I'm planning to start a polymath project-type blog to determine the face lattice of the tridiagonal & cyclic tridiagonal Birkhoff polytopes (i.e., having dimension n-1 and n+1, respectively). There are some published results, but in my opinion, this is a underdeveloped area of math. I have some answers worked out, but I am lacking proofs. I would much prefer someone else proving the answers I've developed in the near future, than eventually proving them myself many years from now.
That is Michael Nielsen, summarizing a bit of his excellent new book Reinventing Discovery. I previously quoted passages from the book here and here.
This quotation from a failed Nature collaborative project from Nielsen's website leaped out to me:
"A small majority of those authors who did participate received comments, but typically very few, despite significant web traffic. Most comments were not technically substantive. Feedback suggests that there is a marked reluctance among researchers to offer open comments. "
Many of the folks here have pet projects where they imagine easy pickings out there and they wonder why nobody is much interested in pursuing them and free software and open source software look like good analogs for making progress on a wide number of subjects. How to make this happen is a great question and I can only encourage Nielsen but I do not put a high probability on finding any simple solutions. Sharing is hard to do without trust, and trust is hard to build across the ether.
I will share with you my own pet idea that could potentially be researched in such a manner to great benefit for humans if not great profit for Silicon Valley entrepeneurs. Garlic. There are a plethora of anecdotes that eating a bunch of garlic is great for your health. Doing a rigorous study that is worthy of submission to New England Journal of Medicine or similar costs a lot of money and there is a lot more money in atypical antipsychotics and antidepressants and other chemicals which a company can patent; the incentives for medical and biology and biochemistry researchers are completely lined up to ignore garlic. There have been a couple of studies (the current wikipedia article on garlic looks pretty good to me) but dependable science looks to me like it may well never get here, short of something like Nielsen advocates: thousands of collaborative networked individuals all pitching in a little piece.
This makes me sad on the one hand. When I eat a couple of cloves of garlic there is an obvious and significant effect. It is as if my muscles and my blood are mildly drunk while my brain and reflexes remain completely sober. I want some more certain knowledge about what is going on here and I am not hopeful that I will ever find it. On the other hand there is an opportunity here, and many other similar opportunities in a wild variety of areas, just like Nielsen is writing and talking about. In theory I agree with him totally. What will come of his vision and those of similarly minded folk remains to be seen.
I unreservedly endorse this talk's message. I now intend to agitate harder to convince my peers and professors.
I should probably read the book. :j
This is a TED talk about open science. It starts with a description of a new math problem which is offered on a blog, and which eventually attracts enough mathematicians working on it to solve, not just the original problem, but a more difficult version of it. It was enough easier than the usual way of doing math that it was described as being like driving a car instead of pushing it.
Then the speaker talks about more ambitious projects-- like a wiki about quantum computing-- which get started, but no one is actually willing to do the work, so that the wiki lies all but vacant.
He suggests that public science isn't what scientists get paid for nor what builds their careers, and has some ideas for pushing the standards of science to change. There's been at least one success involving publishing genomes.
Perhaps the reason the math project succeeded was because the problem was small enough that success was both well-defined and possible, not to mention that working on it was probably more fun than figuring out how to do tolerable and sensibly-linked wiki articles.
There may be a way to get publicly funded science to be open source. We're already got proof of concept for solving math problems if they're interesting enough, so I suggest going public if you've got a math problem people might like to work on.