Program Outline: Approximately 10 teams of two to four students each will work collaboratively with this summer's program faculty.   The Industrial project will be directed by L. Ives (Calabazas Creek Research Inc.) and Hien Tran, Biomathematics projects will be directed by Professors Mette Olufsen, Hien Tran, Mansoor Haider, James Selgrade, Alun Lloyd and Sharon Lubkin, Material Science projects will be supervised by Pierre Gremaud, Financial Math projects by Tao Pang, Symbolic Computation by Erich Kaltofen, Data Mining by Carl Meyer and Computer Vision by Irina Kogan and Dmitri Zenkov.

Program Objectives: This REU will expose undergraduate students to challenging and exciting real-world problems arising in industrial and government laboratory research. This experience benefits its participants in ways not possible with traditional coursework. REU students will learn valuable techniques of applied mathematics and how to work in teams to accomplish common objectives. The fact that there may be no ready-made structures or body of results to cover the initial problem formulation only compounds the excitement. Students will see the value placed on their work by those outside the University, in the "real world" of business and industry. In addition, students will gain practice communicating about mathematics. Each student will be required to give frequent progress reports, both oral and written, and to write a paper and make a poster presentation.

Female, minority and physically disabled students are encouraged to apply.

Project: Cell rearrangements in tissues
Advisor: Sharon Lubkin
Developmental biology is the study of how we go from a spherically symmetric egg to a complex dynamic structure with dozens of differentiated cell types in spatially complex and useful arrangements, such as lungs, hands, and brains. Part of the mechanism of development is cell rearrangements, which are due to forces generated by the cell interacting with forces applied to the cell. We will model the rearrangements of cells in morphogenesis using a stochastic simulation technique. Some computer programming experience is necessary.

Project: Information Retrieval and Web Search
Advisor: Carl Meyer
The project concerns the mathematical technology involved in building various kinds of information retrieval systems and search engines. The material will include classical methods such as latent semantic indexing systems along with various document clustering schemes and will proceed through more recent techniques based on nonnegative matrix factorizations. A primary facet involves studying web search with particular emphasis on the Google technology. It is hoped that upon successful completion of the course students will have amassed enough knowledge to build their own search engines.

The material will be drawn from a variety of recent research papers in conjunction with the two primary references: "Google's PageRank and Beyond: The Science of Search Engine Rankings" by A. N. Langville and C. D. Meyer http://pagerankandbeyond.com/ and "Understanding Search Engines" (Second Edition) by M. W. berry and Murray Browne
http://www.ec-securehost.com/SIAM/SE17.html.

Project: Mathematical Models in Orthopaedic Tissue Engineering
Advisor: Haider

In this project the REU group will formulate, solve, analyze and refine mathematical models relevant to tissue engineering applications for cartilagenous soft tissues. In such applications, natural or bioengineered scaffolds are seeded with cells which then synthesize extracellular matrix constituents in a manner that is highly dependent on time-varying properties of the local cellular environment. The group will develop and analyze mathematical models of an evolving gel-tissue construct with the goal of optimizing its functional properties.

Prerequisites: This project will be well suited to undergraduate students with a solid foundation in calculus, some experience in differential equations or numerical analysis, and with interests in mathematical modeling and/or mathematical biology.

Project: Edge Detection by Multi-Dimensional Wavelets
Advisor: Pierre Gremaud
Following the spectacular applications of the Fast Fourier Transform, many attempts have been made to adapt the efficient methods of one dimensional signal processing to multidimensional data. In the last twenty years, wavelet based algorithms have emerged as the most successful approach in that respect because (i) they provide optimal representations of one-dimensional data and (ii) they are amenable to fast transforms. For instance, wavelets are used in the new FBI fingerprint database ⁴ and as well as in JPEG-2000 ³ , the new standard for image compression.

