As we wrote about 3+ years ago, Tim Gowers ran an experiment on his blog which he called Polymath. The idea was to have an open collaboration on the internet where anyone who was interested could work together on a (hard!) research problem in combinatorics. It worked out spectacularly well and a solution was obtained quicker than anyone expected.
The results were written up in a research paper and recently appeared in the Annals (which, as we talked about here, is the highest level math journal out there). Amusingly, the journal agreed to publish the paper with author “D. H. J. Polymath”! 
Two new polymath projects have started up.
As has recently been the tradition, Terry Tao is organizing a Mini Polymath based on the current year’s IMO questions. This year’s IMO is this July 10-11. It will be held in beautiful Argentina. The questions are notoriously challenging/fun! The mini-polymath will kick off on Thursday, July 12th at 1:00 am (OK time). To join in, or just to lurk as people collaborate on the IMO problems, go here to the offical page of the mini-polymath.
In the meanwhile, a full scale polymath project has started up to try and prove the “Hot Spots Conjecture for Acute Triangles”. It was proposed by Chris Evans and they are working on it here. It is a more geometric/analytic/physics based question this time. Dr. Evans describes the problem like this:
Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary.
– from Chris Evans’s Polymath proposal
A picture of the Hot Spots Conjecture (have they tried applying talcum powder to the problem?)
Also on the polymathprojects.org website you can find several other important unsolved problems in math which might tickle your fancy. Check it out!
