Abstract: The mirror to the quintic in P^4 is a family X_p of Calabi-Yau 3-folds over a thrice-punctured sphere. As p moves in a loop around each of the three punctures, we can parallel transport classes in H^3(X_p), and observe the monodromy. H^3(X_p) is four-dimensional, and the monodromy can be expressed by matrices in Sp(4,Z). These matrices generate a subgroup which is dense in Sp(4,Z), but it was not known whether or not it was of finite index. We showed that the subgroup is isomorphic to the free product Z/5 * Z, from which it follows that it cannot be of finite index. The mirror quintic family is one of 14 similar families of CY 3-folds; our methods establish similar results for 7 of the 14 families. For the other 7, it has recently been shown that the monodromy is of finite index, so our result is best possible. This talk is based on joint work with Chris Brav, arXiv:1210.0523.