Math 389 Projects List [Version 1]

Explorations in Mathematics - Possible Projects

The class will be split into small groups, preferably 3 students but sometimes 2, who will select projects and work on them. During the semester, at least three lab projects must be completed by each group.

Suggestions for projects will be added to the following list during the semester.

Descriptions of some of the possible projects:

• Adiabatic Processes: Pdf or LaTeX
• Continued Fractions: Pdf or LaTeX
• Eigenvalues of the Fourier Matrix: Pdf or LaTeX
• Estimating Timber: Pdf or LaTeX
• Gluing the Edges of a Polygon: Pdf or LaTeX
• Minimizing Functions of Several Variables: Pdf or LaTeX
• Numerical Computation of Roots of Polynomials: Pdf or LaTeX
• Primes Congruent to 1 Modulo 4 p: Pdf or LaTeX
• Random Matrices: Pdf or LaTeX
• Random Walks : Pdf or LaTeX
• Sums of Cubes: Pdf or LaTeX
• Tossing a Coin: Pdf or LaTeX

The projects which have already been taken:

Project number 1

• IRS: Pdf or LaTeX (Corey Kosch, Brian Nixon, Ben Kunji) Mentor: Jeff Lagarias
• Continuous Blackjack: Pdf or LaTeX (Michael Bennett, Greg Affeldt, Chris Link) Mentor: Mark Conger
• Graphing Monomial Relations: Pdf or LaTeX (Rachel Slezak, Seth Wolbert, Josh Levin) Mentor: Andrew Kiluk
• Factoring Polynomials Modulo p: Pdf or LaTeX (Eric Barkley, Sam Faught, Justin Campbell) Mentor: Brian Mann
• Periodic Recurrence Relations: Pdf or LaTeX (Ruthi Hortsch, Feiqi Jiang, Alex Carney) Mentor: Mark Conger

Project number 2

• Kolakoski Sequence: Pdf or LaTeX (Corey Kosch, Brian Nixon, Ben Kunji) Mentor: Mark Conger
• Sums of Cubes: Pdf or LaTeX (Michael Bennett, Greg Affeldt, Chris Link) Mentor: Brian Mann
• Coloring Knots: Pdf or LaTeX (Rachel Slezak, Seth Wolbert, Josh Levin) Mentor: Andrew Kiluk
• Polynomial Images of Circles: Pdf or LaTeX (Eric Barkley, Sam Faught, Justin Campbell) Mentor: Jeff Lagarias
• 3x+1: Pdf or LaTeX (Ruthi Hortsch, Feiqi Jiang, Alex Carney) Mentor: Jeff Lagarias

Project number 3

• Sudoku: Pdf or LaTeX (Corey Kosch, Brian Nixon, Ben Kunji) Mentor:Brian Mann
• Triangular Arrays : Pdf or LaTeX (Michael Bennett, Greg Affeldt, Chris Link) Mentor: Jeff Lagarias
• Decimal Expansions: Pdf or LaTeX (Rachel Slezak, Seth Wolbert, Josh Levin) Mentor: Andrew Kiluk
• Noncommutative Polynomial Relations: Pdf or LaTeX (Eric Barkley, Sam Faught, Justin Campbell) Mentor: Mark Conger
• Substitution Sequences: Pdf or LaTeX (Ruthi Hortsch, Feiqi Jiang, Alex Carney) Mentor: Jeff Lagarias

Advice on choosing your topics

Teams may choose their topics, but approval from staff is required. This is mainly to avoid too much repetition.

1. Implementation of the necessary algorithms on the computer or by hand,
2. Collection of data,
3. Analysis of the data,
4. Preparation of a project report, first in draft form, followed by revision into final form,
5. An oral report to the class.

Part 1.

The implementation will often require writing some programs for the computer. You may use standard software packages such as Maple, Mathematica, Matlab, or C++.

Parts 2,3.

In practice, the collection of data and its analysis go hand in hand. The topics have been chosen because they exhibit interesting phenomena that ask for explanation. Often those phenomena will not be apparent until some experiments are made, and then you may want to do other experiments to refine your understanding of the perceived regularities. Your conclusions should be stated in precise mathematical terms as conjectures, or as theorems if you can find proofs. Proofs will be looked upon with favor, but are not required. Plausibility arguments may also be given.
For some of the projects, relevant material can be found in the literature. Please obtain permission of the staff ahead of time if you wish to consult the literature. In general, it will be much more satisfying for you if you explore your topic experimentally before doing so. References to the literature will not replace your obligation to (1) document your discovery of the phenomena, and (2) explain the underlying mathematics clearly.

Part 4.

A project report should have the following parts:

1. introduction: an outline of the project and of the conclusions drawn,
2. a description in words of the computer implementation, with attention to any problems that arose,
3. (the body of the report) a discussion of the results of the experiments and of the conclusions,
4. an appendix including summaries of data and computer programs.

It may be appropriate to combine parts (b) and (c).

Part 5.

Each group will give an oral presentation to this class once or twice during the semester. We will help each group individually prepare for this.

Your report must make explicit acknowledgement to the literature you consult, and to any help you receive from people other than teammates and course staff.

The course grade will be based (a) on the quality of the project work and (b) on the quality of the written and oral reports, weighted approximately equally.