Possible undergraduate research projects: for more information, contact me at sjm1 AT williams.edu

Here is a summary sheet of the projects.

Some papers for potential topics (REVISING THIS RIGHT NOW!):

• Irrationality and infinitude of primes:
  • Miller-Schiffman-Wieland: Infinitude of primes: there are lots of proofs of the infinitude of the primes. One of my favorite is based on zeta(s) = sum_n 1/n^2 = pi^2/6 is irrational. If you've heard of Euclid's proof, that leads to pi(x) = #{p prime: p <= x} is larger than loglog(x); using some advanced concepts (irrationality measure) we can basically recover this bound by looking at how irrational pi^2 is; unfortunately, the proof of how irrational pi^2 is uses as input pi(x) is about x / log(x). There are ways around this assumption, and we do get something nice.
• Additive and elementary number theory
  • Miller-Roman-Sinnott: y^2= n(n+1)(n+2)(n+3): A nice proof when four consecutive integers can and cannot be a square; can you generalize to products of six consecutive integers? This could make a really nice Monthly article.
  • More sums than differences: See big papers by (1) Nathanson,     (2) Martin and O'Bryant. Gist: given a finite set of integers A, form A+A and A-A, where A+A = {a1 + a2: a_i in A} and A-A = {a1 - a2: a_i in A}. As addition is commutative but subtraction isn't, a generic pair gives two differences and only one sum. Thus expect A+A to be smaller than A-A. Amazingly, Martin and O'Bryant showed that a positive percent of the time A+A is larger! If you use a difference model of choosing sets randomly, however, Hegarty and I showed almost all sets are difference dominated. In the other direction, I and some students constructed the world record for densest families of sum-dominated sets.
    • Hegarty-Miller: in a certain model of sets, almost all sets are difference dominated: Some tough open problems at the end.
    • Miller-Orosz-Scheinerman: constructing infinite families of sum dominated sets: I have some thoughts about trying to extend this construction.
• Continued Fractions (student reports from Princeton): Click here for the homepage, with access to computer programs
  • Digits of continued fractions
    • Algebraic Numbers of Degree 3, 4, and 5: nth roots of primes
    • Algebraic Numbers of Degree 7: All possible Galois Groups
    • Special Values of Special Functions: Riemann-Zeta and Gamma
  • Investigation of lengths of periods of quadratic irrationals
    • Average period result: mariusjp.pdf
    • Conjecture of Chowla on continued fractions of sqrt(prime): alexajp.pdf
  • Closed Form Expansions of Continued Fractions: dfJP2.pdf
  • Almost periodic continued fractions: tdJP.pdf
• Differential equations:
  • Kontorovich-Miller-Roy: virus infection model on star graphs
  • Note by Kontorovich on motivation of problem
• Probability topics:
  • Miller: derivation of the log5 rule for when one team wins
  • Miller: Pythagorean Won-Loss Theorem
  • Miller: Die battles: Applications to the game of risk
  • Miller: Extending the Pidgeonhole Principle
  • Miller: Differentiating identities notes: lots of good math in this, from differentiating identities to matching coefficients of polynomials. This leads to formulas to predict how often you expect to see certain runs / how many runs you expect to see in Heads / Tails.