Research Programs

Our NSF-supported research program explores the interplay of information theory with mathematics, physics, and data science. Information theory was created by Claude Shannon in order to model and solve problems of compression and communication of data, but is now understood to be a fundamental part of mathematics and not just engineering. Indeed, mathematics, physics, and data science are rife with optimization problems, and somewhat surprisingly to the uninitiated, the study of many of these optimization problems as well as their extremal configurations is facilitated by adopting an information-theoretic point of view. The resulting deep connections between a number of apparently disparate fields makes this a very broad and exciting field of research. For example, our research program investigates questions in additive combinatorics (which first arose from number theory), convex geometry, high-dimensional probability, and harmonic and functional analysis. Results emerging from the program not only fundamentally advance our understanding, but also have applications to theoretical computer science, data science, statistical physics, and electrical engineering.

Researcher(s): Mokshay Madiman

Every time you blink, a thin fluid film is left behind that covers the front of your eye. This tear film provides a smooth optical surface, defense against inflammation and foreign particles, and lubricates the eye’s surface. When the tear film is not healthy, a variety of maladies may occur, including a collection of symptoms known as dry eye. Dry eye may arise from a shortage of tear fluid for each blink, from too much evaporation of the tear film, or a combination of both.If this shortage of tear fluid persists, pain and inflammation of the eye follow. Understanding the dynamics of healthy and unhealthy tear films may help lead to better understanding of the progression and treatment of dry eye and other conditions that afflict millions.

Researcher(s): Richard Braun, Tobin Driscoll

Analyzing the structure of a large graph/network by brute force is not feasible. The challenge is to use a small number of parameters who capture the network shape. The eigenvalues of a graph are such parameters and although they do not completely determine a graph in general, they reveal important structural information and play fundamental roles in many instances. My research investigates when the eigenvalues determine a graph and how they relate to its structure.

Researcher(s): Sebastian Cioaba

We develop a mathematical background of ultrasound methodology for the diagnosis of bone brittleness. Such research is usually connected to an evaluation of the microstructure of cancellous/ trabecular bone. We construct artificial CT scans to enlarge our catalogue of bone samples to be used for the inverse problem associated with unknown bone parameters. This entails devising an efficient numerical scheme to produce realistic, orthotropic trabecular bone structures. Our work on bone remodeling involves investigating the entry of 1,25D through the lipid bi-layer membrane of the stem cell and its entry and release into the cytoplasm. This requires modeling the cell's membrane enclosure of 1,25D by adhering to binding sites to form a vesicle, the separation of the vesicle from the lipid bilayer and entry of a coated vesicle into the cytoplasm. In order to understand the mechanical processes in bone remodeling we ned to investigate the flow of ions in vivo bone. To this end, we use homogenization/mixture theory to derive the acoustic response for an in vivo model of wet bone.

Researcher(s): Robert Gilbert, Philippe Guyenne, Yvonne Ou, George Hsiao

Collaborators: George Hsiao

​The field of inverse scattering theory has been a particularly active field in applied mathematics for the past thirty years. The aim of research in this field has been to not only detect but also to identify unknown objects through the use of acoustic, electromagnetic or elastic waves. Although the success of such techniques as ultrasound and x-ray tomography in medical imaging has been truly spectacular, progress has lagged in other areas of application which are forced to rely on different modalities using limited data in complex environments. Indeed it is often said that "Target identification is the great unsolved problem.We detect almost everything, we identify nothing". Until a few years ago, all existing algorithms for target identification were based on either a linearizing weak scattering approximation or on the use of nonlinear optimization techniques. However, as the demands of imaging increased, it became clear that incorrect model assumptions inherent in weak scattering approximations imposed severe limitations on when reliable reconstructions were possible. On the other hand, it was also realized that for many practical applications nonlinear optimization techniques required information that is in general not available. Hence, in recent years, alternative methods for imaging have been developed that avoid incorrect model assumptions but, as opposed to nonlinear optimization techniques, also avoid strong a priori assumptions about the scattering object. Such methods come under the general title of qualitative methods in inverse scattering theory and are based on the development and use of linear sampling methods and transmission eigenvalues, both of which were discovered and developed here at Delaware in collaboration with researchers in Germany and France. In particular, such methods are based on solving a linear integral equation for a range of "sampling points" and frequencies which are not "transmission eigenvalues" and lead to an approximation to the shape of the scattering object together with limited information about the material properties of the scatterer. Such an approach is remarkable since the inverse scattering problem itself is nonlinear but no linearizing assumptions have been made in the derivation of the above mentioned linear integral equation. This research project with AFOSR is devoted to the further development and application of this new approach in inverse scattering theory.

