Prof. dr hab. BronisÅ‚aw Jakubczyk (Institute of Mathematics, Polish Academy of Sciences)**Series of formally noncommutative variables versus functions and vector fields on manifolds**

Taylor coefficients of a function can be calculated as iterated derivatives along constant vector fields (versors), computed at a fixed point. If a constant vector fields is replaced by any fixed set of vector fields on a manifold, a similar operation leads to the definition of a formal series with noncommutative variables.

We will show some interesting facts connected with such series, in particular the so-called *realization theorem* and its consequences. In particular, we will prove that a controlled and observed dynamical system on a manifold can be represented by such a series. Another consequence is a new test of convergence of a series of noncommutative variables.