Ideas for Capstone Projects

Note: The deadline for completing a final draft of your Capstone has been changed starting in the Fall of 2003. Ask your academic advisor for details, or look at this official description of the Mathematics Capstone project: [here] (PDF file). An HTML version of this document should be available via the division's web page under the student section.

Students for whom I have served as supervisor are:

• David Unger (Fall 1999, Markov Chains)
• Angie Miller (Spring 2000, Incidence and Adjacency: Some Algebraic Graph Theory)
• Cara Spencer (Summer 2000, Frieze Groups and Crystallographic Groups, in progress)
• Corrie Christensen (Spring 2001, Summer Program for Women in Mathematics: A Capstone Experience)
• Tim Hudson (Spring 2001, The Classification Theorem for Compact Two-Dimensional Manifolds)
• Margaret Clark (Spring 2001, Continuity: Understanding the Development of the Calculus)
• Patrick Hill (Unknown, Untitled, game theory in the commercial strategy game, Diplomacy).

UMAP Modules

Some Capstone Project Ideas

Below is a list of ideas I've had for a math major's Capstone project. Some are generic ideas (with few details), and others are more specific.

Please do not assume that because I am listing these projects, that I am the best mathematics faculty member to supervise these projects.

These projects are separated into two groups, Summer Research/Internship, and On-Campus Independent Study. The latter group consists of ideas for students who intend to satisfy their Capstone requirement through independent investigation of some mathematical idea. Each idea is followed by a parenthetical indication of the mathematical field I suspect is mostly involved. The former group contains ideas for ways that students can get involved in group related activities that will satisfy most of the Capstone requirement (and probably make them money in the meantime).

On-Campus Independent Study

1. NEW, 9/2003 (Prof. Don Bindner suggests this and is eager to mentor a student on this topic) Learn something about the mathematics involved in the choosing of good computer passwords. [JM: I suppose this also will involve understanding some of the mathematics used in cracking passwords.... Use the knowledge for good, not evil.]
2. Read the paper The Sound of Trigonometry. I'm sure there's a Capstone topic lurking here (but this information appears to be gear toward a non-mathematics major.)
3. Investigate Euler's Constant, "gamma", and the various methods there are for calculating approximations to Euler's Constant. Is the number rational or irrational? Transcendental? What is knowns about the number? Why was it of interest to Euler and others? What methods are there to compute Euler's Constant? (Elementary Real Analysis, a.k.a Advanced Calculus.)
   • A Radical Approach to Real Analysis, by David Bessoud. See section 2.3.
4. Investigate the various methods there are for calculating approximations to Pi (you know, 3.14159...). There are methods of the Greeks, methods by Newton and his contemporaries (including Wallis) and methods of Ramanujan. Of course, there are many others (and perhaps more modern methods, I don't know). (I don't know what techniques this topic would require.)
   • A Radical Approach to Real Analysis, by David Bessoud. See section 2.2 and Appendix A.1.
5. Investigate the mathematics involved in the burgeoning new field of data-mining. (Advanced Calculus, Computer Programming, some Geometry)
   • Geometric Data Analysis, by Kirby.
6. Compare calculus described using limits to calculus desribed using the infinitessimal calculus. (Algebraic Structures, Analysis)
   • Inroductory Calculus from the Viewpoint of Non-Standard Analysis-Derivative of Sine and Cosine, by Jack L. Uretsky.
7. Investigate the theory and applications related to splines (cubic splines, B-splines, thin plate, etc.) and, if you're ambitious NURBS. All this requires is Calculus II and a dose of mathematical maturity (which is gained in Math 357, 461, and other advanced intermediate math courses). This topic could easily evolve into a Undergraduate Student Research Project (see below).
8. What is an algebraic variety? What is a projective variety?(Algebra, Geometry)
   • David Cox, Donal O'Shea, and John B. Little, Ideals, Varieties, and Algorithms : An Introduction to Computational Algebraic Geometry and Commutative Algebra. 2nd Edition (Nov 1996) Springer-Verlag (Undergraduate Texts in Mathematics)
9. Investigate the relationship between a plane curve, its family of parallel curves, and the plane curve's evolute. This is a project that can be carried out by first making several computer experiments using Mathematica, from which you can make conjectures that you can then prove or find counterexamples. (Calculus III, Advanced Calculus)
   • J.W.Bruce, P.J. Giblin. Curves and Singularities : A Geometrical Introduction to Singularity Theory. 2nd editiion (August 1992). Cambridge University Press.
   • John McCleary, Geometry from a Differentiable Viewpoint. (1997) Cambridge University Press.
10. Investigate the ways of classifying curve sigularities using groups. In particular, learn about right-equivalence of functions (and, perhaps, left-equivalence) and how these determine equivalence classes of functions that can be characterized by singularities. It's also interesting to compare the relative "sizes" of the difference equivalence classes using c0-dimensionality in the space of all function. (Geometry, Algebraic Structures, Advanced Calculus)
    • J.W.Bruce, P.J. Giblin. Curves and Singularities : A Geometrical Introduction to Singularity Theory. 2nd editiion (August 1992). Cambridge University Press.
    • John McCleary, Geometry from a Differentiable Viewpoint. (1997) Cambridge University Press.
11. What is a manifold? (Geometry, Analysis)
    • Michael Spivak, Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus. (June 1965). Perseus Press.
    • Barrett O'Neill, Elementary Differential Geometry. (1966) Academic Press.
12. Investigate and present the Gauss-Bonnet Theorem. (Geometry, Analysis)
    • Barrett O'Neill, Elementary Differential Geometry. (1966) Academic Press.
13. Why are there so many different formats for digital images? What are some of their differences and similarities? (Analysis) (I don't know of a good reference for this.)
14. How can you assign a polynomial to a knot, and what information does it give to you? (Topology, Analysis)
    • David Farmer, Theodore Stanford, Knots and Surfaces: A Guide to Discovering Mathematics. (1996) American Mathematical Society.
15. What does it mean for two curves to be transverse to one another? Can this happen in three dimensional space? How does transversality differ from orthogonality? What sort of "stability" does transversality bring into the mix that orthogonality does not? (Geometry, Analysis)
    • Morris W. Hirsch, Differential Topology. (1976) SPringer-Varlag (Graduate Texts in Mathematics).
16. What are the mathematics behind the medical imaging technique called tomography? (Analysis, Physics)
    • [I don't know of a good reference for this.]
17. How can you tell if two matrices are close together? What if anything does this have to do with deciding if two geometric objects being rendered by a computer have collided? (Geometry, Algebra) (Off the top of my head, I don't know of a good reference for this, but it's a well studied question.)
18. What is the Theory of Voting? how can we compare two voting methods (e.g. plurality without run-off and plurality with run-off) and decide which is more fair? What is the Borda method of voting and why is it the most fair method? Why doesn't the united State employ the Borda method? how would it change the landscape of American politics? Is there anybody is the world that regularly uses the Borda method? (Discrete)
    • Phillip D. Straffin, Jr. Topics in the Theory of Voting (1980) Birkhauser.
    • Peter Tannenbaum, Robet ARnold. Excursions in Modern Mathematics. (1995) Prentice Hall.
19. Investigate the Morse Theorem and describe what it says about any twice differentiable function. What is Morse Theory? (Analysis)
    • Jerrold E.Marsden, Michael J. Hoffman, Elementary Classical Analysis. (1993) Freeman.
20. What are differential forms? Should Calculus I students learn about differential forms before they take derivative and antiderivatives? Why would I ask such a question? (Analysis)
    • Barrett O'Neill, Elementary Differential Geometry. (1966) Academic Press.
    • Michael Spivak, Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus. (June 1965). Perseus Press.
21. Read and work through Porteous's paper The Intelligence of curves (see my research bibliography). Flesh out the details, create some computer images (perhaps write a Mathematica notebook that will compute the evolute of a given curve), and/or create a model/sculpture of a given curve's focal surface/curve.

