2023 REU Site in Mathematical Analysis and Applications
Dates: Monday, May 22 - Friday, July 14
The program will start on Monday, May 22, virtually. 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 30, for the remaining seven-week residential component. The program will conclude on Friday, July 14. The participants can move in on Tuesday, May 30, and move out on any of the three days: July 15, July 16, or July 17.
Application Deadline: Sunday, February 19, 2023
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.
2023 Projects
Linear Fractional Transformations
(Mentor: 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:
Shapiro, J.H., Notes on the numerical range.
Psarrakos, P.J., and Tsatsameros, M.J., Numerical range: (in) a matrix nutshell.
Parameter Estimation using Bayesian Procedure
(Mentor: Keshav Pokhrel)
Bayesian parameter estimation is a promising alternative framework for parameter estimation. We will focus on Bayesian parameter estimation procedure for skewed probability distributions. In addition to sampling algorithm Markov Chain Monte Carlo (MCMC), we will discuss about Hamiltonian Monte Carlo (HMC) and No-U-Turn (NUTS) sampling algorithms with dual averaging for posterior weight generation. NUTS uses a recursive algorithm to assemble a set of likely points that spreads to cover the target distribution and help us to automate tuning parameters. We will make sure that the proposed algorithms are compatible with RStan diagnostic tool ShinyStan.
Selected References:
Sharaf, T., Williams, T., Chehade, A., and Pokhrel, K., “BLNN: An R Package for Training Neural Networks Using Bayesian Inference," SoftwareX, Vol 11, 2020.
Hoffman M.D., Gelman A. The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15 (2014), pp. 1593-1623.
Mudholkar G., Srivastava G., and Kollia G. A generalization of the weibull distribution with application to the analysis of survival data. Journal of the American Statistical Association, 91(436): pp.1575–1583, 1996.
Elliptic Dedekind Sums and Bianchi Modular Forms
(Mentor: Tian An Wong)
Dedekind sums and modular forms are classical objects in number theory. They contain a lot of interesting information about arithmetic and have applications to many wide-ranging fields. Elliptic Dedekind sums and Bianchi modular forms are generalizations to imaginary quadratic fields, which are certain extensions of the rational numbers, and again carry rich arithmetic structure. Building on recent works, we shall explore various properties of these functions, formulate conjectures on their behaviour, analogous to the classical ones, and try to prove them. Experience with linear algebra, complex analysis, and/or number theory will be useful.
Selected references:
N. Berkopec, J. Branch, R. Heikkinen, and C. Nunn, and T.A. Wong, The density of elliptic Dedekind sums" Acta Arithmetica (preprint)
Karabulut, Cihan. "From binary Hermitian forms to parabolic cocycles of Euclidean Bianchi groups." Journal of Number Theory 236 (2022): 71-115.
Zagier, Don. "From quadratic functions to modular functions." Number theory in progress 2 (1999): 1147-1178.
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:
A. Boggess. CR Manifolds and the Tangential Cauchy Riemann Complex. Studies in Advanced Mathematics. Taylor & Francis, 1991.
Siqi Fu. Hearing pseudoconvexity with the Kohn Laplacian. Math. Ann., 331(2):475–485, 2005.
Phase Retrieval with Applications to Optical Microscopy
(Mentors: Yulia Hristova and Aditya Viswanathan)
Note that this particular project has a separate application link on ETAP. 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
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-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.