MathOverflow is a remarkable recent platform for research level questions and answers in mathematics. Joe O’Rourke have asked over MO wonderful questions. (Here is a link to the questions) Many of those questions can be the starting point of a research project usually in discrete and computational geometry and sometimes in other areas. Many of the questions remained open, quite a few have led to definite quick solutions, and for many others substantial answers were offered. Usually Joe’s questions (and also his MO answers) contain beautiful and illuminating pictures. Amog the highlights: Joe’s question on Billiard knots; a poetic question about “light reflecting off christman-tree balls“; rolling a random walk on a sphere – with a definite answer by S. Carnahan; Pach animals of high genus; Fair irregular dice (with a nice answer by Bill Thurston); Parabolic envalope of fireworks; Coiling rope in a box; A convex polyhedral analog of the pentagram map ; Random-polycube-shapes ; Which convex bodies roll along closed geodesics and many more. The 100th question is The rain hull and the rain ridge.
Test your intuition: For two n by n matrices A and B, is it always the case that tr(ABAB) = tr(ABBA)?
Polymath is a collective open way of doing mathematics. It started over Gowers’s blog with the polymath1 project that was devoted to the Density Hales Jewett problem. Since then we had Polymath2 related to Tsirelson spaces in Banach space theory , an intensive Polymath4 devoted to deterministically finding primes that took place on a special polymathblog, a miniPolymath leading to collectively solving an olympiad problem, and an intensive Polymath5 devoted to the Erdos discrepency problem which is running as we speak. We have a plan for resuming Polymath3 (which had some initial rounds of discussions) which is devoted to diameter of polytopes over this blog. There are several proposed projects on the polymath blog and on Gowers’s blog. The polymath concept was mentioned as an example of a new way of collaborating and doing science and it got some blogs and media attention.
This post is an invitation for some reflections and thoughts about the polymath idea and how it goes. What do you think?
Polymath5 – The Erdős discrepancy problem – is on its way.
Update (February 2014): Boris Konev and Alexei Lisitsa found a sequence of length 1160 of discrepancy 2 and proved that no such sequence of length 1161 exists.
A SAT Attack on the Erdos Discrepancy Conjecture
The abstract reads: “We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solvers, one can obtain a sequence of length 1160 with discrepancy 2 and a proof of the Erdos discrepancy conjecture for C=2, claiming that no sequence of length 1161 and discrepancy 2 exists. We also present our partial results for the case of C=3. See also this post. So the title of this post could be revised to: Is 2 logarithmic in 1160? (My conjecture is that 2 represents the square root of log 1160.) Followups: See also this post Practically P=NP? on GLL, and Jacob Aron’s New Scientist’s article Wikipedia-size maths proof too big for humans to check. On GLL, Dick and Ken devoted the third Hebrew letter ‘Gimel’ for C.
After several discussion threads, polymath5 devoted to Erdos’s discrepency problem is on its way on Gowers’s blog. While a theoretical post with several possible attacks on the problem is planned, there is intensive experimental activity. The picture above shows a sequence of +/- signs length 1124 with discrepency 2: namely on every arithmetic progression of the form d, 2d, 3d, … where all terms are between 1 and 1124 the gap between the number of ‘+’s and the number of ‘-‘s is at most two. The present hopes from these experiments are described in this paragraph:
So I did try mathoverflow a bit and it is a cool site. Over the few days I spent there I gained 593 reputation points, and no less than 9 bronze badges. The first answer I proposed gave me a badge as “teacher”, and the first question I asked gave me a badge as “student”. In fact, if I will only reveil my age I would gain also a badge as “autobiographer”.
Indeed, mathoverflow is ran by an energetic and impressive group of very (very very) young people (more precisely, very very young men). Here is the link to the 35 users with highest reputations.
Update: Well, I got a little addicted and visited quite a bit this nice site. Among other things, I tried to promote my fundamental examples question. Basic examples can give a quick invitation to wide areas of mathematics. (Maybe to most areas, I find it hard to think about a counterexample.) Overall, there are not that many basic examples and I think there will be a consensus about most of them so lists of different mathematicians will not be so different. More examples of important examples are welcome.
In my profile you can find there the six questions that I asked and the 10 answers or so that I offered (to other questions).