Hope College will host an eight week REU program for six undergraduates per year in mathematics. Pairs of students will work with one of three faculty members. Projects will be chosen from topics in algebra, probability and analysis.
Primary goals are to help talented and motivated students develop as mathematical researchers, to promote mathematics research as a career, and to have the participants achieve significant mathematical results in partnership with a faculty mentor. In particular, students will 1) learn new mathematics, 2) gain mathematical independence by learning to read math texts, journals, and research papers, 3) solve an original mathematical problem, 4) learn to communicate mathematics verbally, and 5) write a mathematical paper suitable for submission to a journal.
The three student pairs will present their problems and results to each other informally throughout the summer at weekly seminars. Students and faculty mentors will give feedback to presenters on how to more effectively communicate mathematical ideas in both formal and informal settings. Towards the end of the program students will write and formally present papers at a conference held in conjunction with other REU programs.
This project brought six students each summer (during summers 2008 through 2012) to Hope College for 8 weeks to collaborate with faculty research mentors on open questions in mathematics. Students were selected through a national application process based on their stated interest in one of our research projects and on their overall mathematical background and ability. Typically in teams of two or three, students were involved in learning mathematical background relevant to their research project, working alongside their faculty mentor and student colleagues on various open questions in mathematics, and preparing talks and research articles based on their results. Topics of the research projects included Representation Theory (3 projects), Graph Theory (4 projects), Dynamical Systems, Geometric Spirals (2 projects), Complex Analysis (2 projects), Modeling Sand Dune Formation (2 projects), Modeling Patterns in Arab Spring, Modeling Chimney Swift Roosting, and Modeling Insect Development in Climate Change. In addition to journal articles that have appeared or are still in progress, these projects have resulted in over 25 talks at mathematics conferences including MathFest, the Annual Michigan Sectional MAA Meeting, the Michigan Undergraduate Mathematics Conference, the Summer Undergraduate Michigan Mathematics Research Conference, the Nebraska Conference for Undergraduate Women in Mathematics, the Young Mathematicians Conference, the Midstates Consortium Undergraduate Research Symposium, and the Ohio State Undergraduate Research Conference. The Representation Theory produced a new geometric method for describing a complete set of irreducible representations of metacyclic groups and wreath products of cyclic groups, and later gave a geometric way of constructing a correspondence between conjugacy classes of certain groups and their irreducible representations. The Graph Theory research groups gave new theorems relating to the winnability of the Lights Out puzzle for certain families of graphs and new threshold information for the pebbling problem on certain families of graphs, including graphs of diameter 2. The Dynamical Systems group came up with necessary and sufficient conditions for a decreasing function on the unit interval to have the shadowing property. The Geometric Spirals group gave new structural theorems for piecewise-linear spirals determined by recursively-defined sequences of points in the plane and in space, including the total length and the point of convergence of such spirals. The Complex Analysis group considered two conjectures and achieved results in each case under additional hypotheses. One conjecture relates the degree of a polynomial to its number of points of extreme curvature, and the second relates to the monotonicity of the number of non-real zeros of higher-order derivatives of an entire function. The Mathematical Modeling groups all came up with new models for their given situation and worked to estimate various model parameters. The project on sand dune dynamics also involved geology and biology students and faculty, and the project on insect development time also involved biology students and faculty.
