Ideas for Capstone Projects

To graduate with a mathematics major from Truman State University you must complete a Senior Capstone Integrating Experience. We math faculty and students just call it "the Capstone". If you're having trouble finding a topic to study, this page might help. It's just a jumble of staring points for a topic search.
If you are looking for a Capstone project, here's some info that might help.

Choosing a Topic (General)

Students can find the process of choosing a topic for their Capstone project somewhat challenging. The source of the challenge isn't the difficulty of the material. The challenge flows form the sheer volume of options to choose from. Identifying viable options and then narrowing them down to a single choice is often a student's first challenge. This section contains my advice on how to choose a topic to study and ends with a piece of advice on what to do once you've chosen your topic.

Talk with faculty about projects they can suggest.
If there are faculty who you like or admire, go talk with them about potential Capstone projects they might know of and be able to mentor you through. These faculty members need not belong to the mathematics department, but they ought to be able to lead you through a project that had significant mathematical content. Note that research-active faculty often have Capstone topics/projects that relate to their research interests, and such topics might lead to opportunities for publication as a co-author with that faculty member. A list of projects that interest me appear below.
Look at textbooks you've had for courses in your (math) major.
The Capstone for the Truman mathematics major requires you to learn material that is new to you. One great way to find potential material, that is interesting and not too far removed from what you have studied, is open up a textbook for on of your courses and explore sections/topics that you didn't cover in your course.
Attend departmental colloquia or seminar series.
Most departments have a regular seminar series which bring in professionals from other schools or companies to talk about important or cutting-edge ideas. Use topics from these talks as candidate Capstone topics. And if you time things right, you can have conversations with speakers about opportunities for Capstone topics that relate to their talk topic.
Consult UMAP Modules
Our library has a collection of books that contain meaty bits of mathematics in application. The books are called UMAP Modules and have Pickler call number QA 11 A1 U5 browse through their introductions to see if any of the teaching modules deals with or relates to a topic that interests you.
Undergraduate Research Experiences
This take a bit more planning, but the payoff is considerable. If you secure a summer or academic year research experiences with a faculty member (n.b., summer research experiences are often at other schools or Universities), you can use that experience as the basis of their Capstone project. Most summer research experiences in science or mathematics will pay you a generous stipend, so this is a way to get paid for your Capstone project. You'll still need to write up a paper on your topic and give an oral presentation on your topic, but these are generally the most manageable aspects of Truman's Capstone. If you did research with a faculty member from another institution, you should identify a Truman faculty member as your local Capstone mentor to help you negotiate the Capstone requirements.

Capstone Project/Topic Ideas

Below is a list of topics of interest to me that would be suitable for a Capstone project in mathematics at Truman. Each idea interests me enough to make time to supervise a student that is curious about topic. Each idea is followed by a parenthetical indication of the mathematical field I suspect is mostly involved.

I will only work with students who will meet with me weekly, do at least five hours of independent work per week on the project, and will agree to register for course credit (either using the course MATH 489 or MATH 497) each semester we work together. Below I list some of the past students I've worked with on projects like these.

