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 Program, 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 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 student’s advisor and the Math Honors Director) may be done with faculty in other departments.
My current interest is focused on “supermathematics’”. This involves so called supernumbers. Some such numbers satisfy xy=yx (even supernumbers) while others satisfy xy= – yx (odd supernumbers). Vectors of such supernumbers then have certain of their components of one type supernumber and others of the other type. So a supervector space has dimension (p,q) reflecting the two types of components. All of linear algebra is transformed. Calculus also is transformed. For example, the second derivative of a function of odd variables is always zero while derivatives of functions of even variables satisfy familiar rules of ordinary calculus. The subject is about 40 years old and is of great interest to physicists who are interested in super electromagnetism, super gravity, super gauge theory, super quantum theory, ect. The subject was developed and is being developed by both mathematicians and physicists. There are enumerable projects at an undergraduate level.
Prerequisites: MA414 or MA555.
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.
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.
Geometric concepts, such as symmetry and invariance, find applications in many areas of mathematics, engineering and science. I am particular interested in applications to differential equations and computer vision. The prerequisites is MA 425 and MA 405. Experience with or willingness to learn programming in Maple, Mathematica or Matlab will be helpful.
Singular perturbation technique used on traveling wave problems in various areas.
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.
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.
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 MA341, MA401 and some programming efficiency in Matlab or a higher level language such as fortran 77/fortran 90 or C/C++.
I would be happy to advise students in reading courses, projects, etc. in the areas of analysis, differential equations and related topics.
Prerequisites: MA425, MA426 and a little differential equations.
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.
Applied Algebra material– coding, cryptography, Markov Chains, graphs, counting (Polya Theory), least squares.
Prerequisites: MA405 and MA407.
(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 MA341, MA401, some programming efficiency in MATLAB or a higher level language such as FORTRAN or C.