Math research projects for undergraduates

Introduction            

These are some of the mathematical research projects for University of Delaware undergraduates in 2016. Please contact the associated faculty member directly for more information. 

Funding
  • Funds are available to support some students during the summer, either from the faculty member or from the UD Undergraduate Research Office. 
  • The deadline for applications to the UD Undergraduate Research Office is March 1, 2016 and the applications must be accompanied by a letter of support from the faculty member you will be working with. 
Projects
  1. Models for the tear film
    Dr. Richard Braun and Dr. Toby Driscoll
    A collection of projects that studies various aspects of the tear film are under way. In collaboration with ocular scientists who make tear film measurements, we develop mathematical models to understand and explain those experiments.  Those models range in difficulty from single ODEs or PDEs to large nonlinear systems.  There are a limited number of funded opportunities from external funding agencies, as well as UD Summer Research Scholars and Fellows.  More information is available at the the following site

  2. Decomposing Proof in Secondary Classrooms


    Dr. Michelle Cirillo
    Slides for her presentation.

  3. Patient-specific circulation models for infant heart defects

    Dr. Tobin Driscoll  and Dr. Gilberto Schleiniger
    Our long-term goal is to create real-time, patient-specific models of the circulation in newborns with a ventricular heart defect, there are a range of opportunities available, including curating and summarizing our (first of its kind) data collection, application of data-mining or machine learning techniques, and trying existing or emerging techniques for matching the parameters of a fixed ODE model to the outputs of bedside monitors. Depending on the problem being targeted, prior experience with linear algebra, ODEs, and/or scientific programming is strongly recommended. The work will be with a team of faculty, physicians from the local children's hospital, and a variety of graduate and undergraduate students.
    Slides for their presentation.

  4. Modeling a medical device for measuring reaction rates
    Dr. David Edwards
    Dr. Edwards is looking at the BIAcore optical biosensor, which can measure chemical reactions in real time by weighing a reacting surface.  Though this provides data which are easy to interpret when only one reaction occurs, the data is more difficult to interpret when multiple reactions (each of which affects the mass of the surface) are occurring simultaneously.  Obtaining reliable rate constants in this situation requires mathematical modeling (ODEs, PDEs, and integral equations) as well as ideas in experimental design.  Depending on their area of interest, students may use analytical (by-hand) techniques, as well as computer software such as Mathematica, Maple, or Matlab.  Students should have completed linear algebra and ODEs.
    Slides for his presentation.

  5. Vessel Remodeling in Atherosclerosis: Modeling and Simulation
    Dr. Pak-wing Fok
    Cardiovascular disease (CVD) affected about 80 million Americans in 2006 and was responsible for 800,000 deaths. One common form of CVD is atherosclerosis which leads to histopathological changes in a vessel, often accompanied by growth or resorption. These changes in the vessel wall are known as remodeling. Currently, our understanding of vessel remodeling is still quite qualitative and could greatly benefit from mathematical modeling.

    For the project this summer, interested students will work on modifying an existing Matlab code (see Fok and Sanft, Mathematical Medicine and Biology 2015 for details) to simulate the growth and elastic response of a vessel cross section. The ultimate aim is to incorporate some auto-regulatory features into the model (e.g. vasodilator release and response) to see what effect they have on the remodeling process. Specifically, there have been human clinical trials and animal experiments where vessel dimensions are measured over time. The goal of this summer research is to see if the model can capture features of these data sets.

    Suitable students would have an interest in mathematical biology, applied mathematics and have experience in Matlab.
    Slides for his presentation.

  6. Image compression
    Dr. Mahya Ghandehari
    The fundamental feature of vector spaces is the existence of basis, i.e. a set of vectors which uniquely represent every other vector through linear combinations. The collection of "nice" real-valued functions on the 2-dimensional plane is an important example of vector spaces, which has a basis of infinite size. Although this vector space is huge, we are usually interested in special elements in there.

    In this project, we are interested in understanding and analyzing 2-dimensional medical images. We think of an image as a function which assigns to every point in the plane a colour. We suppose that we have 10 colours, 0, 1, ..., 9, which represent different shades of grey. In most cases, scans of body parts have very simple forms: they consist of a few smooth contours nested in each other, and each area in between two consecutive contours is coloured by only one colour. Our goal is to find a basis, which represents medical images sparsely, i.e. their linear combination representation have few nonzero coefficient. Finding such a sparse representation for signals, has significant applications, as it reduces the cost of saving or analyzing the image.
    Slides for her presentation.

  7. Numerical Methods for Scattering from Thin Structures
    Dr. Peter Monk
    In photonics and the management of microwaves there is engineering interest in using special interfaces between materials to produce novel electromagnetic wave propagation patterns.  These interfaces are typically complicated involving, for example, a layer of small metallic scatterers.  When these are periodically distributed homogenization theory shows how to predict the interaction of the material with an electromagnetic wave by an equivalent medium.  However if the structure is not periodic it would be desirable to have a numerical scheme that automatically performs a local form of homogenization to again compute the solution without the cost of solving a detailed model of the interface.  

    In the summer I hope that 1) a few model problems can be identified, 2) full finite element solutions can be constructed to provide an “exact” solution of the problem, 3) homogenization can be tested. Much of this work would be computational and involve coding finite element methods within existing packages.

  8. Mathematical Modeling of Plankton: Sink or swim


    Dr. Lou Rossi
    Plankton are crucial to ocean health, CO2 absorption and oxygen production.  Commonly reported problems in the media involving plankton and waterway health are 'algal blooms' where plankton concentrations run amuck.  Various species of plankton are well studied in the marine science literature, but they are not well understood mathematically.  What makes certain species interesting is that they can sustain themselves by using photosynthesis and by grazing on other species of plankton.  Predation is more efficient and effective than photosynthesis so it is the preferred means of sustenance.  Understanding the tradeoffs between swimming, grazing and non-grazing behavior is an open problem, and mathematical modeling and analysis may shed some important light on the processes. 

     A number of accessible undergraduate projects are possible to 'shine some light' on these processes.  Undergraduate and graduate projects can involve mathematical analysis, modeling, computation and/or experiments.
    Link 1,  link 2,  link 3

    Slides for his presentation.


  9. Exploring waves in elastic media

    Dr. Francisco Sayas
    Slide for his presentation.

Last modified on February 21, 2016, by Rakesh