2021 REU Site in Mathematical Analysis and Applications

Dates: Monday, May 24--Friday, July 16 (8 weeks, full participation required. We intend to run a residential program here in Michigan. However, we have to follow the university and state guidelines, and it is not yet clear if we will have a residential program or a virtual program. We will inform all the applicants and participants as soon as we have more information on this issue. We had our 2020 program in the virtual setting, and it was a blast.)
Application Deadline: Sunday, February 21, 2021. 
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:

2021 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. 
  • Selected References: 

  • 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:

  • Linear Fractional Transformations (Advisor: John Clifford): 
  • The focus of the summer REU project will be the study the numerical range of a composition operator on the Hardy space.  The three main ideas are the numerical range, composition operator, and Hardy space.  We will start by studying the basic properties of the numerical range; the numerical range of a 2 by 2 matrix is an ellipse, the numerical range is invariant under unitary conjugation, the numerical range is convex and the numerical range of a normal matrix is the convex hull of the eigenvalues. The goal of the project will be to characterize the numerical range of a composition C_{A} acting on the Hardy space where A is an n by n matrix that is a self-map of the unit ball in C^n.  In the case of n=1, the result is known and interesting. We will work through the n=1 case carefully as we learn about the numerical range, composition operators, and the Hardy space.  For the case n>1 the results are not known.  Our first goal will be to characterize the numerical range of the composition operator C_{A} when A is 2 by 2.  We will then investigate what results for n=2 generalize for all n>1. Once we are done with n=2 we will have many interesting questions to explore.  
  • Selected References: 

  • Phase Retrieval with Applications to Optical Microscopy (Mentors: Aditya Viswanathan and Yulia Hristova): 
  • Note that this particular project has a separate application link on Mathprograms.org. Please use the specific link for this project.
  • 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.
  • Selected References:
    • (a survey article) Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao and M. Segev, "Phase Retrieval with Application to Optical Imaging: A contemporary overview," in IEEE Signal Processing Magazine, vol. 32, no. 3, pp. 87-109, May 2015, doi: 10.1109/MSP.2014.2352673
    • (mathematical framework) M. Perlmutter, S. Merhi, A. Viswanathan and M. Iwen, "Inverting spectrogram measurements via aliased Wigner distribution deconvolution and angular synchronization," in Information and Inference: A Journal of the IMA, Oct 2020, doi: 10.1093/imaiai/iaaa023
    • (related work from the 2019 REU) C. Cordor, B. Williams, Y. Hristova and A. Viswanathan, "Fast 2D Phase Retrieval using Bandlimited Masks," in Proceedings  of  the  28th European  Signal  Processing  Conference  (EUSIPCO), Amsterdam, Netherlands, pp. 980-984,  Aug. 2020 https://www.eurasip.org/Proceedings/Eusipco/Eusipco2020/pdfs/0000980.pdf
    • (physics behind the application) J. M. Rodenburg, "Ptychography and Related Diffractive Imaging Methods," in Advances in Imaging and Electron Physics, vol 150, pp. 87-184, May 2008, doi: 10.1016/S1076-5670(07)00003-1

  • TBA

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-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.