
Research Areas and Faculty Mentors of the RoseHulman REUEach of the program faculty has a different area of interest. The faculty participate on a rotating basis with three or four faculty each summer. The projects in a given summer will depend on which two of the faculty are participating. For the summer of 2013 the research areas will be high dimensional data analysis, optical tomography, and computational biology, with Professors Eichholz, Inlow and Reyes, and Holder, respectively. * = areas of research for the current year. Inverse Problems (Kurt Bryan, 20022006, 2008, 2009, 2011, 2012, 2014, 2016)Professor Bryan's REU interests are in the areas of inverse problems and nondestructive testing (some recent REU groups have worked on the closely related topic of cloaking and invisibility.) The goal of nondestructive evaluation (NDE) is to determine the interior structure of an object without damage to the object. This involves applying of some kind of energy to the exterior of the object  electromagnetic, thermal, mechanical, or other  and then measuring some aspect of the object's response. The behavior of the energy in the object, termed the "forward" or "direct" problem, is typically governed by a partial differential equation, with the internal condition of the object manifest as a coefficient in the governing differential equation or boundary conditions. The "inverse" problem is to determine these coefficients from knowledge of the solution(s) to the differential equation on some portion of the exterior of the object. Physically, this means observing the object's behavior to the input and using this information to infer internal structure. Two NDE methods that have recently been the subject of much mathematical investigation are thermal imaging and electrical impedance imaging. In the case of impedance imaging the forward problem is governed by some variation of Laplace's equation, while for thermal imaging the forward problem is governed by the heat equation. These methods show promise for the purpose of shape identification, essentially determining the shape of an object (including interior holes or cracks) from limited access to the exterior boundary. This is approach is often used to model corrosion or interior damage to an object. We'll consider the mathematical inverse problem of shape identification, especially the imaging of interior voids or cracks and the governing boundary conditions, using thermal and electrical impedance imaging. These inverse problems have applications as varied as nondestructive testing in aircraft, medical imaging, the testing of soldered connections in circuit boards, and the structural assessment of composite materials. Several questions naturally arise:
This last point will involve the implementation of simulation and reconstruction algorithms using Matlab. No prior knowledge of partial differential equations, numerical methods, or Matlab is required, although participants should have some background in basic (ordinary) differential equations and some programming experience. Our research groups in years 20022011 had great success in analyzing open inverse problems of interest (including papers accepted by professional journals). For more information see the Inverse Problems Research Home Page.
Computational Number Theory (Joshua Holden, 2007, 20092011, 20152016)Just a few decades ago, cryptography was considered a domain exclusive to national governments and militaries. However, the computer explosion has changed that. Every day, millions of people trust that their privacy will be protected as they make online purchases or communicate privately with a friend. Many of the cryptographic algorithms they use are built upon a common transformation, namely discrete exponentiation modulo an integer "n". For instance, DiffieHellman key exchange, RSA and the BlumMicali pseudorandom bit generator all use discrete exponentiation. It is thought that the inverse of this transformation, the " discrete logarithm problem", or "DLP" is computationally intractable. This is part of the basis for assuming the cryptographic security of the algorithms referred to above. However, there is no known proof of this fact. In particular, it would be interesting to know if there were patterns in this transformation that can be exploited. One way to determine this would be to construct the "functional graph" associated with the transformation. Any unexpected characteristics of this functional graph might lead to new progress in breaking the discrete logarithm problem. Questions of interest regarding this functional graph include:
For more information see the Discrete Logarithm Home Page.
Computational Biology (Allen Holder, 2010, 2012, 2013, 2015)
Driven by modern advances like the human genome project, much of biology is
undergoing a dramatic change from wetlab experimentation to
computational investigation. Indeed, many of the most interesting and
relevant challenges in biology are addressed by interdisciplinary teams
that include biology, chemistry, computer science, physics, statistics
and mathematics. In particular, the interplay between mathematics,
theoretical computer science and biology is paramount in the area of
Systems Biology. Biologists have spent years collecting detailed
genetic information that describes what are called biological pathways.
To a mathematician, these pathways are graphs (or networks) that
describe the regulatory mechanisms of a cell. Previous biological
research is largely based on pairwise comparisons between different
components of the network. From these pairwise comparisons,
generalizations about the whole cell are suggested. For example,
pairwise comparisons might show that decreasing the level of one
protein increases the level of another. If the second protein is
beneficial in large quantities, then one could attempt to design a drug
that decreases the level of the first protein. For more
information see the Systems Biology Home Page.
