The Polymath Project is a collaboration among mathematicians to solve important problem in mathematics by providing a platform for mathematicians to communicate with each other on how to find the best route to the solution.
It began in January 2009 when Tim Gowers posted a problem on his blog and asked readers to reply with partial ideas or answers. This experiment resulted in a new answer to a difficult problem, proving the benefits of collaboration.
Previous Polymath projects that have successfully led to proofs incude the density version of the Hales-Jewett theorem and the Erdös discrepancy problem, as well as famously reducing the bound on the smallest gaps between primes.
Recently the 12th Polymath Project has started; Timothy Chow of MIT has proposed a new Polymath Project – resolve Rota’s basis conjecture.
What is the Rota’s Basis Conjecture?
The Rota’s Basis Conjecture states:
“If B1, B2,…., Bn are n bases of an n-dimensional vector space V (not necessarily distinct or disjoint), then there exists an n x n grid of vectors (vij) such that:
- the n vectors in row i are the members of the ith basis Bi (in some order), and
- in each column of the matrix, the n vectors in that column form a basis of V.”
Although easy to state, this conjecture has revealed itself hard to prove (like Fermat’s Last Theorem)!