Solvability and unsolvability of equations in finite terms
Seminar Room 2, Newton Institute Gatehouse
My talk will be dedicated to the question of unsolvability of equations in finite terms. This question has a rich history.
First proofs of unsolvability of algebraic equations by radicals were found by Abel and Galois. Thinking on the problem of explicit indefinite integration of an algebraic differential form, Abel founded the theory of algebraic curves. Liouville continued Abel's work and proved the non-elementarity of indefinite integrals of many algebraic and elementary differential forms. Liouville was also the first to prove the unsolvability of many linear differential equations by quadratures.
The relationship between the solvability by radicals and the properties of a certain finite group goes back to Galois. The notion of finite group introduced by Galois was motivated exactly by this question. Sophus Lie introduced the notion of continuous transformation group while trying to solve differential equations explicitly and to reduce them to a simper form. With each linear differential equation, Picard associated its Galois group, which is a Lie group (and, moreover, a linear algebraic group). Picard and Vessiot showed that this particular group is responsible for the solvability of equations by quadratures. Kolchin developed the theory of algebraic groups and elaborated the Picard--Vessiot theory.
Arnold discovered that many classical mathematical questions are unsolvable for topological reasons. In particular, he showed that the general algebraic equation of degree at least 5 is unsolvable by radicals exactly for topological reasons. While developing Arnold's approach, in the beginning of 70s, I constructed a peculiar one-dimensional topological variant of the Galois theory. According to this theory, the way how the Riemann surface of an analytic function covers the complex plane can obstruct the representability of this function by explicit formulas. In this way, the strongest known results on non-expressibility of functions by explicit formulas are obtained. Recently, I succeeded to generalize these topological results to the multivariable case.