Noncommutative Mather-Yau theorem

I will survey several Mather-Yau type theorems in noncommutative differential calculus. They can be viewed as implicit function theorems for functions with singularities in noncommutative geometry. In the categorical approach of birational geometry, noncommutative Mather-Yau theorem can be used to lift certain isomorphism of cohomology algebra to the chain level. This is joint work with Gui-song Zhou and with Bernhard Keller.