Final Project Topics

This course has a final project that counts for 10% of
the total grade for the course. Some possible topics for the
project are listed below. Most of the topics are for research
papers, but there are a few other possibilities. You are **not**
limited to the topics in this list. You are welcome to design some
other project, in consultation with me. Your project must be
related to mathematics, but does not have to have a lot of
specifically mathematical content. It could, for example,
be primarily historical, sociological, or artistic in nature.

You will work on your project in several stages. Part of Assignment 8 is to do a little preliminary research on two possible topics for the project. In the middle of April, as part of another assignment, I will ask you to turn in an outline or plan for the project. Your completed project is due on the last Wednesday of classes, April 30. I will grade the project and return it on the last day of classes, and you will have a chance to revise the project and try to improve your grade. If you choose to revise the project, the final version is due at the final exam period on May 10.

**Famous solved problems.**In the last third of the course, we will be looking at the Poincaré Conjecture, a famous mathematical problem that has been solved recently. You might want to write a paper that looks at some other famous problem that was solved after many years of trying. For example:**The Four-Color Problem**: Is it always possible to color a map using just four colors?**Fermat's Last Theorem**: Are there positive integers x, y, z, and n, with n >2, such that x^{n}+ y^{n}= z^{n}?**Hilbert's Tenth Problem**: Is there an algorithm for solving Diophantine equations?

**Famous unsolved problems.**There are also many famous mathematical problems that remain unsolved. You might want to look into one of these. For example:**The Riemann Hypothesis.**Do all the non-trivial zeros of the Riemann zeta function have real part equal to 1/2? (Like the Poincaré Conjecture, this is one of the problem on the Clay Institute's list of millennium prizes, so it's worth $1,000,000 to the person who solves it.)**Goldbach's Conjecture.**Can every even integer greater than 2 be written as a sum of two prime numbers?**Twin Prime Conjecture.**Is there an infinite number of twin primes? (Two integers N and N + 2 that are both prime are said to be twin primes.)

**Millennium Prizes**. You could write a paper on the whole phenomenon of the millennium prizes. See http://www.claymath.org/millennium/.**Famous Numbers**. Recently, it seems like a lot of books have been devoted to individual numbers and their history. In addition to the book we read,*ZERO*, there have been books about infinity, π,*e*, and*i*. (*e*, the base of the natural logarithms, is about 2.71828, and it turns up in unexpected places almost as often as &pi.*i*is the "imaginary" number equal to the square root of -1.) You could certainly write a paper about one of these numbers. Other possible number topics include the golden ratio and the Fibonacci numbers. You could also look at the role of the Fibonacci numbers in nature.**Homework Hotline**. Watch the TV show*Homework Hotline*at 6:30 PM, weekdays, on WCNY TV Channel 24 from Syracuse. Elementary school children call in to get help on their homework -- almost always on their math homework. Analyze the way math problems are solved on this show, and compare it to Keith Devlin's ideas about learning math in*The Math Instinct*.**Math Attitudes**. Do a survey of attitudes about math among your friends, and analyze the result in the light of Keith Devlin's ideas in*The Math Instinct*.**Innumeracy.**There has been a lot of concern expressed over "innumeracy," which is something like illiteracy only with numbers instead of words. You might report on this concern or analyze it in the light of*The Math Instinct*.**Gemetria**. Write a paper about gemetria, the form of number mysticism that is based on numerical values of letters in words. Gemetria is mostly associated with the Jewish Kaballah, using the Hebrew alphabet, but it is also possible using the Greek alphabet.**Fractals**. Learn more about fractals and their role in nature and art.**Escher's Art**. The artist M.C. Escher was noted for using the mathematics of symmetry in his artwork. We will be looking at this later in the course. You might want to look at this in more depth, or you might want to try to produce some art based on his ideas. (We will do a little of that on the computer.)**Flatland.***Flatland*is a famous nineteenth century book about a two-dimensional world. We might watch an animated film based on this book in class. You could read the book and report on it.**Perspective in Art**. This is discussed briefly in*ZERO*, but it would be interesting to find out more about the introduction of the illusion of three dimensions into art during the Renaissance. Alternatively, you might want to find out more about the mathematics of perspective.**The Abacus**. Write a paper about how this ancient computing device works or how it has been used historically.**Math and Gambling**. Write a paper about probability theory and its application to gambling. You could also look at how probability theory arose historically out of an interest in gambling odds.**How to Lie with Statistics.**I have a copy of a book called*How to Lie With Statistics*that would make a good basis for a report.**The Pythagoreans**. Investigate the philosophy and history of the Pythagoreans in more depth.**Music and Numbers**. Look into the mathematical nature of music. How are numbers related to harmony and musical scales?**Mayan numbers and calendar**. These were discussed briefly in*ZERO*, and would certainly be worth a close look.**Perfect Numbers**. A perfect number is one that is equal to the sum of its proper divisors. Write a paper about the search for perfect numbers.**Calculating π**. Look at the quest to calculate more and more digits of the number π.**The Search for Large Primes.**Look at the search for very large prime numbers. The largest known prime number as of 2006 was 2^{32582657}-1, which has 9,808,358 digits. See http://primes.utm.edu/largest.html.**Newton vs. Leibnitz.**Investigate the historical controversy over who really invented calculus.**The Enigma Machine.**The Enigma was a cryptographic machine used by the Germans in World War II. A British code-breaking group was able to crack the Enigma code, and the intelligence gained from decoded German messages made a major contribution to the war effort. You might want to read about and report on this history.