Topics in Mathematical Biology

  1. A few years ago, I was working on a project with a colleague in the Biology Department, M. Scott Burt (great guy!). We made a bunch of ultrasonic recordings of free-flying bats, and we have a bunch of measurements of those recordings. I would like to get a student (or a group of students) to help sift through those recordings, clean them up a bit (I'll train you to do it), and them do some statistical analysis of the measurements (I'll train you to do that, too).
  2. A hot topic in modern science and mathematics is the 'human-computer interface'. This refers to pluggin computers into people or people into computers. Here is a list of nice, introductory papers along with some comments on them; the list and comments come second-hand from an expert at Johns Hopkins.
    • Haynes and Rees ("Decoding mental states from brain activity in humans." Nature Reviews Neuroscience 7, 523-534. July 2006) reviews the decoding of mental states from brain activity using multivariate statistics to do it. It covers both sensory perception and cognitive states.
    • O'Toole et al. ("Theoretical, Statistical, and Practical Perspectives on Pattern-based Classification Approaches to the Analysis of Functional Neuroimaging Data." J. of Cognitive Neuroscience. 19:11, pp.1735-1752. 2007) put together an excellent review of current methodologies used in decoding fMRI data, with a particular focus on why people should use multivariate classifiers.
    • Formisano et al. (2008) puts the fMRI classification problem into a mathematical framework that really doesn't add anything new, but presents it in a nice "mathy" way.
    • Mitchell et al. ("Learning to Decode Cognitive States from Brain Images." Machine Learning, 57, 145–175, 2004.) uses different machine learning algorithms to analyze previously-recorded data and shows that Gaussian Naive Bayesian classifiers and Support Vector Machines have the best performance.
    • Kay et al. ("Identifying natural images from human brain activity." Nature. Vol 452|20. March 2008.) showed that fMRI could be used to determine which image (out of a set of 100 natural images) a person was looking at. To do this, they first trained a model (using a training set of images) specifying parameters for each voxel in V1, V2, and V3 in terms of its receptive field, its spatial frequency tuning, and its orientation tuning (see supplementary material). Then, for each test image, they computed the expected response from each voxel based on the model, and compared it to the actual response observed through fMRI, and the closest match was selected. The overall performance was very good using data from 500 voxels for each decision.
    • Shinkareva et al. ("Using fMRI Brain Activation to Identify Cognitive States Associated with Perception of Tools and Dwellings." PLoS One. January 2008. Issue 1. e1394) trained Bayesian classifiers on whole-brain fMRI activation patterns resulting from looking at images of either tools or dwellings, and found that the could both identify the identity (i.e., which tool) and the class (e.g. tool) of an image being presented exclusively from the fMRI data. In addition, and possibly more interestingly, they showed that activation patterns generalized across subjects, so that by training a classifier on Subject 1 and looking at the fMRI data from Subject 2, they could accurately infer which object/class was being observed from Subject 2's fMRI data.

    These papers would form the basis for a great capstone project that has the potential to be interdisciplinary in nature (e.g., with psychology, political science for the public policy angle). Mathematical topics appear to include machine learning, classification methods.

  3. 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 on a paper that is currently being written.
    • Classification and Regression Trees, by Leo Breiman, Jerome Friedman, Charles J. Stone, R.A. Olshen. Chapman & Hall/CRC; 1 edition. January 1, 1984.

Topics in Applied Mathematics

  1. When are two matrices close together? The answer to this question could have many applications including to computer graphics. (How do we decide if two geometric objects being rendered by a computer have collided?) My interest lies in a study of the metric space of real symmetric matrices and a stratification of the space based on the eigenvalues of the elements. These strata come together in interesting ways, and I would like to be able to visualize these intersections. The project would start with visualizing the space of 2x2 real symmetric matrices and then extend that visualization (somehow) to the space of 3x3 real symmetric matrices. (Differentiable Geometry, Algebra)
  2. Investigate the mathematics involved in the burgeoning new field of geometric informatics. What type of data has patterns that can be understood in a geometric way? (Advanced Calculus, Computer Programming, some Geometry)
    • Geometric Data Analysis, by Kirby.
    This would be an exploratory project.
  3. I don't exactly know how to tie this into a mathematics project, but there's this computer language called Processesing that seems to provide a novel and easy way to display data (i.e., support visual data representation and exploration). A project that uses this software would interest me. This could happen in conjunction with the classification trees project, the vascular network project, or the data mining project, above.
    • Getting Started with Processing, by Casey Reas and Ben Fry.

Topics in Pure Mathematics

  1. 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. The project will involve work with digital images, large datasets, and the R statistical programming language.
  2. 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 Matlab, from which you can make conjectures that you can then prove or find counterexamples. (Calculus III, Linear Algebra, 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.
  3. In Calculus II, we learn that planar and spatial curves are either smooth or not smooth. The points at which they fail to be smooth are called singularities if they are interesting. And if you look at things properly, there are only a few different singularities that are possible. This capstone would involve learning about these curves, an easy equivalence relation on curves called right-equivalence, and the relationship between singularities and the equivalence classes you get. (Calculus II, 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.
  4. One of the coolest mathematical is the manifold. No, it's not that thing on your car. This is an abstract mathematical object that versatile and everywhere. 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.
  5. Investigate and present the Gauss-Bonnet Theorem. (Geometry, Analysis)
    • Barrett O'Neill, Elementary Differential Geometry. (1966) Academic Press.
  6. 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).
  7. 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.
  8. 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.
  9. 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.

Past Students

Let me summarize my experience with the discipline's Capstone. Since the Fall of 1998, I have supervised seven Capstones (one of which is still in progress) and I have taught the Capstone Seminar (Math 497, a one credit seminar designed to help students start their project) several times.

Students for whom I have served as supervisor are:

I have also worked with many students on unrelated research projects through Truman's STEP Program, Mathematical Biology Program, and TruScholars program.

Jason Miller
Last modified: Wed Dec 6 08:25:00 CST 2009