Possible projects for Math 162:

• Continued Fractions (Gauss-Kuzmin theorem, size of largest digit, how the algebraic structure of the number effects the expansion, ...).
• Galois groups (what is the size of the Galois group as we vary polynomials -- is there a nice distribution?)
• Increasing sequences: look at all permutations of {1,...,N}; there is a lot known about the distribution of the largest increasing subsequence. Can anything be said about the second largest? (Note: for {5,2,3,6,4,1} the largest increasing subsequence is 3, the second largest is 1.)
• Cryptography data from Professors Hoffstein, Pipher and Silverman.
• Benford's law. Lots of fun stuff here, ranging from the results of googling various terms and seeing how long it takes and how many hits are found to iterates of dynamical maps or adding points on elliptic curves or studying recursive functions.
• Several people have noticed more women pursuing applied math over pure math - gather data and track trends.
• lots of questions about distributions of primes, generalized primes, random primes, ....
• Kevin Bacon game (Erdos game): how is the number of links to get from A to B distributed? (VERY tough question to solve, often approximate answer by random walks on graphs.)
• Randomness: if people are asked to construct random sequences, how random are they? Do they have biases? For example, ask them to choose 5 coin flips.
• Baseball statistics: lots of fun things here.
• NCAA basketball tourneys: find an algorithm to do a good job predicting who will win.
• TV: how long in a generic half-hour show (say a sitcom) does one go before seeing the first commercial break? Is there a nice distribution to this?
• Economics: Mandelbroit and others believe standard random walk models are flawed and don't have enough large / small deviation events, and fractal geometry does a better job with changes.
• Which sneaker (left or right) do people put on first? How is that influenced by sex, race, major, ....
• From Professor Silverman: Consider {primes p < X : gcd(2^p-1,3^p-1) = 1}. I have a not-very-convincing heuristic argument that says the size of this set should look like
  pi(X) * (1 - O(1/\log X)), where pi(X) = X / logX. Not clear from curve fitting.
   

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