Suggested Projects for Math 640: EXPERIMENTAL MATHEMATICS, Spring 2014

First posted: March 23, 2014

Last Update: April 15, 2014

First draft due Tue., May 6, 2014, 10:00pm . It should be in the same public-secret directory (em14) where your homework resides. As with the homework, if you prefer not to have it posted, please say so.

Feel free to propose your own project! As long as it uses Maple (or even Mathematica, or SAGE) in a non-trivial way it is fine. I prefer that it would involve Numerical Analysis, but you are welcome to do it on anything mathematical. You must pick a project (and write a short description in the usual directory) by April 13, 2014, 10:00pm.

NOTE: Any project can be done by n people, with 1 ≤ n ≤ 3 . The optimal size of a team is 2. Most of the projects are open-ended, and would hopefully lead eventually to a published paper, and who knows? perhaps to a whole thesis.

Untaken Projects

• Use Experimental Mathematics to develop the theory of Orthogonal Polynomials ab initio (i.e. from scratch), in the style of Lectures 8 and 9, possibly using WZ theory

• Develop a theory of multi-variable orthogonal polynomials along similar lines.

• Look up the vast literature on Runge-Kutta methods (some of which uses combinatorics), and see whether you can improve it, and find new Runge-Kutta methods.

• Use Experimental mathematics to find explicit expressions for solutions of finite-difference equations in `nice domains'

• Experiment with higher order versions of finite-difference operators of differential operators to reduce both the local truncation errors and the global error.

• Systematically investigate the meta-Fibonacci recurrences discussed in Douglas Hofstadter's fascinating Experimental Mathematics seminar lecture: part 1, part 2,

Taken Projects

• Systematically investigate the conjectures discussed in Douglas Hofstadter's fascinating Colloquium lecture: part 1, part 2,
  [Taken by Katie McKeon and Anthony Zaleski, see their proposal]

• Find codes with neat properties.
  [suggested and claimed by Ross Berkowitz (with help from Tom Szinigir). See Ross Berkowitz's proposal ]

• Implement, and possibly extend to PDEs the interesting Interpolation method suggested in Tom Sznigir's message
  [Claimed by Frank Wagner]

• Investigate the gambler's ruin problem in two (and possibly even three!) dimensions, along similar lines to this gem.
  [Taken by Jeff Ames and Kafung Mok, see their proposal

• Investigate subtraction games (continuing from last year, soon to be a seminal paper) [Taken by Nathan Fox]

• See which sets of lines drawn in the plane with over and under crossings can come from a projection of lines in three space to the plane. [Suggested and taken by Cole Franks and Andrew Lohr]

• Implement the neat hypermatrix algebra described in Edinah Gnang's beautiful paper
  [Taken by Ha Luu and Somwya Srinivasan, see their proposal]

• Investigate enhanced Enumeration schemes for pattern avoiding permutations [Taken by Nathaniel Shar]

• Independently of this class' official requirement,Anthony Zaleski hopes to investigate nonlocal isoperimetric poblem
  [see his proposal]