  This article describes local normal forms of functions in noncommuting
variables, up to equivalence generated by isomorphism of noncommutative Jacobi
algebras, extending singularity theory in the style of Arnold's commutative
local normal forms into the noncommutative realm. This generalisation unveils
many new phenomena, including an ADE classification when the Jacobi ring has
dimension zero and, by suitably taking limits, a further ADE classification in
dimension one. These are natural generalisations of the simple singularities
and those with infinite multiplicity in Arnold's classification. We obtain
normal forms away from some exceptional Type E cases. Remarkably these normal
forms have no moduli, and the key new feature is that the noncommutative world
affords larger families.
  This theory has a range of immediate consequences to the birational geometry
of 3-folds. The normal forms of dimension zero are the analytic classification
of smooth 3-fold flops, and one outcome of NC singularity theory is the first
complete list of all Type D flopping germs, with variants covering Type E. The
normal forms of dimension one have further ramifications to divisorial
contractions to a curve. In addition, the general techniques also give strong
evidence towards new contractibility criteria for rational curves.
