature, their variety, and their relevance to the undergraduate curriculum, tiling questions continue to be a lively subject for undergraduate mathematics research. Find out more here.