In spite of their remarkable success, multidimensional wavelets are far from being optimal. Even though they outperform Fourier methods, they fail to capture intrinsic geometrical features of multidimensional phenomena. For instance, while tensor products of 1-D wavelets efficiently capture point discontinuities, they do poorly at dealing with distributed discontinuities such as edges, shock waves and moving fronts. The directional information is lost and consequently many coefficients have to be computed to ensure minimal accuracy requirements.

The focus of the project will be the application of new multidimensional wavelet techniques to edge detection problems for images and numerical data. As part of the project, the students will
*familiarize themselves with some basic tools in scientific computing such as the Fast Fourier Transform, wavelet transforms and their applications; existing software packages and routines such as those found in MATLAB will be used;
*learn about state-of-the-art multidimensional wavelets techniques such as shearlets ¹ and curvelets²
*be exposed to existing techniques and methods in edge detection;
*explore the application of advanced wavelet techniques to edge detection.

References:
⁴ C.M. Brislawn, Fingerprints go digital, Notices Amer. Math. Soc. (1995), 1278-1283.
³ C.M. Brislawn and M.D. Quirk, Image compression with the JPEG-2000 standard, in Encyclopedia of Optical Engineering, R. G. Driggers (Ed.),New York: Marcel Dekker, 2003, 780-785.
² E.J. Candes and D. L. Donoho, New tight frames of curvelets and optimal representations
of objects with piecewise C² singularities, Comm. Pure and Appl. Math. (2004),216--266.
¹ G. Easley, D. Labate and W. Lim, Sparse Directional Image Representations using the Discrete Shearlet Transform, pp. 26, preprint (2006).

Project: Design of Electron Devices using Computer Optimization
Advisors: Lawrence Ives (Calabazas Creek Research, Inc.)
Hien Tran (NC State University)

The performance of vacuum electron devices producing radio frequency (RF) power depends on the operational characteristics of several subcomponents. For example, the efficiency of a traveling
wave tube depends on the quality of an electron beam, the parameters of the RF circuit, the energy recovery characteristics of the spent beam collector, and the matching characteristics of the output waveguide system. Each subcomponent is related to one or more of the others in determining the overall performance of the device. Currently, technology exists to optimize each component's
performance in isolation from the others. Unfortunately, high performance in some components, such as the circuit, leads to reduced performance in others, such as the spent beam collector.
Therefore, it is proposed to determine a mechanism for optimizing the performance of various components based on the total performance of the device rather than the performance of single
components. A starting point would be to optimize both the circuit and collector together. Following a successful demonstration of this capability, the design of the output waveguide system and electron gun could be added. Dr. Ives will be visiting NCSU several times during the REU program to discuss the physical aspects of the problem. 

Project: Modeling Toxicant Exposure in Different Species
Advisors: Marina Evans (U.S. Environmental Protection Agency (EPA))
Karen Yokley (U.S. Environmental Protection Agency (EPA))

The harmful effects of chemical exposure can be evaluated through the aid of mathematical modeling.  Physiologically based pharmacokinetic (PBPK) models involve describing the amounts of toxic chemicals in the regions body of a rat, human, etc., following exposure, and knowing how
much of the chemical is present in a susceptible tissue can help in assessing the health risk.   A specific PBPK ordinary differential equation model for rats will be considered and updated for humans, and how to change parameters from species to species will be investigated through sensitivity analyses.  Additional study may include fitting a model to available data, modeling data from exposure to a mixture of chemicals, and/or variation of physiological parameters in order to
describe specific human subpopulations.   Expectations of students include a beginning knowledge of ordinary differential equations and a willingness to learn computational methods.

