A conjecture on square-zero upper triangular matrices and Carlsson's rank conjecture

A well-known conjecture states that if an elementary abelian p-group acts freely on a product of spheres, then the rank of the group is at most the number of spheres in the product. Carlsson gives an algebraic version of this conjecture by considering a di erential graded module M over the polynomial ring A in r variables: If the homology of M is nontrivial and nite dimensional over the ground eld, then N := dimAM is at least 2r. In this thesis, we state a stronger conjecture concerning varieties of square-zero upper triangular N N matrices with entries in A. By stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when N < 8 or r < 3. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture as well as novel results for N > 4 and r = 2.