One of the most intriguing conjectures arising from mirror symmetry states that various geometric invariants, some classical and some more homological in nature, agree for any two Calabi-Yau manifolds which are birationally equivalent to one another. I will discuss how new methods in equivariant geometry and geometric invariant theory have shed light on this conjecture over the past few years, leading to interesting representations of generalized braid groups as a bonus. Combined with the new theory of "Theta-stratifications" and "Theta-stability" -- a generalization of geometric invariant theory -- we have recently been able to establish the first cases of the D-equivalence conjecture for compact Calabi-Yau manifolds in dimension >3.