Abstract : I will discuss a method for constructing a Haar unitary $u$ in a subalgebra $B$ of a $II_1$ factor $M$ that's “as independent as possible” (approximately) with respect to a given finite set of elements in $M$. The technique consists of “patching up infinitesimal pieces” of $u$. This method had some striking applications over the years:
1. vanishing of the 1-cohomology for $M$ with values into the compact operators (1985);
2. reconstruction of subfactors through amalgamated free products and axiomatisation of standard invariants (1990-1994).
3. first positive results on Kadison-Singer type paving (2013);
4. vanishing of the continuous version of Connes-Shlyakhtenko 1-cohomology (with Vaes in Jan. 2014) and of smooth 1-cohomology (with Galatan in June 2014).