Optical Tomography (Joe Eichholz, 2013, 2014)Optical tomography is an emerging biomedical imaging technique in which optical light (nearinfrared) is passed through a biological medium. The intensity of light is measured as it exits, and from these measurements one attempts to discern the internal structure. This methodology has numerous potential advantages over current imaging modalities, including low cost, use of nonionizing radiation, and potential for high resolution. Reconstruction of the internal structure of the medium leads to an illposed inverse problem. Some questions of interest:
Symmetry and Asymptotic Analysis (Vin Isaia, 2014, 2016)Asymptotic behavior for time dependent systems is sometimes the main interest for a scientist or engineer. Some asymptotic behavior may be difficult to determine, for example secular terms for partial differential equations, which are not as tidy for ODEs. In addition such behavior may be delicate to simulate numerically (e.g. crossover problems in physics). There is thus considerable interest in developing approaches that can bypass these issues. This research project is concerned with ascertaining the asymptotic behavior of ordinary or partial differential equations by making use of symmetry (Lie Group) approaches. These approaches offer evidence that they can bypass common pitfalls with traditional adhoc methods. In addition, their structure allows for wellestablished proof techniques such as fixed point methods. Some classes of problems have been analyzed for ODEs with a small parameter dependence, but little has been proved for the PDE case.
Chemistry of Dimerization (David Goulet, 2015, 2016)A dimer is a molecule formed by the binding of two similar or identical smaller molecules. The specific example we're interested in involves the dimerization of two estrogen receptor proteins. Abstractly, this can be written as the chemical equation M1+M1>D1. Our protein exists in a second form, and two molecules of this can also form a dimer, M2+M2>D2. These homodimers consist of two identical monomers. If both monomer types are present in solution then the reaction M1+M2>D3 can occur, forming a heterodimer. Building mathematical models of this process is simple using the principle of mass action, but fitting the model to experimental data is challenging due to disparities in time scales and the extremely low concentration of monomers at equilibrium. There is some evidence that chromatographic data might facilitate data fitting, even though monomer concentrations and equilibration time scales are below the detection limit. Simulation of a deterministic chromatography model requires numerically solving systems of stiff nonlinear partial differential equations. Additionally, various parameter fitting algorithms should be considered, so that experimental data and differential equations can be seamlessly wedded; this involves solving nonlinear optimization problems. Specific questions to be explored are
High Dimensional Data Analysis (Mark Inlow, Eric Reyes, 2013, 2015)Modern research and predictive analytics efforts increasingly require the analysis of huge, highdimensional data sets in which the number of variables often exceeds the number of observations. Statisticians and computer scientists are developing sophisticated alternatives to traditional analysis methods which cannot handle such data. The REU projects directed by Eric Reyes and Mark Inlow will consist of collaborative highdimenstional data methods research with the Indiana University Center for Neuroimaging and the Li Shen Laboratory. In particular we will investigate new methods for neuroimaging genomic analysis and multidimensional data mining with applications to biomarker discovery. Alzheimer's disease (AD) is the 6th leading cause of death in the United States. Currently AD is unique among the top 10 causes of mortality by virtue of having no known cure and no known preventative; see this clip. The Indiana University Center for Neuroimaging is the genomics core of the Alzheimer's Disease Neurimaging Initiative (ADNI). The mission of the genomics core is to facilitate the investigation of genetic influences on disease onset and trajectory as reflected in structural, functional, and molecular imaging changes; fluid biomarkers; and cognitive status. Carrying out this mission requires the analysis of large, highdimensional data sets to determine genetic correlates and biomarkers of Alzheimer's disease. The REU projects directed by Reyes and Inlow will consist of the development and application of new highdimensional data analysis methods to ADNI neuroimaging and genomic data to investigate genetic correlates and biomarkers of Alzheimer's disease. Professor Reyes' project will consist of developing and applying complete least squares methods to ADNI biomarker data. Complete least squares is a new variable screening approach devised by Reyes and colleagues at North Carolina State University; see this presentation for more information. Professor Inlow's project will consist of extending and refining new nonparametric alternatives to standard Gaussian random field theory methods. These methods are based on results by Inlow motivated by leftspherical distribution theory results in multivariate statistics. Preliminary results (presented at the Joint Statistical Meetings last summer) show these methods are competitive with and possibly superior to current random field theory methods; see this presentation for more information.


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