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**Abstract:**

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.