Sorin Popa - University of California, Los Angeles (UCLA), Math I will discuss a method for constructing unitary elements u in a subalgebra B of a II1 factor M that are "as independent as possible" (approximately) with respect to a given finite set of elements in M. This technique had most surprising applications over the years, e.g., to Kadison-Singer type problems, to proving vanishing cohomology results for II1 factors (like compact valued derivations, or L$^2^-cohomology), as well as to subfactor theory (notably, to the discovery of the proper axiomatisation of the group-like objects arising from subfactors). After explaining this method, which I call {\it incremental patching}, I will comment on all these applications and its potential for future use.