2019 REU Site in Mathematical Analysis and Applications

Dates: Tuesday, May 28--Friday, July 19 (8 weeks, full participation required).
Application Deadline: Sunday, February 24, 2019. 
Application Website: Mathprograms.org  
Eligibility: Undergraduate student participants must be citizens or permanent residents of the United States or its possessions. Students from underrepresented groups in mathematics, women, and students from two-year college programs are strongly encouraged to apply. 
Participant Support: Qualified participants will receive a stipend, as well as housing and meal allowance. 
Tentative List of Research Projects:

2019 Projects
  • Spectral Theory and Complex Analysis (Mentor: Yunus E. Zeytuncu): 
  • The spectrum of an operator contains a lot of information about the operator and also the domain/manifold it is defined on. A famous example of this is the phenomena of hearing the shape of a drum as explained in the first two articles below. Domains and manifolds in several complex variables come with some canonical operators defined on them, and spectral properties of these operators can be used to answer many geometric questions. One of the particular geometric questions is the embeddability of abstract manifolds in the complex Euclidean space. This question can be answered by studying the eigenvalues of the d-bar Neumann problem. In this project, we will learn more about spectral theory, complex analysis, CR manifolds, and complex geometry. The project will be based on the work of the groups from the last two years. Prerequisites include a course in linear algebra, some familiarity with complex analysis, and basic real analysis/advanced calculus. 
  • Selected References: 

  • Eigenvalues of Markov Matrix of the Asymmetric Simple Exclusion Process (Mentor: Hyejin Kim):
  • Asymmetric Simple Exclusion Process (ASEP), which was introduced in 1970 by Frank Spitzer, is a Markov model for random particles. Each particle hops independently and randomly to the nearest neighboring site in any preferred direction, but two particles cannot occupy the same position, that is, the jumping to nearest sites is not allowed if the site is already occupied by other particle. It is known that the exact solution to the dynamics of the ASEP with reflecting boundaries can be obtained by using a generalized Bethe ansatz. Using Bethe equations for the eigenvalues of the Markov matrix of the ASEP, we can obtain qualitative information for ASEP such as the stationary state of the process and the longest relaxation timescale from the eigenvalues of the Markov matrix of the ASEP. In this project we will study the general form of eigenvalues of the Markov matrix. A course in linear algebra and probability theory would be helpful for this project.  
  • Selected References:

  • Reproducing Kernel Hilbert Spaces (Mentor: Alan Wiggins):
  • The focus will be on problems involving either the theory or applications of Reproducing Kernel Hilbert Spaces. Some avenues that may be pursued are: 
    • How can one characterize subsets of the dual of a Hilbert space that induce a "nice" reproducing kernel structure?
    • Analysis of kernel techniques in dimension reduction problems.
  • Selected References:

If you have any questions about REU at the University of Michigan-Dearborn, please contact us at um-dearborn-math-reu@umich.edu.

This REU program is supported by the National Science Foundation (DMS-1659203), the National Security Agency, the College of Arts, Sciences, and Letters, and the Department of Mathematics and Statistics at the University of Michigan-Dearborn. 
Special thanks to Al Turfe for the Turfe Summer Research Fellowships.

Overview of Program

The University of Michigan-Dearborn REU Site in Mathematical Analysis and Applications is an eight-week summer program. The research projects are related to Fourier analysis, complex analysis, operator theory, spectral theory, and mathematical music theory. Selected participants will have a unique interaction with experienced faculty mentors in a rich intellectual environment, where they will learn how to use mathematical ideas to solve real-life problems. The program will introduce participants to a large network of mentors and peers, which will assist them in career planning and in commitment to the scientific community. The program organizers will help students to find appropriate venues to present and publish their results. Additionally, students will learn more about the graduate school application process and career opportunities in academia and in industry.