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Let k be an algebraically closed field of characteristic 2, A be a polynomial ring in m variables with coefficients in k, and (M,d) be a free, finitely generated DG-A-module. Carlsson conjectured that if the homology of M is nontrivial and finite dimensional, then the dimension of M is greater than or equal to 2^m. In this talk, we will state a new conjecture which implies Carlsson's conjecture and we wil provide some evidence for this conjecture. Then considering the fact that the differential d can be represented by a strictly upper triangular, square zero matrix, we will also discuss Borel orbits which contain such matrices and state a conjecture for the case in two variables.