The following projects have direct application to research being carried out in the Physics department and would be ideal for a math/physics double major, or for a math major interested in a project in applied math.

1. (Original Research Problem) Solve a system (which would be given to you) of coupled first-order differential equations in order to model population transfer among quantum energy levels in a molecular gas, working within the constraints of the physical system under study.  The model would ultimately be used to extract population transfer rates from experimental data.
2. . Explore the mathematics describing electromagnetic waves confined to a cavity - a boundary value problem - and apply the equations to microwave radiation within photonic crystal cavities.  (Photonic crystals are a unique material composed of a periodic array of alternating dielectric materials, which have the property of prohibiting the transmission of electromagnetic radiation over wide frequency ranges.)

Summer Research/Internship

Please note that the below summer activities do not, in and of themselves, satisfy the Capstone requirement. All students are required to submit a proposal for approval describing their summer experience, and all students are required to present their "experience" orally and in written form the following school year.

1. (VIGRE dummer research in Iowa) I just received this email from the Division Head concerning oportunities for undergraduate research in a way that appears to integrate statistics with other scientific topics. You read the email and be the judge as to whether or not this interests you. Remember that you might be able to sue this experience as the centerpiece for a mathematics Capstone project. (12 Jan: I don't know if this program will runduring the summer of 2003, but if you are interested in participating, talk with me or talk with Prof. Dean DeCock.)
2. Have an idea for a research project of your own, talk to a faculty member who can sponsor your for a research stipend of $2000. You will present your work at Truman's Undergraduate Research Symposium during the Spring semester.
3. Participate in an off-campus specialized summer research experience. For example, the Summer Program for Women in Mathematics, Summer Undergraduate Applied Mathematics Institute, and the MASS program at Penn State. Web links to information about these programs can be found in the Student section of the MAA's home page, www.maa.org.
4. I have some data that I acquired in a Industrial Mathematical Modeling Workshop, and I would like to see the data (data points on two dimensional slices of a nose) to be assembled into a three dimensional data set. To do this, one would need to know how to use splines to interpolate surfaces through the data points. Which type of spline? I don't know: thin plate, B-spline, cubic? We would rely heavily on Matlab, so some programming skills would be helpful; expect to learn some during the course of the project if you haven't already.