A $2n^2-log(n)-1$ lower bound for the border rank of matrix   multiplication
  Let M_n denote the matrix multiplication tensor for nxn matrices. We use the border substitution method combined with Koszul flattenings to prove the border rank lower bound of 2n^2-log(n)-1 for M_n. 
J.M. Landsberg and Mateusz Micha{\l}ek
cs.CC math.AG