[For the entire collection see Zbl 0562.00001.]
The theory of actions of formal groups on formal schemes is developed. Important applications are given. Since the theory of formal groups (over an arbitrary field k) is well established but the same techniques apply also to the theory of actions on formal schemes, the first paragraphs present a survey without proofs of the theory of formal groups. They are defined as functors G as well as by their contravariant (representating) algebras A and covariant Hopf algebras H. Also the necessary concepts of linearly compact spaces are discussed. Of prime interest is the relation between formal groups and their associated Lie algebras, which generates a category equivalence in characteristic zero, if one restricts oneself to infinitesimal formal groups. An additional restriction, the author calls it positively filtered Lie algebra, is important for the applications discussed below.
The Lie algebra of the formal scheme represented by the algebra of formal power series \(k[[x_ i| i\in I]]\), is generated by the partial derivatives \(\frac{\partial}{\partial x_ i}|_ 0\), so an intimate connection with analytical questions arises. The relation between variables and partial differential operators is studied under the concept of parameter systems. Effective tools for their calculation are developed. They are used to give a recursive algorithm to find the (unique) solution of the system of differential equations: \[ \Phi_ j(0)=0;\quad \frac{d}{dt}\Phi_ j=\sum_{i}u_ i(t)(\frac{\partial}{\partial y_ i}|_ 0\circ y_ j)(\Phi) \] with \(\Phi_ j\in k[[t]]\), \(u_ i\in k[[t]]\) coordinate components of a curve in a positively filtered Lie algebra and \(y_ i\) the parameter system of the associated infinitesimal formal group A. - This is then applied to give a solution of the differential system \[ \Psi (x,0)=x,\quad \frac{d}{dt}\Psi (x,t)=\sum u_ i(t)A_ i(\Psi (x,t)) \] from control theory as discussed by M. Fliess [Bull. Soc. Math. Fr. 109, 3-40 (1981; Zbl 0476.93021) and Invent. Math. 71, 521-537 (1983; Zbl 0513.93014)].
The concepts of controllability, observability and realization of control theory are also expressed and studied in the context of formal groups. In particular the existence of a minimal realization is shown by the fact that orbits exist in formal geometry. A different area of applications treated here is umbral calculus in combinatorics (of Rota and Roman), the interplay between polynomials and power series to obtain combinatorial identities. It is derived from and generalized to actions of the additive group \(G_ a\) and the multiplicative group \(G_ m\) on the affine line \({\mathbb{A}}\). So this technique opens a possibility to obtain also a generalization of the multivariate umbral calculus of Joni.