2021 REU Site in Mathematical Analysis and Applications

Dates: Monday, May 24 - Friday, July 16

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. 

• 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.

• 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.  

• 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.