Approximating freeness under constraints

Sorin Popa
University of California, Los Angeles (UCLA)

I will discuss a method for constructing unitary elements $u$ in a subalgebra $B$ of a II$_1$ 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 II$_1$ 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.

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