(based on joint work with Roozbeh Hazrat, Alexei Stepanov and Zuhong Zhang)

In an abstract group, an element of the commutator subgroup is not necessarily a commutator. However, the famous Ore conjecture, recently completely settled by Ellers-Gordeev and by Liebeck-O'Brien-Shalev-Tiep, asserts that any element of a finite simple group is a single commutator.

On the other hand, from the work of van der Kallen, Dennis and Vaserstein it was known that nothing like that can possibly hold in general, for commutators in classical groups over rings. Actually, these groups do not even have bounded width with respect to commutators.

In the present talk, we report the amazing recent results which assert that exactly the opposite holds: over any commutative ring commutators have bounded width with respect to elementary generators, which in the case of SL_n are the usual elementary transformations of the undergraduate linear algebra course.

Technically, these results are based on a further development of localisation methods proposed in the groundbreaking work by Quillen and Suslin to solve Serre's conjecture, their expantion and refinement proposed by Bak, localisation-completion, further enhancements implemented by the authors (R.H., N.V, and Z.Zh.), and the terrific recent method of universal localisation, devised by one of us (A.S.)

Apart from the above results on bounded width of commutators, and their relative versions, these new methods have a whole range of further applications, nilpotency of K_1, multiple commutator formulae and the like, which enhance and generalise many important results of classical algebraic K-theory.

In fact, our results are already new for the group SL_n, and time permitting I would like to mention further related width problems (unipotent factorisations, powers, etc.) and connections with geometry, arithmetics, asymptotic group theory, etc.