Jordan Blocks of the Quantum Monodromy

The small quantum connection of a Fano variety is among the most accessible and fundamental objects in enumerative geometry. In this talk, I’ll survey recent results on the structure of the quantum connection, with an emphasis on bounds for the sizes of Jordan blocks of the “regularized monodromy”. These bounds can be viewed as mirror analogues of classical results by Borel, Katz, and Varchenko concerning the Jordan blocks of monodromy for Gauss–Manin connections associated to families of varieties. This is joint work—partially in progress—with P. Seidel.