Problem suggestions for polymath for undergraduates research - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:37:17Z http://mathoverflow.net/feeds/question/31153 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31153/problem-suggestions-for-polymath-for-undergraduates-research Problem suggestions for polymath for undergraduates research Chao Xu 2010-07-09T05:04:16Z 2010-11-21T14:42:20Z <p>I'm inspired by the polymath project. It might be great for few undergraduates to work together on a research topic.</p> <p>What are some research problems with the following properties(Experimental mathematics is a field containing problems with the criteria below):</p> <ol> <li>Accessible to undergraduates</li> <li>There can be many reasonable approaches to the problem</li> <li>People with computer science, applied math or other related backgrounds can also contribute</li> </ol> http://mathoverflow.net/questions/31153/problem-suggestions-for-polymath-for-undergraduates-research/31155#31155 Answer by Bruce Westbury for Problem suggestions for polymath for undergraduates research Bruce Westbury 2010-07-09T06:18:04Z 2010-07-09T06:18:04Z <p>I don't know about the polymath project but here is one thought:</p> <p>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).</p> <p>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.</p> <p>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?</p> <p>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.</p> http://mathoverflow.net/questions/31153/problem-suggestions-for-polymath-for-undergraduates-research/31162#31162 Answer by Alon Amit for Problem suggestions for polymath for undergraduates research Alon Amit 2010-07-09T09:06:00Z 2010-07-09T09:06:00Z <p>Pick any of the problems in the archives of <a href="http://www.azspcs.net/" rel="nofollow">Al Zimmermann's Programming Contests</a>, and make progress either on the theoretic side (tighter upper bounds / lower bounds / asymptotics) or the computational side.</p> <p>A specific nice example could be <a href="http://www.azspcs.net/Contest/PointPacking" rel="nofollow">Point Packing</a>. </p> http://mathoverflow.net/questions/31153/problem-suggestions-for-polymath-for-undergraduates-research/31169#31169 Answer by Doug Chatham for Problem suggestions for polymath for undergraduates research Doug Chatham 2010-07-09T10:05:28Z 2010-07-09T10:05:28Z <p>Consider this generalization of the $N$-queens problem:</p> <blockquote> <p><strong>The $N + k$ Queens Problem</strong>: 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?</p> </blockquote> <p>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 <a href="http://npluskqueens.info" rel="nofollow">http://npluskqueens.info</a> .</p> http://mathoverflow.net/questions/31153/problem-suggestions-for-polymath-for-undergraduates-research/46823#46823 Answer by To be cont'd for Problem suggestions for polymath for undergraduates research To be cont'd 2010-11-21T14:42:20Z 2010-11-21T14:42:20Z <p>I hope the question about sign matrices <a href="http://mathoverflow.net/questions/40451/sign-matrices-1-1-square-matrices" rel="nofollow">here</a> 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. </p>