2022 REU Site in Mathematical Analysis and Applications

Dates: Monday, May 23 - Friday, July 15

Research Projects:

• Spectral Theory and CR Geometry (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 phenomenon 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. The famous Weyl's law is also a great example of such a connection. In the previous years, we worked towards an analog of this result on CR manifolds. 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. 

• 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 particles. 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.  

• Mathematical Music Theory: Balanced Uniform Chord Transformations (Advisor: Thomas Fiore, joint work with Thomas Noll, Moreno Andreatta, and Sonia Cannas) or a Related Topic: 

  • Balanced Uniform Chord Transformations combine several recent group-theoretic perspectives on musical transformations to analyze classical works. This project explores a new mathematical construction of generalized interval systems and how it relates to recent characterizations of chord transformation groups, revisits earlier musical passages in the music theory literature with this new tool, and considers new musical examples. Desired mathematical results are a characterization of Lewinian duality in the new construction, characterizations of relevant group structures, and an exploration of associated musical group action orbits. REU students will either join this project or start a separate project on a related topic that ties into the 2018 Math&Music REU project, depending on the status of this project and student results. Students will be provided with the relevant background, and will also be introduced to the results already achieved by coauthors Thomas Noll, Moreno Andreatta, and Sonia Cannas. Either way, REU students will work within the fascinating circle of ideas created, discovered, and cultivated in the literature cited below and in many other articles and books.  

• Value distribution of elliptic Dedekind sums (Mentor: Tian An Wong)

  • • Elliptic Dedekind sums are a particular generalization of the classical Dedekind sum, which arise in the disparate fields of number theory and topology. A typical question in number theory asks how the values of number theoretic functions such as Dedekind sums are distributed. It was shown long ago that the values of Dedekind sums are dense on the real line; the same was also proved more recently for elliptic Dedekind sums. Very recently, a new and short proof of this result of Dedekind sums was obtained, and the goal of this project is to apply the same ideas to the case of elliptic Dedekind sums. Besides this, we will also ask questions about the equidistribution of elliptic Dedekind sums. There are no strict prerequisites, but some familiarity with complex analysis, real analysis, linear algebra, and/or number theory will be very useful.

• Phase Retrieval with Applications to Optical Microscopy (Mentors: Aditya Viswanathan and Yulia Hristova): 

• • • The phase retrieval problem involves the reconstruction of signals (i.e., vectors, matrices, functions, or other quantities of interest) from magnitude-only (or phaseless) measurements, which unavoidably arise from the underlying physics of certain measurement processes. This is a challenging yet fascinating problem since a unique (up to an equivalence class) solution exists only under certain conditions, and developing a theoretically rigorous yet computationally efficient reconstruction procedure is non-trivial. The focus of the proposed project will be to develop an efficient, noise-robust and mathematically rigorous phase retrieval algorithm for a high-resolution optical microscopy setup designed by our collaborators (see https://smartimaging.uconn.edu/ for more details). The project will incorporate elements of Fourier analysis, applied linear and matrix algebra, numerical analysis, signal processing, inverse problems, and scientific computing. In addition, algorithms developed during this project will be validated on real-world datasets through numerical simulations using a software package such as MATLAB. The prerequisite for this project is a course in linear algebra. Some familiarity with programming and numerical analysis would be beneficial but not necessary.

This REU program is supported by the National Science Foundation (DMS-1950102), the National Security Agency, the College of Arts, Sciences, and Letters, and the Department of Mathematics and Statistics at the University of Michigan-Dearborn. 

- 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, algebraic coding 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 industry.