the development of the theory of mirror symmetry in high energy physics has led to deep conjectures regarding the geometry of a special class of complex manifolds called calabi-yau manifolds. one of the most intriguing of these conjectures 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 have shed light on this conjecture over the past few years, leading to the first substantial progress for compact calabi-yau manifolds of dimension greater than three. the key technique is the new theory of ``theta-stratifications," which allows one to bring ideas from equivariant morse theory into the setting of algebraic geometry.