2024 REU Site in Mathematical Analysis and Applications

Dates: Monday, May 20 - Friday, July 12

The program will start remotely on Monday, May 20. We will have the first week virtually to accommodate various academic calendars and quarter-system schools. The participants will arrive at Dearborn on Tuesday, May 28, for the remaining seven-week residential component. The program will conclude on Friday, July 12. The participants can move in on Tuesday, May 28, and move out on any of the three days: July 13, July 14, or July 15.

Application Deadline: Sunday, February 18, 2024

Apply here on the NSF ETAP portal
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.

2024 Projects

Error-correcting codes coming from toric varieties

(Mentor: Kelly Jabbusch)

- - Coding theory examines efficient ways of packaging data so that errors in the transmission of data can be corrected. In 1977 Goppa defined algebraic codes, thus allowing a wide range of algebraic geometry to be applied. Goppa built code from a nonsingular projective plane curve and his theory can be extended to higher-dimensional varieties, though many of his algebraic-geometric tools don't apply directly. In 1998, Hansen introduced a family of codes from higher dimensional varieties, those arising from toric varieties. Since toric varieties have a related combinatorial structure coming from a lattice polytope, one can compute a basis for the code and estimate its parameters more easily than a general higher dimensional algebraic variety. In this project, students will learn some algebraic-geometric techniques in the context of curves and toric varieties over a finite field and see how they apply to coding theory.
- Selected References:
  - S. H. Hansen, Error-Correcting Codes from Higher-Dimensional Varieties.
  - V. D. Goppa, Geometry and Codes.
  - J. P. Hansen, Toric Surfaces and Error-correcting Codes.

Period polynomials of Bianchi modular forms and their zeros

(Mentor: Tian An Wong)

- - Modular forms are central objects in number theory. Defined over the rational numbers, their extension to imaginary quadratic fields are known as Bianchi modular forms. In this project our goal is to study period polynomials associated to Bianchi modular forms, and in particular the zeros of such polynomials. In the classical case, it was shown that almost all zeroes of a period polynomial lie on the unit circle, sometimes called the "Riemann hypothesis for period polynomials." What is the analogue for Bianchi period polynomials? The goal of this project will be to explore this question. Experience with linear algebra, abstract algebra, and complex analysis is strongly recommended.
- Selected references:
Conrey, John Brian, David W. Farmer, and Ozlem Imamoglu. "The Nontrivial Zeros of Period Polynomials of Modular Forms Lie on the Unit Circle." International Mathematics Research Notices 2013, no. 20: 4758-4771.
Jin, S., Ma, W., Ono, K., & Soundararajan, K. (2016). Riemann hypothesis for period polynomials of modular forms. Proceedings of the National Academy of Sciences, 113(10), 2603-2608.
Karabulut, Cihan. "From binary Hermitian forms to parabolic cocycles of Euclidean Bianchi groups." Journal of Number Theory 236 (2022): 71-115.

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.

- Selected References:

Convolutional blurring and PSF estimation

(Mentor: Aditya Viswanathan)

- - Many of us are guilty of capturing photographs on our phones with blemishes such as motion or out-of-focus blurs. This occurs in many other imaging modalities too - such as astronomy, microscopy, radar and medical imaging. In several such important applications, this phenomenon is mathematically modeled as a convolution; the observed image is a convolution of the true underlying image and a second image or function which characterizes the type of blurring artifact - referred to as a blurring function or a point spread function (psf). This project is concerned with the estimation or recovery of such psfs from potentially noise-corrupted and incomplete measurements. This is an important precursor to recovering the underlying unblemished image (which we may consider if time permits). We will utilize results from approximation theory, Fourier analysis, scientific computing, and statistical estimation theory in developing our theoretical results. These will be validated using computational simulations and real-world datasets.

- Selected References:
  - - The Image Deblurring Problem: Matrices, Wavelets, and Multilevel Methods; Austin, D., Espanol, M., and Pasha, M.; Notices of the American Mathematical Society; vol. 69; no. 8; 2022; doi: 10.1090/noti2534
    - Blind image deconvolution; Kinder, D. and Hatzinakos, D.; IEEE Signal Processing Magazine; vol. 13; no. 3; pp. 43-64; 1996; doi: 10.1109/79.489268
    - (Last summer's REU poster) Edge-Informed Estimations for Blurring Parameters in Convolutional Models; Hume, J., McDonald, D., Newman, A.; 2024 Joint Math Meetings; San Francisco, CA [pdf]

If you have any questions about REU at the University of Michigan-Dearborn, please contact Professor Zeytuncu at zeytuncu@umich.edu.

This REU program is supported by the National Science Foundation (DMS-2243808), 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.