Behaviour of slope stable sheaves under a Fourier-Mukai transform

Over the years, Fourier-Mukai transforms have been used to construct birational maps or isomorphisms between moduli spaces, explain symmetries in counting invariants, and even construct Bridgeland stability conditions on Abelian threefolds.  All these problems involve the question of how stable objects behave under a Fourier-Mukai transform.  In this talk, I will consider this question for slope-stable torsion-free sheaves on a product elliptic threefold.  If time permits, I will also discuss this question for stable 1-dimensional sheaves.