Research Programs
Our research
The faculty and students of the Department of Mathematical Sciences conduct research in a wide variety of areas of applied mathematics and mathematics. These projects are carried out in multiple ways and the involvement of students at the both the graduate and undergraduate levels is a priority for the department. In following the links below you will find descriptions of the areas of focus of the faculty, current research projects, specific projects carried out by undergraduates, the facilities available for research in the department, as well as news about the latest research achievements of our faculty and students.
Undergraduate Research
The Department of Mathematical Sciences offers many opportunities for undergraduates to get involved in research. The department participates in two main summer research programs:
Summer Scholars/Summer Fellows
This is a universitywide program that provides the opportunity to work with a faculty member during the summer. For more details, please visit the Undergraduate Research Program website.
Research Discovery Days
This opportunity happens in February to provide more information about the summer research programs. During these events, faculty members will talk about their research to help students interested in doing research find a mentor and a research project they would like to work on. Watch your emails for more information. We also encourage you to visit the individual websites of our faculty members to learn about their research areas.
Graduate Research
Graduate Research is done at several levels within the Department. Research related to the Ph.D dissertation formally starts after students have passed their candidacy exam. Our faculty work in several areas of research as seen further below on this page.
Precandidacy, students can take the courses MATH 868 (Research) and MATH 870 (Reading) as part of their coursework requirement.
There are also opportunities for summer research through the GEMS and UniDel programs.
GEMS
GEMS summer research experiences provide graduate students with the opportunity to become involved in mathematical research early in their graduate careers. In addition, participating graduate students will gain valuable experience serving as mentors to undergraduate researchers. The program is open to graduate students in the Mathematical Sciences who are in their first year of studies at UD. The program provides a stipend of $6,000 for a 10week commitment during the summer following your first year of classes.
Center for Applications of Mathematics in Medicine (CAMM)
The mission of the Center for Applications of Mathematics in Medicine (CAMM) is to coordinate research and education that fosters the application of mathematics and computation to biomedical research and clinical practice in order to improve human health in the state, region, and beyond.
By facilitating interaction and collaboration between the members of these communities, CAMM enhances the application of current mathematical knowledge to medicine and accelerates the understanding of societally important intellectual challenges that will shape new generations of mathematical researchers.
For the most uptodate info on CAMM, please go to mathandmedicine.org.
Research Areas
Analysis
Analysis is a wide research area that constitutes the backbone of rigorous modern mathematical thinking, providing solid foundations to external areas like partial differential equations or quantum mechanics, in addition to exploring abstract constructions that generalize classical ideas of the theory of functions and their connections with geometry, topology, or combinatorics. The interests of the UD Analysis group include operator theory, graph limits, harmonic analysis, and functions of complex variables.

Discrete Mathematics
Every member of the Discrete Mathematics group is actively engaged in research; for more specific information concerning the research activities of any particular member, please visit that individual's home page. Approaches to conduct suitable collaborative work are welcome. All members of the group are interested in working with and advising graduate students of suitable standard. Should you be considering working with any of the members, you are encouraged to approach the member to discuss the matter as soon as is practicable. For those students who are US citizens and interested in employment with a government agency, the National Security Agency is about one hour's drive away from campus and is one of the world's largest employers of discrete mathematicians.

Fluid and Material Sciences
The fluid mechanics and materials science group within the Department is very active. Faculty members are interested in a wide range of modern problems that originate from or have applications in industry. In fluid mechanics, many of us are interested in viscoelastic fluids, thin film flows and transonic aerodynamics. In materials science, our faculty members strive to understand the mathematics of phase transformations and to develop novel descriptions of composite materials.

Industrial and Applied Math
There are a wide range of problems being solved in collaboration with industry and national labs. Research ranges from basic numerical analysis (finite element, boundary element and finite difference convergence theory) and fast methods (multigrid) to applications in materials science (foam evolution, phase transformations in crystalline alloys, approximation of microstructure, viscoelastic phenomena) and electromagnetism (scattering, inverse scattering and ferromagentism). Graduate students are welcome!

Mathematical Medicine and Biology
Research in Mathematical Biology is extremely diverse in the Department of Mathematical Sciences, with faculty members working on a wide variety of modern problems ranging from the molecular scale to the organism level. Current research projects include the fluid mechanics of tear films and the effect of the blink cycle, models of atherosclerotic plaque, transport in bone and osteoporosis, imaging of bone marrow and the collective dynamics of ants.

Mathematics Education
Mathematics education research involves disciplined inquiry into the teaching and learning of mathematics at all grade levels – preschool though college. Such research has flourished over the past several decades. As outlined below, faculty in the Department of Mathematics Sciences are actively engaged in a number of research projects.

Numerical Analysis and Scientific Computing
Numerical Analysis and Scientific Computation is one of the largest and most active groups within the Department, with most of the major research areas represented. Our group is interested in a wide range of topics, from the study of fundamental, theoretical issues in numerical methods; to algorithm development; to the numerical solution of largescale problems in fluid mechanics, solid mechanics, electromagnetism and materials science.

Probability and Stochastic Methods
In probablility theory a stochastic process, or sometimes random process (widely used) is a collection of random variables; this is often used to represent the evolution of some random value, or system, over time. This is the probabilistic counterpart to a deterministic process. Instead of describing a process which can only evolve in one way, in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
Students wishing to study and do research in Probability theory and its applications are advised to apply directly to the graduate program of the department.

Scattering and Inverse Scattering
Research in scattering theory is mainly focused on developing and analyzing algorithms for computing approximations to scattered acoustic and electromagnetic fields. There is also a significant activity in antenna modeling. Research on inversion techniques includes acoustic inverse problems in ocean acoustics, and electromagnetic inverse problems in complex environments. Graduate students are welcome!

Topology
Topology is a major area of mathematics concerned with the most basic properties of space, such as connectedness. More precisely, topology studies properties that are preserved under continuous deformations, including stretching and bending, but not tearing or gluing. Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.

Research Projects
Our NSFsupported 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 informationtheoretic 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, highdimensional 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 bilayer 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 xray 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 3D 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 mixedmethods 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 subgoals 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 posttest 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 smallgroup 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 universitybased 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