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.