Dr. Bryan Dorner

Some capstones can deal with the intersection of computers and mathematics. Capstones along this line can take many forms.

1. Famous problems.

  • The Four Color Theorem. This theorem says that you only need 4 colors to color any map so that adjacent countries have different colors. The only proofs known require a computer to check all the possibilities. The big ideas in the proof and a complete proof of the 5 color theorem have made good capstones in the past.
  • The Halting Problem. This problem from theoretical computer science says that its impossible to write a computer program that takes as input any other computer program and outputs whether or not that other program will terminate after a finite number of steps. The ideas a somewhat similar to Goedel’s proof that there are statements that are true, but cannot be proved, and also to proofs that there are numbers that cannot be computed.
  1. Games and Puzzles. Sudoku has become a very popular pastime. One way to really understand the strategies would be to write a computer program to solve (and then to generate) such puzzles. A capstone would require presenting the logic of such a solution clearly and engagingly.
  1. Algorithms
  2. In the past, two students worked on methods for factoring large integers which lies at the heart of efforts to break the RSA security code. There are other methods which they did not cover.
  3. You could work on historical and new methods for computing square and higher roots as well as values of trig functions.I know a non-standard way of computing sine, cosine, sinh, and cosh that suggests a geometric way to approach the CORDIC method of computing these functions that is simpler than the approach usually given. (Many calculators use the CORDIC method.)

Dr. Jessica Sklar

Past topics students have worked on include computational group theory, the Rubik’s cube, Galois theory (an advanced topic in abstract algebra), tilings, error-correcting codes, and topology. I am good good-to person on topics involving abstract algebra, topology, or recreational mathematics(e.g., games or puzzles).

Dr. Chris Meyer

1. A surface in space on which the natural geometry is non-Euclidean

Student should have had non-Euclidean geometry (M321) and be proficient with multivariable calculus.

2. Geometric properties of various map projections.

Mapping a portion of the spherical earth on a flat paper always introduces distortions; this leads to some interesting mathematical analysis. Student should be proficient with multivariable calculus.

3. Quadric surfaces

Ellipsoids, paraboloids, hyperboloids, etc. Linear algebra can be used to classify them.

4. Stochastic Processes

This subject combines probability theory with using time as a variable; it has applications to physics (Brownian motion) and economics (the stock market). I do not know much about this subject, but I would enjoy learning with a student.

5. Topology

Topology is the study of the properties of shapes that remain the same even under continuous distortion.

6. Information Theory

This is an attempt to quantify what information is. It uses probability.

Dr. Amy Shell-Gellasch

Any topic in the history of mathematics would make an interesting and creative capstone project. Topics can range from biographical pieces, overviews of the history of a topic in mathematics, the history of an era or trend in mathematics, or the history of an idea over time. Any other historical topic is also welcome. Note that any capstone project must contain significant mathematical content. Pick an area of math that interests you and we can narrow down the topic together.

Dr. David Muller

Math education topics

  • Writing lessons that integrate math and art.
  • Teaching math through visits to art museums.

Dr. Jeffrey Stuart

I have directed 19 capstone projects at PLU since the Fall of 2001. Prior to arriving at PLU, I directed several undergraduate and masters projects that included an honors thesis and a master’s thesis.

Titles of capstones, projects and theses that my students have done:

