I was told by departments at MIT and Stanford that outside funding does not affect admission decisions. However, Berkeley, Michigan and Austin openly stated that outside funding will affect decisions.
I wonder if this is true for all departments within those schools. I know of a number of professors within my field at MIT that would only take students with full external funding.
I wouldn't expect it to be true in other departments. Math is different from many other disciplines in that graduate students are normally funded by the department; few students get funding from their adviser.