Project: Modeling Blood Flow and Pressure Regulation during Postural Change
Advisors: Mette Olufsen and Hien Tran

During posture change from sitting to standing blood is pooled in the legs as a result of increased gravitational force. As a result, blood pressure in the trunk and upper extremities drop. To compensate for this drop in pressure, the body experiences parasympathetic withdrawal, while sympathetic activity is increased. The response to parasympathetic withdrawal is an increase in heart rate and a small increase in cardiac contractility. The responses to the increased sympathetic activity
are increases in heart rate, cardiac contractility, and systemic resistance. Simultaneously, cerebral autoregulation decreases cerebrovascular resistance to maintain constant blood flow to the brain. The interaction between the two types of regulation is not well understood and cannot easily be investigated using conventional clinical methods. A basic model that can predict blood flow and blood pressure has already been developed. The goal of our research is to refine and to further improve the model by including effects of changes in parasympathetic and sympathetic activity as well as effect of cerebral autoregulation, and to validate the model against experimental data.

Project: Mathematical Modeling of Infectious Diseases
Advisor: Alun Lloyd

Mathematical models have contributed much to our understanding of the transmission dynamics of infectious diseases and attempts to control such infections. Models provided many valuable insights
during the recent SARS epidemic, and have aided planning in the bioterrorism and homeland security arena. One question of particular interest involves an examination of the conflicts that arise when the use of a vaccine has some risk. If enough people are vaccinated, the population benefits from so-called herd immunity to the infection. From an individual's standpoint, their risk is then minimized if sufficiently many other people are vaccinated, but not themselves. Clearly, this selfish strategy cannot work if it is adopted by all of the members of a community. Such conflicts have been of great interest in recent years with the threat of the reintroduction of smallpox by terrorists. These issues become heightened in populations in which there are groups of individuals who are more susceptible to the infection and groups who are more prone to any negative side-effect of vaccination, particularly if there is overlap between these groups.

Project: Posets of Involutions in Weyl Groups
Advisors: Loek Helminck (NC State University)
Ruth Haas (Smith College)

Weyl groups generalize permutation groups and are useful in many branches of mathematics. Understanding which elements are in the set of involutions is important for many applications
to symmetric spaces, representation theory, network problems, etc. The elements of the set of involutions are characterized by sequences in the set of reflections which induce  a poset relationship between them called the Bruhat poset.  There exists an  algorithm to compute this poset which is inefficient due to the fact that in Weyl group packages elements are expressed as reduced words in the generators and there can be many words representing the same element.   A new data structure which gives a unique representation of the  elements  of  Weyl groups has recently been introduced.  To use this data structure for Weyl group computations all old algorithms for Weyl Groups  must be rewritten. This requires proving many results about  this data structure, since there is no simple direct translation. Early results indicate that this data structure is at least 100 times faster to compute the Bruhat poset as above. Many related theoretical questions about the structure of this poset need further study as well.

Project: Geometric Invariants in Computer Image Recognition
Advisor: Irina Kogan

I this project students will apply geometric and algebraic methods to the problems of object recognition in computer vision. Invariants under the actions of the Euclidean, affine and projective groups are widely used in image processing and computer vision. Differential invariants, such as Euclidean curvature and torsion for space curves, are the most classical. The practical utilization of differential invariants is limited, however, due to their high sensitivity to noise. This motivates interest in other invariants, in particular various types of integral invariants. Since integration diminishes the effects of noise, integral invariants have advantage in applications. Their properties, however, are much less studied. The group will study the properties of integral invariants, their numerical approximations, and develop new methods for object recognition.

Prerequisites:  solid foundation in multivariable calculus, some experience with abstract algebra (mainly group theory) and some experience with computer software, such as Maple, Mathematica or Matlab.  

References: 

J.  Sato and R. Cipolla, “Affine integral invariants for extracting symmetry axes,”  Image and Vision Computing 15, pp. 627--635, 1997.
C. Hann and C. E. Hickerman,  “Projective curvature and integral invariants,” Acta applicandae mathematicae,  74, pp. 177--193, 2002.
S.Feng, I. A Kogan, H. Krim, “Integral Invariants for Curves in 3D: Inductive Approach”, to appear in the proceedings of ISPIE Conference on "Visual Communications and Image Processing” 2007.