Rose-Hulman Institute of Technology
NSF-REU Site in Mathematics

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a Research Experience for Undergraduates

REU Faculty Research

Research Areas and Faculty Mentors of the Rose-Hulman REU

Each 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.

Faculty Mentor Research Area
Kurt Bryan Inverse Problems
Joe Eichholz Optical Tomography *
Allen Holder Computational Biology *
Vin Isaia Symmetry and Asymptotic Behavior
David Goulet Chemistry of Dimerization
Mark Inlow High Dimensional Data Analysis *
Eric Reyes High Dimensional Data Analysis *
Josh Holden Computational Number Theory

* = areas of research for the current year.

Inverse Problems (Kurt Bryan, 2002-2006, 2008, 2009, 2011, 2012, 2014, 2016)

Professor Bryan's REU interests are in the areas of  inverse problems and non-destructive 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:

  1. What is a reasonable mathematical model for such defects  (including  the boundary conditions that hold on the defect)?
  2. What can one say (theoretically) about what can and cannot be determined  with these methods?
  3. How can one reconstruct the unknown coefficients in the relevant PDE from boundary data?

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 2002-2011 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.

Kurt M. Bryan
(2002-2003, 2010-2012 senior investigator, 2004-2009, 2013-2016 - PI, program director)
Kurt M. BryanProfessor Kurt Bryan received a B.A. (1984) from Reed College and Ph.D. (1990)  from the University of Washington. While pursuing his doctorate degree he worked at Blount Industries, as a statistician/mathematician and later had a post-doc at the Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, before coming to Rose in 1993. He is interested in mathematical models and inversion algorithms for non-destructive testing of materials using electromagnetic and thermal methods. Professor Bryan mentored REU research groups in the summers of 2002 to 20. You can find out more about Professor Bryan through his Home Page.

Computational Number Theory (Joshua Holden, 2007, 2009-2011, 2015-2016)

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, Diffie-Hellman key exchange, RSA and the Blum-Micali 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:

  • Under what circumstances can we find a "fixed point" for the discrete exponentiation problem? Some results have been obtained, but the approaches are very computationally intensive. It would be interesting to find more efficient approaches.
  • Holden and Moree investigated the ``two-cycles'' of discrete exponentiation modulo a prime p. They provided much evidence to suggest that two-cycles occur about as often as one might expect, including proofs in some special cases, but no conclusive general theorems. There is still much interesting work to be done in the two-cycle case.
  • There are a number of statistics of interest derived from functional graphs, including number of connected components, number of cyclic nodes, number of terminal nodes, average cycle length, maximum cycle length, average tail length, and maximum tail length, among others. Flajolet and Odlyzko have shown how to compute the expected values of these statistics for a random functional graph.

    However, an undergraduate (Dan Cloutier, in his Senior Thesis, written under Dr. Holden) showed by extensive computational examples that these statistics for random functional graphs do not agree with those produced from discrete exponentiation. However, he was able to compute the expected values of the statistics for the cases of binary functional graphs (where every node has outdegree 2), and collect computational evidence that discrete exponentiation graphs which are binary functional graphs do behave like random binary functional graphs. In addition, he was able to find the statistics in the literature for permutations (functional graphs where every node has in-degree 1) and conclude that also in this case the discrete exponentiation graphs appear to behave like random functional graphs.

    Using the same techniques it should be possible to extend to the cases of ternary and perhaps quaternary functional graphs, and it would be of much interest to do so. It is not clear how to extend to the general case and more work should also be done here. In addition, these results might extend to other statistics of random graphs and these should be investigated.

For  more information see the Discrete Logarithm Home Page.

Joshua Holden
(2007, 2009 - senior investigator, 2010-2012 program director)
Joshua HoldenProfessor Holden received his A.B. (1992) from Harvard and his Ph.D. (1998) from Brown University. He held post-doctoral positions at both the University of Massachusetts (1997-1999) and Duke University (1999-2001) before coming to Rose in 2001. He is interested in algebraic number theory, specifically class fields and class fields towers, and also computational and algorithmic number theory. You can find out more about Professor Holden through his 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 wet-lab 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.

While pairwise comparisons are useful in building biological networks, their ability to predict cellular regulation has been limited. Indeed, the modern biological paradigm dictates that the network in its totality needs to be studied with respect to adjusting parameters, and it is these investigations that constitute the area of Systems Biology. Mathematically, the problems reduce to studying parametric graph theory and mixed integer optimization. The underlying graphs along with their associated optimization problems are enormous, and extracting information is a computational hurdle. Mathematical theory plays a significant role since it establishes exact results without computation. Many of the problems are well suited to undergraduate research and are based on coursework in (introductory) analysis, combinatorics, and optimization.