Researcher(s): David Colton, Peter Monk

As the use of 3-D printers has exploded in recent years, so have several fundamental design challenges. For instance, a cold feedstock must be heated so it is pliable enough to be extruded through the printing head. We want to print as fast as possible, but at excessive speeds, the polymer doesn't get hot enough and clogs the head. What is the ideal balance between heating temperature and velocity? Similarly, the strength of the bonds between printed layers depend on how hot the layers are. But what is the exact relationship? And can it be optimized? These are the types of questions we attempt to answer mathematically.

Researcher(s): David A. Edwards

We develop reliable methods for a variety of applications to science and engineering such as fluid flow models, acoustics, diffusion through heterogeneous porous media, and others. The research focuses on multilevel finite element discretization and analysis of discretizations using modern results in functional analysis and approximation theory. Our techniques apply also to solving singularly perturbed models, and designing numerical methods for fractional reaction diffusion convection problems with applications to atmospheric prediction and ocean fluid flow behavior.

Researcher(s): Constantin Bacuta

It is often hard to construct interesting combinatorial objects (graphs, set systems, difference sets, etc.) directly. In such cases, we often turn to other areas of mathematics, such as geometry, number theory, and various parts of abstract algebra, to name just a few. In particular, one may define graphs by using affine Lie algebras, or as systems of certain polynomial equations in many variables over finite fields. Also some graph theory coloring problems can be approached by studying certain polynomials associated with the graphs, like chromatic polynomials. My recent research is connected to these objects.

Researcher(s): Felix Lazebnik

This design and development research study focuses on secondary students' success with mathematical proof. The goal of this mixed-methods project is to develop a new and improved intervention to support the teaching and learning of proof. This is important because despite the fact that there have been ongoing calls to focus on reasoning and proof in school mathematics, success with proof has remained elusive. The project makes use of pilot study data and findings that suggest a promising approach to scaffolding the introduction to proof in high school geometry. Understanding that reasoning and proof should also be taught outside of geometry, a goal of this study is to leverage the existence of proof in geometry to explore alternative ways to teach it across the grades. This study takes as its premise that if we introduce proof by first teaching students particular sub-goals of proof, then students will be more successful with constructing their own proofs later on. A pedagogical framework informed the development of 16 detailed lesson plans that serve as the experimental intervention for the study. The effect of the intervention will be determined through comparison of pre- and post-test assessments in control and experimental groups. Additionally, professional development, classroom observations, and teacher and student interviews will be conducted over three years of the study.

Researcher(s): Michelle Cirillo

This IUSE Development and Implementation for Engaged Student Learning project addresses the urgent need to improve the preparation of undergraduate prospective secondary mathematics teachers. The University Teaching Experience (UTE) is an assisted teaching experience in an undergraduate mathematics course that serves as an early field experience as part of a methods course prior to student teaching. The UTE begins with preservice teachers (PSTs) actively observing a mathematics course for several weeks to learn the curriculum, classroom norms, and instructor's teaching style. PSTs are asked to join small groups of students to observe student thinking, get to know the students, and practice facilitating small-group discussions. Eventually PSTs take on responsibility for some of the instruction in the course, while receiving support from their methods course instructor and the mathematics course instructor. The UTE is designed to help prospective teachers understand the nature and complexities of teaching mathematics in an authentic context and enact specific practices that are specified in the learning goals of the methods course. This project involves collaborative research across three different teacher preparation programs (Michigan State University, The Pennsylvania State University, and University of Delaware) to achieve two broad aims: (1) to assess the portability of the UTE model to a wide range of university-based teacher preparation institutions and (2) to gather both quantitative and qualitative data to investigate the development of PSTs' knowledge about teaching and their teaching practice as a result of participating in the UTE. Collaborators include Kristen Bieda at Michigan State and Fran Arbaugh at PSU.

Researcher(s): Michelle Cirillo