  • Graph Theory
  • Maximum Matchings for Complete, Multipartite Graphs
  • Developing Algorithms for Finding Hamiltonian Cycles in Complete, Multipartite Graphs
  • Ramsey Theory
  • The Optimal Pebbling Number for Various Graphs
  • The Four Color Theorem
  • Shortest Path Algorithms
  • Matrix Theory
  • Who is #1? Ranking Round Robin Tournaments
  • An Introduction to Subspace Iteration and the QR Algorithm
  • Homotopy Methods for Finding Eigenvalues of Tridiagonal Matrices
  • Generalized Inverses and Least Squares
  • Markov Chains and the Perron-Frobenius Theorem
  • Mathematical Statistics
  • AR(p) Models in Time Series
  • Logistic Regression and Categorical Data Analysis
  • Improved Confidence Intervals for Binomial Probabilities
  • Surveys and Stratified Sampling
  • Probability, Blackjack and Card Counting
  • Combinatorics
  • Generalizations of Pascal’s Triangle
  • Optimization
  • Three Interior Point Methods and Their Performance on Small, Dense Problems
  • The Traveling Salesman Problem
  • Lemke’s Algorithm
  • Financial Mathematics
  • Options Pricing and the Black-Scholes Merton Model
  • Financial Time Series

I would be happy to be the go-to person for any topic that interests you and that broadly falls into one or more of the areas of graph theory, matrix theory (numerical or algebraic), combinatorics (clever counting and interesting sequences), mathematical statistics and its applications, optimization (numerical, analytic or combinatorial), financial mathematics. If you do not have a particular topic in mind, but these areas interest you, we can discuss possible topics.

Other topics you may want to consider:

  • Generalized inverses of matrices (algebraic matrix theory)
  • Bootstrap methods in statistics (mathematical statistics)
  • Using asymptotic series and Pade approximation to efficiently compute the trigonometric and exponential functions (numerical analysis)
  • Linear and integer programming (optimization)

Some software you may want to use for a project of the types listed above:

  • Minitab
  • MATLAB

Dr. Mei Zhu

1. Mathematical Models using differential equations and/or numerical analysis.

  • Developmental Biology: Why zebra has stripes and leopard has spots?Pattern formation on Butterfly wings. Nonlinear partial differential equations.
  • Infection in placenta. Nonlinear partial differential equations.
  • Skin cancer. Nonlinear ordinary differential equations.
  • Mathematical Model of Marriage.
  • Population models (Logistic model. Nonlinear Models of Interactions of multiplePredator-prey Model. Linearization and stability of the models.)
  • f. Infectious Disease models.Models for studying epidemics such as the Black Death and AIDS.Elementary Epidemic Models: the SIR (susceptible-infective-removed) model(chickenpox), the SI model (AIDS), and the SIS model (gonorrhea, common cold).Multiple Populations (sexually transmitted disease, etc).

2. Numerical analysis.

  • Solving f(x) = 0 for x.
  • Numerical solution to differential equations.

3. Fractals.

Dr. Ksenija Simic-Muller

Logic

  • Building on the Halting problem: a study of Turing machines and other formalizations of the informal idea of algorithm (e.g. computable functions, lambda calculus etc.), showing that they are all equivalent, which leads to Church's thesis
  • Godel's Incompleteness Theorems
  • Godel's Completeness Theorem and its consequences (this would require learning first order logic first)
  • Study of first order logic and its relationship to mathematics – more precisely, looking at proof systems in logic and seeing how well proofs in mathematics satisfy the general structure
  • Axioms of set theory and some of their consequences
  • Nonstandard analysis
  • Intuitionist logic -- seeing what you can prove if you are not allowed proofs by contradiction
  • Hilbert's problems (this would be more of an exposition problem)

Category Theory

Math education

  • Literature review and curriculum development (writing a lesson or a series of lesson) pertaining to social justice issues (teaching mathematics for social justice is in David Muller’s and my area of interest)
  • Comparing textbooks in algebra, geometry, or calculus, across curricula (reform vs. non-reform) or countries. For example, an international student could do a comparison of textbooks from her/his country and from the U.S. It is also possible to focus on a particular topic, e.g. how are transformations taught across curricula.
  • Students' perceptions of, for example, functions (or a different concept): doing a literature review plus a small study in a school, if possible
  • The gender gap in mathematics -- literature review and possibly conducting a study

Ethnomathematics

Anne Cook

1. The Student's t distribution and its relationship to the normaldistribution

2. Proof of the Central Limit Theorem

3. Order statistics (