"Groups, Graphs, and Geometry," hosted at the University of North Carolina, Asheville, provides eight talented college math majors with the opportunity to engage in original research in mathematics. Faculty possessing substantial experience in mentoring undergraduate research projects lead this summer REU program. Students focus on problems which lie at the intersection of several related fields of mathematics (group theory, graph theory, and geometry), and a number of the projects have practical implications as well as theoretical ones. For instance, several of the students in past years of the program have investigated models for communication networks, relationship networks, and other dynamically changing structures.
The program does more than give participating students an authentic research experience. It also trains them in professional skills such as library research, abstract-writing, and technical communication. Furthermore, through their participation in the program, the students are ushered into the mathematical community as they meet and consult with top researchers in the program's focal fields and are given support and encouragement to attend and present at regional and national conferences. By these means the students are set on a path leading to successful careers in mathematics.
," held at the Unviersity of North Carolina, Asheville during each summer from 2007 to 2012, provided eight students each year an opportunity to further our understanding of various mathematical objects and entities. The focus of these students' study was the structure of graphs, mathematical models for networks that are used in a number of fields and applications ranging from chemistry and physics (in investigating bonds strengths in molecular structures, for instance) to computer science. Graphs consist of vertices (or nodes) that represent objects joined to one another by edges that model connections between the objects the nodes represent. Much of the REU's student participants worked on problems related to independence, the presence of cycles, and connectivity. A set of nodes in a graph is said to be independent if no two of them are connected by an edge. Independent sets therefore represent collections of nodes that are "well-distributed" throughout the network the graph models. Students in the REU were able to discover much about the structure of independent sets of nodes in a variety of families of graphs, answering questions such as how many independent sets are there in a given graph? How large are such sets? Are there useful patterns in the numbers of independent sets of various sizes? A cycle in a graph is a collection of edges that creates a "loop," beginning at a single node, tracing a path made up of consecutive edges from one node to the next and so forth, ending up at the node where the cycle began. In a sense the presence of cycles guarantees a measure of redundancy or integrity in the network a graph models, for if there is a cycle that contains two nodes in it, there is always more than one way to reach one of the nodes from the other. Students in the REU were able to further our understanding of the presence of cycles, in particular developing conditions on a graph that guarntee the presence of cycles of various lengths, provided the graph has sufficiently many edges. Somewhat related is the notion of a graph's connectivity. Connectivity is a technical measure of the "integrity" of a graph, found by counting the number of different paths one can take between any two nodes in the graph. Several of the student participants worked closely with a variation on this theme known as asymptotic connectivity, a measure of the integrity of infinite graphs. Over the lifetime of the REU, several students worked to determine the asymptotic connectivity of a wide variety of families of graphs, many of which have very interesting geometric structure. Some students worked on other problems related to graph theory, including topics in the areas of number theory, combinatorial and geometric group theory, and Ramsey theory. Several publications resulted from the students' work, and nearly every student presented their work at regional and national mathematics conferences, including various regional meetings of the Mathematical Association of America and the last several Joint Mathematics Meetings, held annually in January. Students were given substantive instruction in communicating mathematics both orally and in writing, and were given other professional development (e.g. grad school preparation) as well. During recruitment, care was taken to attract members of groups traditionally underrepresented in mathematics. Student participants included Asian-American students, Latino students, and students of nontraditional college age. Additionally, 50% of the student participants were women; this percentage is much higher than that of most similar programs.
