A.N. Kolmogorov and V.I. Arnold proved the following amazing result: Every continuous function of n variables can be represented as a composition of continuous functions of one variable and the operation of addition.
In this result one can not replace continuous functions by entire algebraic functions. In 1969 (being an undergraduate student) I proved the following theorem.
Theorem. If an entire algebraic function can be represented as a composition of polynomials and entire algebraic functions of one variable, then its local monodromy group at each point is solvable.
I tried to solved the Hilbert problem by extending this result. Unfortunately, no one has succeeded in proving that some particular algebraic function cannot be represented as a composition of algebraic functions of two variables.
Using similar ideas I developed the Topological Galois Theory, which provides topo-logical obstructions to solvability of equations in finite terms.
In the talk I will discuss this theory. I also will discuss a topological approach to Klein’s problem (due to V.I. Arnold and Yu. Burda) which provides a visual proof of J.P. Serre’s result.
A. Khovanskii. Topological Galois theory. Solvability and non-solvability of equations infinite terms. Series: Springer Monographs in Mathematics. Berlin: Springer. 2014.