REU Site: USM Research Experience for Undergraduates in Computational and Applied Mathematics
2020 REU at USM (pending for approval)
Department of Mathematics
The
University of Southern Mississippi
Introduction 
The goals of this new REU site are to provide undergraduate researchers an encouraging experience in terms of successful research and a memorable summer. We will organize multiple scholarly activities to inspire undergraduate researchers. One of the important goals is to spark the undergraduate researchers' interest in mathematical research as early as possible through studentfaculty and studentstudent interaction. During the summer, the undergraduate researchers will generally accomplish a lot of learning and research in their projects as they work fulltime. If they complete a sufficient amount of work during the summer, they can opt to spend time during the subsequent academic semesters conducting additional research and polishing up a paper for journal submission. 
How to apply 

Project 1: Super Central Configurations of the nbody Problem with the General Homogenous potential 
A central configuration plays an essential role in understanding the global structure of solutions of the nbody problem. The question on the number of central configurations for a given mass vector is one of challenge problems for 21st century's mathematicians. It is well known that for any given three positive masses, there are three collinear central configurations. Three bodies on the vertices of an equilateral triangle is the only planar central configuration. The exact number of central configurations for 4body problem is still unknown. In general, one can investigate the existence of central configurations with certain type of configurations and finds the relationship between the masses and the shape of configuration. There are hundreds of research papers on central configurtions. Here are some papers you can start to read. We propose to investigate the new phenomenon in the nbody problem with general homogenous potential. Initially, students will focus on learning about central configurations and their properties, and learn about super central configurations and their contributions to the counting of central configurations for some given masses. Progress has been made in the effort to classify super central configurations in the collinear 3body and 4body problem under Newtonian potential by previous undergraduate research groups. We propose the following research problems for undergraduate students to investigate: (A) Classify super central configurations in the collinear 3body or 4body problem with a homogenous potential. Conduct comparison analysis and find how the potential power affects the existence of super central configurations. (B) Find the exact number of central configurations for the collinear 3body or 4body problem with general homogenous potential. (C) Existence of Super Central Configurations: Can one get similar results as in paper 9 for collinear fivebody problem or general nbody problem? (D) Find an example of a super central configuration in the planar $n$body problem.
2. Richard Moeckel, Lecture Notes on Central Configurations. 3. Tiancheng Ouyang, Zhifu Xie, Collinear Central Configuration in Fourbody probelm 4. Mervin Woodlin, Zhifu Xie, Collinear Central Configuration in the nbody..... 5. F.R. Moulton, The straight line solutions of the problem of n bodies. 6. Alain Albouy, Richard Moeckel, Inverse problem for collinear central configurations 8. J. Llibre, L. Mello, E. PerezChavela, New Stacked CC For Planar 5body problem 9. Zhifu Xie, Super Central configurations of the nbody problem. 
Project 2: Computational Study of Random Maps 
So far, much of the current work has focused on absolutely continuous invariant measures associated with a single transformation, which is basically the numerical analysis of the corresponding FrobeniusPerron operator. In real world problems, however, several interrelated transformations often appear according to some probability distribution to form a complicated dynamical system, so theoretical and computational studies of the resulting random maps are essential for the statistical understanding of the resulting dynamics. In computational ergodic theory, the major task is the numerical analysis of the more general Foias operator, corresponding to a regular stochastic dynamical system associated with the random maps, which involves FrobeniusPerron operators with respect to the participating transformations. More specifically, we propose that undergraduate students in the REU program study the Foias operator for the random maps that is basically a combination of FrobeniusPeron operators associated with every involved transformation with the given probabilities. 
More Projects 

Any Questions? Please Contact the Program Director: Dr. Zhifu Xie.  

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