Rank 1 transformations are a long studied, flexible class of measure preserving transformations. King proved a variety of deep results about rank 1 rank transformations including:

A) A rank 1 transformation’s centralizer is the weak closure of its powers.

B) The factors of rank 1 transformations are rigid and thus mildly mixing rank 1 transformations are prime. (Mild mixing means no rigid factors.)

C) Mixing rank 1 transformations have minimal self-joinings.

Thouvenot asked if mildly mixing rank 1 transformations automatically have minimal self-joinings and the point of this talk is to say that they don’t. Open problems will be presented and no previous knowledge of joinings nor rank 1 systems will be assumed. This is joint work with Donald Robertson.