Pure Math Colloquium

"What is a non-commutative desingularization?"

A resolution of singularities replaces an algebraic variety with a smooth variety (manifold) sharing the same rational functions.  Also called a desingularization, resolutions of singularities are fundamental tools for working with singular spaces. They are known by Hironaka to exist for varieties defined over the complex numbers, but the question of existence is still open in general.  One approach to the annoyance is to give a purely algebraic definition of desingularization in terms of ring theory.  The world of commutative rings turns out to be too small for this purpose, so we are led to the possibility of ``non-commutative desingularizations.''  I will briefly describe what's known about commutative desingularizations, and what they're good for, then some progress and results on what non-commutative desingularizations are, or at least should be.

M3 3103