Let \(k\) be an algebraically closed field of characteristic \(2\) and \(A\) the
polynomial algebra in \(r\) variables with coefficients in \(k\). G. Carlsson conjectured that for any \(DG\)-\(A\)-module \(M\) of
dimension \(N\) that is free as an \(A\)-module, if the homology of \(M\) is nontrivial
and finite dimensional as a \(k\)-vector space, then \(2^r\leq N\).
In this talk, we
discuss a stronger conjecture about varieties of square-zero upper-triangular
\(N\times N\) matrices with entries in \(A\). Using stratifications of these
varieties via Borel orbits, we show that the stronger conjecture holds when
\(N < 8\) or \(r < 3\). This is joint work with Özgün Ünlü.