This REU is an eight-week summer research experience in mathematics to be held at SUNY Potsdam for 14 undergraduates mathematics students (9 supported by the NSF site grant) working in groups of 3 or 4. The areas of the research for 2010 include Applied Graph Theory, Functional Analysis, Real Analysis and Topological Graph Theory. The student participants will be treated as colleagues by the faculty advisors as they jointly pursue unanswered mathematical questions. We have ambitious goals for our students. We would like to help them build their confidence in their ability to do research independently, and to help them build their appreciation and understanding of the vast field of mathematics. We also want to help them to improve their oral and written communication skills. Each group will present their results to the other students, at least twice during the eight weeks. Students will be encouraged to present their results at a national meeting. Each group will also use LaTex to prepare a paper on their work. We also hope to expand our students' mathematical horizons through hosting a weekly guest speaker.
SUNY Potsdam is widely known for its undergraduate mathematics program, and Clarkson University has an international reputation in scientific research. Together, these institutions create for students a unique atmosphere of learning and research. We strongly encourage applications from women, minorities, students with disabilities, and other underrepresented groups.
For three summers, 2010-2012, we ran an eight-week summer Research Experience for Undergraduates program in mathematics at SUNY Potsdam, with the cooperation of Clarkson University. We mentored 42 students (40 undergraduates) in eleven different projects in algebraic graph theory (3), analysis (3), topological graph/knot theory (2), algebra (1), card shuffling (1) and stochastic differential equations (1). Much of the research was basic theoretical mathematics, though the topics studied have applications to many areas, including quantum computing (applied graph theory), population modeling (stochastic differential equations), and chemistry (topological graph theory). During the program, students presented their findings to their peers in mid-program and in end of program talks. At least 18 of the students presented their work at national conferences, including Mathfest, the Joint Mathematics Meetings, the Young Mathematicians Conference, the Council on Undergraduate Research, the Nebraska Conference for Women in Mathematics and the Undergraduate Knot Theory Conference. They also wrote up their work in a final report. Many of these reports have evolved or will evolve into published articles. In particular, three papers have been published in peer-reviewed journals. Two have been submitted for publication, and at least two more papers will soon be submitted for publication. Of the 42 students involved, thirteen are now in graduate programs in mathematics. Four are teachers or will soon be. One is a Wall Street trader, two are unknown, and the rest are competing their undergraduate studies. We served a variety of students from throughout the United States, as well as four students from our partner institution in Mexico, the Unversidad Autonoma del Estado de Hidalgo. To give the reader a flavor of some of the work we did, we include a more detailed description of one of our thirteen projects: Joel Foisy, topological graph theory (2012): We studied planar and spatial embeddings of graphs. For us, a graph is a collection of vertices (think of them as points) and edges that connect them. Planar graphs can be drawn in the plane with no crossings. A spatial embedding of a graph is a way to place the graph in 3-dimensional space so that edges meet only at vertices. Planar graphs are useful models of circuits on a board, and spatial graphs can serve as models of molecules. We call a graph flat if it can be embedded in space such that every cycle C bounds a disk that intersects the graph only along C. Roberston, Seymour and Thomas proved that a graph is flat if and only if it has a linkless embedding (every pair of disjoint circles in the graph can be pulled apart). In many ways, the property of being flat is a 3-dimensional analog of a graph being planar. Robertson, Seymour and Thomas further proved "Sachs' Linkless Embedding Conjecture:" and characterized all intrinsically linked (nonflat) graphs. One problem we studied was an attempt to characterize flat permutation graphs. Chartrand and Harary introduced the notion of permutation graph. Given a graph G, a permutation graph is the graph obtained by taking the disjoint union of two copies of G, and connecting a vertex from one copy to the vertex in the second copy by an edge, where vertices are connected if they are in a given one-to-one correspondence (permuation). Chartrand and Harary established that the permutation graph of a 2-connected outer planar graph is planar if and only if the permutation is in the symmetry group of the outer cycle. We looked at when the permutation graph of a given graph has a linkless (flat) embedding. It is relatively easy to show that the permutation graph of a non-planar graph is intrinsically linked. We established several results about permutation graphs, including: (a) The smallest n such that the path of n vertices has an intrinsically linked permutation graph in 9. (b) A necessary condition for the permutation of a cycle to be intrinsically linked is for the permutation to contain at least 3 inversions. This necessary condition is not sufficient. (c) For a maximal planar graph, G, with 6 or more vertices, every permutation graph on G is intrinsically linked. One of the students is working on editing this work, which will soon be submitted for consideration for publication in a peer-reviewed journal.