For more information see the Systems Biology Home Page.

Allen Holder
(2010, 2012 - senior investigator)
Allen HolderProfessor Holder received his B.S. (1990) from the University of Southern Mississippi and his Ph.D. (1998) from the University of Colorado at Denver. He taught at Trinity University in San Antonio, Texas, before coming to Rose in 2008. He is interested in mathematical programming and applications in medicine and biology, among other mathematical topics. You can find out more about Professor Holder through his Home Page.

Optical Tomography (Joe Eichholz, 2013, 2014)

Optical tomography is an emerging biomedical imaging technique in which optical light (near-infrared) 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 non-ionizing radiation, and potential for high resolution. Reconstruction of the internal structure of the medium leads to an ill-posed inverse problem. Some questions of interest:

  • As with many inverse problems, current techniques rely on solving the associated forward problem repeatedly. For our purposes the forward problem is the radiative transport equation, a five dimensional integro-differential equation. To date, most methods rely on simplification of the domain/geometry, approximation of the radiative transport equation, or massive parallel computing resources. Due to the sheer size of the discretized problem it seems unlikely that parallel computing can be entirely avoided. However, we propose to investigate fast numerical methods for solving the radiative transport equation, in particular, adaptive techniques applied to discontinuous Galerkin discretizations.
  • When solving any optical tomography problem, well-posedness is an issue. In optical tomography we seek to reconstruct the scattering and absorption characteristics of the medium from input/output optical data. The scattering and absorption properties are regarded as functions in some function space. In general, this input/output data does not uniquely determine these absorption and scattering properties. This can be partially remedied by taking multiple measurements, measuring multiple wavelengths, multiple source configurations, etc., but this can not fix the ill-posedness when the potential absorption/scattering functions are allowed to come from general spaces. However, if the function space is suitably small, unique solutions exist. We propose to study the exact relationships between the number/types of measurements taken and the largest possible function space in which we expect to have a unique solution.
Joe Eichholz
(2013, 2014 - senior investigator)
Joe Eichholz Dr. Eichholz received his B.S. in Mathematics and Computer Science from Western Illinois University in 2005. He earned a Master's Degree in Computer Science and a Ph.D. in Applied Mathematics and Computational Sciences in May 2011, and joined Rose in August 2011. His research interests include biomedical imaging, inverse problems, numerical analysis of partial differential equations, and high performance computing. His website can be found here.

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 ad-hoc methods. In addition, their structure allows for well-established 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.

Vin Isaia
(2014, 2016 - senior investigator)
Vin Isaia Here goes stuff about Professor Isaia.

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

  • Some research has shown that the shape of the chemical peak detected with chromatography contains information about the chemical reactions occurring in the column. Under what conditions does peak shape reveal the nature of the chemical reactions? Can peak shape act as a fingerprint, giving a unique description of the underlying chemistry?
  • If peak shape acts as a fingerprint under only a narrow range of chemical parameters, are there ways to modify the experimental protocol so that these parameters can be achieved? For example, if the current experimental data falls outside of the domain where peak shape gives a unique description of the chemical reaction, can new experiments be done, perhaps by modifying PH or temperature, so that peak shape reveals more information?

David Goulet
(2015, 2016 - senior investigator)
David Goulet Professor Goulet received his B.S. (1999) from Caltech, his M.S. (2002) from the Courant Institute, and his Ph.D. (2006) from Caltech. He held a post-doctoral position and ran a small business before coming to Rose in 2011. He is interested in mathematical biology, specifically deterministic and stochastic modeling of biochemistry and synthetic biological networks. You can find out more about Professor Goulet through his Home Page.

High Dimensional Data Analysis (Mark Inlow, Eric Reyes, 2013, 2015)

Modern research and predictive analytics efforts increasingly require the analysis of huge, high-dimensional 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 high-dimenstional 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, high-dimensional 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 high-dimensional 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 left-spherical 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.

Mark Inlow
(2013, 2015 - senior investigator)
Mark Inlow Here goes stuff about Professor Inlow.

Eric Reyes
(2013, 2015 - senior investigator)
Eric Reyes Here goes stuff about Professor Reyes.

This document was last modified: 11/26/17
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