REU Site in Algebra, Analysis, Geometry, Matrix Theory and Statistics

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The students in the REU Sites program at Central Michigan University will be presented with open questions in several areas of mathematics, including algebra, analysis, geometry, matrix theory, and statistics. Only a modest mathematical background at the sophomore level will be needed to understand research projects that will be investigated by teams of two or three students in collaboration with a faculty advisor. We provide a brief overview of several of the project topics. In the area of Combinatorial Matrix Theory, there is a natural way to associate graphs to symmetric or Hermitian matrices. The general problem is to determine the minimum rank among various classes of matrices associated with a given graph. In the area of Linear Algebra, we present a project called "Frames in Finite Dimensions." Frames are redundant spanning sets. The open question is to find a method to construct frames with desirable properties such as symmetry or sparsity or a given distribution of their angles and lengths. A project in geometry studies the relationship between a surface and its associated graph, as well as properties of its embedding into a projective space. A project in statistics aims to classify the hazard flexibility of parametric lifetime distributions based on total time on test transform curve and to rank the well known one- and two-shape parameter lifetime distributions. The students will learn mathematics beyond what is needed for their research work. For example, students will be exposed to some of the applications of frames. When a signal is transmitted over a channel it is possible that some information may be lost during transmission and the original signal cannot be recovered. The redundancy in frames makes it possible to "reconstruct" the original signal. In the geometry problem students will come to know the deep connection between Riemann surfaces, which are difficult to understand, and combinatorial objects such as graphs, which are easier to understand. Statistics projects will help students understand how to model data and how to estimate certain parameters. The intellectual merit of this REU site is based on the challenging mathematical problems students will attempt to solve. The students will tackle unsolved problems whose solutions (or partial solutions) will improve human understanding of the very structure of mathematics and its applications. Eleven faculty members will be active in the three-year program. Four faculty mentors will coordinate the effort each summer. Students will work in groups of two or three with one faculty mentor. Students will make presentations of their progress during weekly seminars. Each student will submit a written report at the end of the program. Students will be encouraged to publish their work and make presentations at professional meetings. The broader impact of the proposed REU site include training undergraduates in research in fundamental mathematics, increasing the participation of women and minorities in science and math activities, and involving undergraduates as young professionals in the broader disciplinary field.