Jurij Volčič: Germs of noncommutative functions

This talk addresses advances in the local theory of noncommutative functions, which branches in two directions. First we will see that the ring of noncommutative functions analytic about a scalar point admit a universal skew field of fractions, whose elements are called meromorphic germs. We also obtain a local-global rank principle for matrices of analytic noncommutative functions. On the other hand, if Y is a semisimple (non-scalar) point, then there exist nilpotent analytic noncommutative functions about Y. Nevertheless, the ring of germs about Y can be described as the completion of the free algebra with respect to the vanishing ideal at Y. This is a consequence of our second main result, a free Hermite interpolation theorem: if f is a noncommutative function, then for any finite set of semisimple points and a natural number L there exists a noncommutative polynomial that agrees with f at the chosen points up to differentials of order L.