There are two fairly well-known conjectures in the theory of Hilbert-Kunz functions and tight closure:
(a) Hilbert-Kunz multiplicities must be rational numbers;
(b) Localization commutes with tight closure.
My talk will describe examples that refute (b) and make (a) seem very unlikely. The results on (a) arose from joint work with Teixeira; those on (b) are joint work with Brenner.