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I'm inspired by the polymath project. It might be great for few undergraduates to work together on a research topic.

What are some research problems with the following properties(Experimental mathematics is a field containing problems with the criteria below):

  1. Accessible to undergraduates
  2. There can be many reasonable approaches to the problem
  3. People with computer science, applied math or other related backgrounds can also contribute
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    $\begingroup$ If people know about such problems, aren't they going to keep them for their own students? $\endgroup$
    – Qiaochu Yuan
    Commented Jul 9, 2010 at 6:23
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    $\begingroup$ @Qiaochu: in many cases I suspect they will, yes. But some people are in the opposite situation: they have elementary research problems but not enough (or any) undergraduate students interested and qualified to work on them. It certainly doesn't hurt to ask -- to ask around, I mean. It's not so clear to me that this is an on-topic MO question. $\endgroup$
    – Pete L. Clark
    Commented Jul 9, 2010 at 6:36
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    $\begingroup$ Yemon- I'm not optimistic about the probability of a good polymath arising from this question, but I think trying is a perfectly reasonable use of MO. $\endgroup$
    – Ben Webster ♦
    Commented Jul 9, 2010 at 10:27

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Pick any of the problems in the archives of Al Zimmermann's Programming Contests, and make progress either on the theoretic side (tighter upper bounds / lower bounds / asymptotics) or the computational side.

A specific nice example could be Point Packing.

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I don't know about the polymath project but here is one thought:

A long knot is an embedding of $\mathbb{R}\rightarrow\mathbb{R}^3$ which as $t$ tends to $\pm\infty$ approach the line $x=y=z$. Examples are given by $t\mapsto (x(t),y(t)z(t))$ where $x(t)$, $y(t)$, $z(t)$ are monic polynomials of degree $2r+1$. In fact all long knots arise this way. However when I have implemented this you get pretty unsatisfactory pictures. The problem is to find a way to get better pictures (not exactly cutting edge research, I know).

One possibility would be to define an energy functional and then take the gradient flow to find a local minimum. If we fix $r$ this all takes place on a finite dimensional manifold.

Another direction is to apply a Mobius transformation that moves the point at infinity. This gives a knot parametrised by rational functions. I haven't tried this, but I doubt it gives a pretty picture. Can these pictures be improved?

You could also investigate this from the point of view of Vassiliev theory (which is how it came up when I heard about it). That is, look at the discriminant, the polynomials whose long knots have self-intersections.

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  • $\begingroup$ @Bruce, have you seen the stuff by Don O'Shea and Alan Durfee? I did an REU with them on polynomial knots four years or so ago, and we got some nice tricks for parameterizing knots starting with a knot diagram in the plane (so I guess whether this has been solved depends on what "unsatisfactory" means) $\endgroup$
    – Charles Siegel
    Commented Jul 9, 2010 at 11:18
  • $\begingroup$ @Charles, No, I have not seen anything in writing. Where do I find it? $\endgroup$
    – Bruce Westbury
    Commented Jul 9, 2010 at 12:40
  • $\begingroup$ Here's a survey they wrote front.math.ucdavis.edu/0612.5803 and the REU websites are at mtholyoke.edu/acad/math/past_projects.html I do recommend extra care be taken with any of the project papers...I remember finding mistakes in some of the older ones I read, and I vaguely remember looking back at my own a year later and being rather embarrassed by it... $\endgroup$
    – Charles Siegel
    Commented Jul 10, 2010 at 20:43
  • $\begingroup$ We just had a beautiful colloquium by Greg Buck about exactly this: gregorybuck.com Lots of beautiful movies of knots relaxing via gradient flow to their minimal-energy configuration. (But at present, it seems very hard to prove that knots can't get trapped at a local minimum for the potential Buck uses.) $\endgroup$
    – JSE
    Commented Nov 21, 2010 at 19:15
  • $\begingroup$ The first link in Charles' comment is broken, here is a replacement: arxiv.org/abs/math/0612803 $\endgroup$
    – David Roberts ♦
    Commented Mar 29, 2022 at 7:36
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Consider this generalization of the $N$-queens problem:

The $N + k$ Queens Problem: Let $N > 0$ and $k \geq 0$ be integers. On an $N \times N$ chessboard, can you place $N + k$ queens and $k$ pawns so that any two queens on the same row, column, or diagonal have at least one pawn between them?

We've had many math and computer science undergraduates working on projects related to this problem. For more information, please see the $N + k$ Queens Problem Page at http://npluskqueens.info .

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I hope the question about sign matrices here maybe of interest to some undergraduates like me. It certainly also offers a programming experience. Let me know if any get interested. I will be happy to correspond.

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