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, algebraic graph theory)
• Cara Spencer (Summer 2000, frieze groups and crystallographic groups)
• 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).
• Dan Appelbaum (Spring 2002, geometry of curves)
• Janelle Brinkeley (Spring 2004, tilings of the plane)
• Kathleen Field (Spring 2005, graph theoretic measurements on vascular networks)
• Adam Bezinovich (Spring 2006, Sabermetrics)
• Cassie Remmert (Fall 2008, theory of voting)
• Heather Beal (Spring 2009, computer assisted medical image analysis)
• Clayton Davis (Spring 2009, computational analysis of bat echolocation calls)
• Calvin Johnson (ongoing, geometry of curves and projections)

UMAP Modules

Some Capstone Project Ideas

Below is a list of ideas I've had for a math major's Capstone project. Each idea interests me enough to make me want to find time to help a student learn about the topic in question.

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. Classification trees (CT) are a statistical technique used to cluster data and find patterns in data. Matlab and R have implemented CT techniques, and I have used them to some extent with data I have on bat echolocation calls. A student interested in computational statistics, programming in a mathematical and biological context, or Baysian statistics might find this topic a very interesting one for a capstone. There may also be an opportunity for a publication.
   • Classification and Regression Trees, by Leo Breiman, Jerome Friedman, Charles J. Stone, R.A. Olshen. Chapman & Hall/CRC; 1 edition. January 1, 1984.
2. What can algebraic graph theory tell us about the two- or three-dimensional structure of vascular networks? This relates to an interdisciplinary project I've been involved in which a colleague at A.T.Still University of Health Sciences for several years. Lots of good stuff, here, for an enterprising mathematics major to play with.
3. Investigate the mathematics involved in the burgeoning new field of data-mining. (Advanced Calculus, Computer Programming, some Geometry)
   • Geometric Data Analysis, by Kirby.
4. 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).
5. 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)
6. 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 edition (August 1992). Cambridge University Press.
   • John McCleary, Geometry from a Differentiable Viewpoint. (1997) Cambridge University Press.
7. Investigate the ways of classifying curve singularities 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.
8. 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.
9. Investigate and present the Gauss-Bonnet Theorem. (Geometry, Analysis)
   • Barrett O'Neill, Elementary Differential Geometry. (1966) Academic Press.
10. 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.)
11. 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).
12. What are the mathematics behind the medical imaging technique called tomography? (Analysis, Physics)
    • [I don't know of a good reference for this.]
13. 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.)
14. 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.
15. 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.
16. 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.

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.