Some Suggested Projects in Mathematics
Last updated on: Jan. 21, 2010
This is by no means an exhaustive list. Look for a topic or topics that interest you. Do some research before picking a particular topic.
Theory of Voting (government)
- Study various methods of election.
- Come up with "preference schedules" such that the winner is different for the various methods presented.
- Learn what Arrow's Impossibility Theorem states and explain its implications.
- Find other areas of application.
Four-Color Theorem (maps, graph theory)
- Investigate the historical background and the statement of the theorem.
- Try to "color" various maps to show that four colors are necessary.
- Study the relationship between maps and their "duals."
- Find out how the theorem has been "proved."
Apportionment Problem (government)
- Learn the motivation, history, and definition of the problem.
- Explain the first few solutions suggested in history.
- Find out what the "Alabama Paradox" is and measures to avoid it.
- Find other areas of application.
Relations (discrete mathematics)
- Define what a relation is in terms of sets.
- Find out what is meant by reflexivity, symmetry, transitivity, anti-symmetry, etc.
- Find every-day relations satisfying these properties.
- Prove some statements on how these are related to one another.
Cantor Set and Infinity (real analysis)
- Define what the Cantor set is.
- Study how to "measure" various levels of infinity and apply it to the Cantor set.
- Learn what the "measure" of a set is and find how "big" the Cantor set is.
- Extend the discussion to other "Cantor-like" sets.
Jordan's Theorem, Hairy Ball Theorem, Ham Sandwich Theorem (topology)
- Find out what Jordan's Theorem states on the plane and on the sphere.
- Find out what the Hairy Ball Theorem and the Ham Sandwich Theorem state.
- Explain some key steps in the proof of one of these theorems.
- Come up with other applications of these theorems.
Quaternions and Beyond (algebra)
- Define the following: real number, complex number, and quaternion.
- Explain how to do the four basic operations on quaternions.
- Study the historical development of quaternions and applications.
- Extend it one more step—to a larger set of numbers.
Knots (knots)
- Define what a knot is and get familiar with a few basic ones.
- Study various methods invented to distinguish different knots.
- Explain what an invariant is and give some examples.
- Learn to apply some methods to distinguish knots (and links).
Bridges of Konigsberg (graph theory)
- Review a brief history of the problem of Konigsberg.
- Explain Euler's solution to the problem.
- Define various types of graphs and check the numbers of vertices and edges.
- Discover some patterns among these numbers.
Orientability of Surfaces (topology)
- Find out what is meant by terms like the Mobius band, Klein bottle, and torus; learn how to "construct" them.
- Explain why some are orientable while others are not.
- Learn the statement (and the idea behind the proof) of the Classification Theorem.
- Extend the discussion to other dimensions and explore various possibilities.
Miscellaneous Topics:
Can Dogs Do Calculus?
Newcomb's Paradox
Matrix Multiplications and Linear Transformations on the Plane
How to Predict a Pitcher's ERA
Knight's Travel in Chess
Clever Applications of the Pigeonhole Principle
An Experiment to Find the Value of pi
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