Şentürk, Berrin2018-09-182018-09-182018-092018-092018-09-17http://hdl.handle.net/11693/47886Cataloged 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.vii, 68 leaves : charts ; 30 cm.Englishinfo:eu-repo/semantics/openAccessRank ConjectureProjective VarietyBorel OrbitThesisB159012