12th Polymath Project

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:

  1. the n vectors in row i are the members of the ith basis Bi (in some order), and
  2. 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)!

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