Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson�??s �??four-six�?� question and the related Stabilizing Conjecture by Ruan. In the other direction, I will also discuss more supporting evidence via Gromov-Witten invariants.

Among the first significant results of algebraic topology is the computation, by Thom, Milnor, Novikov, and Wall among others, of the bordism groups of stably complex and oriented manifolds. After reviewing these results, I will discuss the notion of derived orbifolds, and briefly indicate how the bordism groups of these objects appear as universal recipients of invariants arising in Gromov-Witten theory and symplectic topology. Finally, I will state what is known about them, as well as some conjectures about the structure of these groups.