A. assumption that
B. assumption about
2 uncountable formal when someone starts to have control or power
A. assumption of
My calculations were based on the assumption that house prices would remain steady.
We are working on the assumption that the conference will take place as planned.
B. People make a lot of assumptions about me.
underlying assumption (=a belief that is used as the basis for an idea, but which may not be correct)
2. the assumption of responsibility