Cataloged from PDF version of article.Thesis (Ph.D.): Bilkent University, Department of Mathematics, İhsan Doğramacı Bilkent University, 2018.Includes bibliographical references (leaves 66-68).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.by Berrin Şentürk.Ph.D