REU 2020
2020 REU Site in Mathematical Analysis and Applications
Dates: Tuesday, May 26--Friday, July 17 (8 weeks, full participation required).
Application Deadline: Friday, February 21, 2020.
Outcomes:
The CR Geometry group presented two talks at the Young Mathematicians Conference at OSU.
The CR Geometry group finished and submit their first paper: A Tauberian Approach to Weyl's Law for the Kohn Laplacian on Spheres. ArXiv version.
The error-correcting codes group presented a talk at the Young Mathematicians Conference at OSU.
The phase retrieval group presented a talk at the Young Mathematicians Conference at OSU.
The numerical range group presented a talk at the Young Mathematicians Conference at OSU.
2020 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 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. 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:
Daniel M. Burns and Charles L. Epstein. Embeddability for three-dimensional CR-manifolds. J. Amer. Math. Soc., 3(4):809–841, 1990.
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.
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, 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.
Phase Retrieval with Applications to Ptychography (Mentor: Aditya Viswanathan):
At the core of many scientific breakthroughs, including the imaging of cells, viruses, and nanocrystals, is the algorithmic solution of a challenging inverse problem known as the phase retrieval problem. This fascinating 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. For example, in x-ray crystallography, the acquired measurements correspond to the squared magnitude of the Fourier transform of the specimen, with the inverse problem then being the reconstruction of the specimen from these phaseless measurements. Only recently has the engineering and mathematics community started to address the theory of phase retrieval, providing measurement maps and associated reconstruction algorithms with provable error guarantees. However, many open problems -- concerning practically realizable measurement constructions, computationally efficient algorithms, robustness to measurement errors and model imperfections -- remain. The focus of the proposed project will be to address some of these open problems in relation to a molecular imaging modality known as ptychography. Here, large molecular specimens are imaged by combining multiple individual scans of small local regions of the specimen. The proposed project will aim to (i) develop large-scale ptychographic phase retrieval algorithms, (ii) provide mathematically rigorous theoretical recovery and error guarantees for these algorithms, (iii) validate the same through extensive numerical experiments implemented using a software package such as MATLAB, Python, Julia etc., and (iv) apply these techniques to real-world imaging datasets.
Selected References:
Balan, Casazza, Eddin, On signal reconstruction without phase.
Candes, Strohmer, Voroninski, PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming.
Rodenburg, Ptychography and Related Diffractive Imaging Methods.
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:
Shapiro, J.H., Notes on the numerical range.
Psarrakos, P.J., and Tsatsameros, M.J., Numerical range: (in) a matrix nutshell.
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.