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)