Rose-Hulman Math REU

Research Areas and Faculty Mentors of the Rose-Hulman REU

Each of the program faculty, Kurt Bryan, David Finn, and Joshua Holden, has a different area of interest. The faculty participate on a rotating basis with two faculty each summer. The projects in a given summer will depend on which two of the faculty are participating. For the summer of 2009 the research areas will be inverse problems and computational number theory, with Professors Bryan and Holden, respectively.

Faculty Mentor	Research Area
Kurt Bryan	Inverse Problems
David Finn	Geometric Analysis *
Josh Holden	Computational Number Theory *

* = areas of research for the current year.

Inverse Problems (Kurt Bryan, 2002,2003,2004,2005,2006,2008,2009)

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

What is a reasonable mathematical model for such defects (including the boundary conditions that hold on the defect)?
What can one say (theoretically) about what can and cannot be determined with these methods?
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 2002-2004 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 kurt.bryan@rose-hulman.edu
(2002-2003 senior investigator, 2004-2009 - program director, on leave 2007)

Professor 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 1990. 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 2005. You can find out more about Professor Bryan through his Home Page.

Geometric Analysis (David Finn, 2006, 2007, 2008)

David Finn's research concerns the applications of partial differential equations and the calculus of variations to problems in differential geometry, i.e. geometric analysis. The goal of the project to be investigated is modelling the shape of a drop sugar cookie. The basic heurestic model to be investigated is that the shape of a sugar cookie (homogeneous cookie dough) is given as a surface with prescribed mean curvature. The reasoning behind the heurestic model is that the cookie dough becomes a liquid when it is heated and attains its equilibrium shape as a liquid before it solidifies later in the baking process.

Drop of Cookie Dough
(becoming liquid)

Cookies Baking
(attaining shape)

Baked Cookie
(cooling to final shape)

After the cookie is baked, a natural question is: Can one recover the shape of the cookie from knowing the wetted domain (the region on the cookie sheet the cookie is in contract with) and properties of the cookie dough and the cookie drop (stiffness, density, volume)?

Some questions to be investigated are:

How does the shape of the cookie depend on the shape of the wetted domain?,
How does the shape of the cookie depend on the physical parameters of the dough and the drop?
Sometimes when baking cookies, the cookie dough runs together forming a double bubble cookie with a wetted domain as showm below. The cookie shape is then no longer smooth, as in the picture below. How does the shape of the cookie vary in this situation?

A lot of the investigation will be done by numerical computation of solutions to generate conjuctures, No prior knowledge of partial differential equations, differential geometry, and numerical analysis is necessary. Some exposure to ordinary differential equations and/or basic analysis (advanced calculus/vector calculus) is extremely helpful. For more information see the Shape of a Cookie Page

David Finn david.finn@rose-hulman.edu
(2006 - senior investigator, 2007-2008 co-director)

Professor David Finn received his B.S. (1989) from Stevens Institute of Technology and his Ph.D. (1995) from Northeastern University. He taught at both Merrimack College in Massachusetts and Goucher College in Maryland before coming to Rose-Hulman in 1999. He is interested in nonlinear partial differential equations, especially applications to differential geometry, mathematical physics, and image processing, as well as geometric modelling and the mathematics of (bicycle) tire tracks. Professor Finn has run an NSF-funded project, "Motivating Geometry through Computation and Visualization" during the summers of 2002-2005, to develop a course in Geometric Modelling, and mentor undergrads who are developing interactive web-based materials for such a course. You can find out more about Professor Finn through his Home Page.

Computational Number Theory (Joshua Holden, 2007,2009)

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 joshua.holden@rose-hulman.edu
(2007, 2009 - senior investigator)

Professor 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.