Jim Geelen, Bert Gerards, and Geoff Whittle Solved Rota’s Conjecture on Matroids

Originally posted on Combinatorics and more:

gcrota

Gian Carlo Rota

Rota’s conjecture

I just saw in the Notices of the AMS a paper by Geelen, Gerards, and Whittle where they announce and give a high level description of their recent proof of Rota’s conjecture. The 1970 conjecture asserts that for every finite field, the class of matroids representable over the field can be described by a finite list of forbidden minors. This was proved by William Tutte in 1938 for binary matroids (namely those representable over the field of two elements). For binary matroids Tutte found a single forbidden minor.  The ternary case was settled by by Bixby and by Seymour in the late 70s (four forbidden minors).  Geelen, Gerards and Kapoor proved recently that there are seven forbidden minors over a field of four elements.  The notices paper gives an excellent self-contained introduction to the conjecture.

This is a project that started in 1999 and it…

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