I will give some background on the Gamma Conjecture, which says that mirror symmetry does *not* respect integral cycles: rather, the integral cycles on a complex manifold correspond to integral cycles on the symplectic mirror, multiplied by a certain transcendental characteristic class called the Gamma class. In the second part of the talk I will explain a new geometric approach to the Gamma Conjecture, which is based on the SYZ viewpoint on mirror symmetry. We find that the appearance of zeta(k) in the asymptotics of period integrals arises from the codimension-k singular locus of the SYZ fibration. This is based on joint work with Abouzaid, Ganatra, and Iritani.