Department of Mathematics - NC State University

Projects and Mentors: The following math faculty have volunteered to mentor MA 491 independent study and research projects and have outlined some possible projects for honors students. Since MA 491 is one of the courses that can be used to satisfy the writing requirement for NCSU's General Education Requirements, a student taking MA 491 is expected to write a paper summarizing the results of his/her work. Students may also wish to present their results at the Undergraduate Research Symposium sponsored by Sigma Xi and held in early April. This list is prepared as an aid for students considering possible MA 491 projects but students should not consider themselves limited to working with these faculty nor on these projects. In particular, projects (with the approval of the studentís advisor and the Honors Director) may be done with faculty in other departments.

J. Franke:
(a) Something dealing with Chaos for students who have taken MA 537.
(b) Learn about the mathematics behind Fractals and design something using them.
Prerequisite: MA 426.

R. Fulp: A study of the Òpath-spaceÓ of various surfaces such as spheres, cylinders, tori, etc.
In particular, what algebraic structure do these spaces have in case you do not require
homotopy-equivalence of paths, as you do when you define a group operation on the
fundamental group. Prerequisites: MA 414 or MA 555.

P. Gremaud: Numerical methods for Solid Mechanics problems.
I have several projects that deal with the resolution of equations modeling plasticity problems and problems involving granular materials. Those projects involve
programming using either MATLAB or another high level language. Modeling
and mathematical issues will also be discussed. Prerequisites:
differential equations at the level of MA341 or MA401.

M. Haider:
(a) Modeling cell and matrix mechanics in orthopaedic soft tissues. Prerequisites: MA341
(b) Numerical simulation of wave propagation in random media. Prerequisites: MA341 and some programming experience.

M. Kang: Probability has a relatively short history among various areas of mathematics, but more and more nowadays there are interesting interfaces between probability and other areas of mathematics. I would like to propose a few projects that lie on interface between probability and PDE that have natural connections to statistical physics, financial mathematics, mathematical biology, combinatorics and other related areas. The projects will be chosen based on the student's interest, strength in mathematics and their ambition as well as time constraint. The mathematical background for the projects include (preferably, not absolutely) basic analysis courses (rigorous), basic probability course, some linear algebra or abstract algebra and a curious mind.

C. T. Kelley: I have a project on optimization and application. Our group includes Ph.D. students and undergraduate research assistants. We do fundamental algorithmic research, parallel computing, and work directly with industry, national laboratories, and other academic disciplines on real-world problems (automotive engineering, gas transmission pipeline optimization, semiconductor modeling, calibration of instruments). My NSF funding supports one honors student as an undergraduate RA. The two undergraduates who have worked on this project so far have worked on parallel implementation, algorithm design, and on applications. Both undergraduates will be co-authors on journal papers. The web page for this project is http://www4.ncsu.edu/~ctk/undergrad.html . Prerequisites: MA 405 and MA 580.

I. Kogan: I can offer a project in classical invariant theory -- a study of intrinsic, or geometric, properties of multivariable polynomials. This project provides a natural, hands on introduction to algebraic geometry and computational algebra. Prerequisites MA 405 (Introduction to Linear Algebra and Matrices) and MA 407 (Introduction to Modern Algebra for Mathematics Majors). For examples, click here.

X. B. Lin: Singular perturbation technique used on traveling wave problems in various areas.

S. Lubkin: Unclogging the carpool line twice a day, 180 days a year, parents and children sit in a carpool line which is hopelessly jammed. This wastes time and energy, ties up traffic in the neighborhood, and contributes to pollution. In this project, we will study traffic flow models and collect data on local carpool flows. Using a variety of approaches, we will analyze the models and data, test scenarios for improvement, and make recommendations to the local school system. The math required is minimal, but some programming experience would be helpful. The best qualification is an ability to think analytically and creatively. This project is suitable for a team.

K. Misra: Partition identities go back to the eighteenth century mathematician Euler. These identities form an important area of research in number theory even today because of its importance in other areas of mathematics and mathematical physics. One such area of application is the representation theory of Lie algebras. Projects in this direction may be appropriate for students interested in algebra and number theory. Prerequisites: MA 405 and MA 407 and programming skill in maple or similar software.

M. Olufsen: Mathematical Biology projects:
(a) Modeling control of blood flow in the brain.
(b) Modeling wave-propagation of the pulse wave in the arterial system.
Prerequisites: knowledge of differential equations MA 341, MA 401
and some programming efficiency in Matlab or a higher level language such as
fortran 77/fortran 90 or C/C++

J. Rodriquez: I would be happy to advise students in reading courses, projects, etc. in the
areas of analysis, differential equations and related topics. Prerequisites: MA 425, MA 426 and
a little differential equations.

J. Selgrade: Analyzing the Behavior of Nonlinear Dynamical Systems Discrete and continuous nonlinear dynamical systems exhibit surprising and interesting behavior asssociated with bifurcation, chaos and fractals. Such systems arise from models in disciplines as diverse as meteorology,
medicine, and economics. Investigating this behavior requires graphical, analytical and numerical tools.
Prerequisites: MA 405, MA 532, and MA 537

M. Shearer: Traveling wave solutions of partial differential equations. This involves analysis
and use of an ordinary differential equations computer package. The objective is to explore
the existence of traveling waves that approximate shock wave solutions of nonlinear partial
differential equations. The equations model fluid flow in porous media, and other
applications in mechanics could be discussed depending on the interests of the student.
Prerequisite: MA 341.

E. Stitzinger: Applied Algebra material-- coding, cryptography, Markov Chains, graphs, counting (Polya Theory), least squares.
Prerequisites: MA 405 and MA 407.

H. T. Tran:
(a) Population Dynamics Models based on physiological age or size.
(b) Modeling and control of mechanical systems including vibrational analysis.
(c) Modeling of physiological systems such as the respiratory system and the cardiovascular
system. Prerequisites: differential equations such as MA 341, MA 401, some programming efficiency in MATLAB or a higher level language such as FORTRAN or C.

R. E. White:
(a) Parallel Preconditioners for Three Dimensional Problems:
Preconditioners, M, for the conjugate gradient and GMRES iterative methods are important in the solution of large algebraic systems, Ax = d. This topic will focus on parallel versions of preconditioners, which are related to the least squares models for minimizing the matrix residual R = I - AM. Parallel implementation, using MPI, for the choice of M based upon (i) selected components of M, (ii) optimal multisplittings and (iii) Krylov "matrix" approximation will be examined. MA 580 and MA/CSC 583 would be a good preparation for this work.
(b) Image Restoration and Quasilinear Differential Equations:
Image restoration modesl are often based on attempts to minimize the toatal variation of an image. The classical approach is to minimze a regularized version of the total variation so as to generate a quasilinear partial differential equation. Numerous perturbations of this have been introduced to reduce different types of noise associated with the image. This project will focus on the efficient numerical solution of the resulting quasilinear equation. MA 402 and MA 580 would be a good preparation for this work.