​ ​ 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     (More available upon request) ​