Topological Galois Theory

What is Galois Theory? Most mathematicians would probably give an answer about field extensions based on polynomials and the group of automorphisms of a field that fix all of the elements of a given subfield — these are in facts the questions classically considered by Galois and Abel. More recently, people such as Picard and Vessiot started applying the same ideas to differential equations, and the idea of considering the group of symmetries of solutions to a given differential equation has led to the branch of mathematics often called Differential Galois Theory. This naturally leads one to wonder what other settings might lend themselves to this kind of analysis; as one such example, V. I. Arnold explored the connection between the solvability of algebraic equations in radicals and the solvability of the monodromy groups of certain algebraic functions. This example led his student Askold Khovanskii to develop a topological version of Galois Theory.

As Khovanskii writes in the introduction to his new book Topological Galois Theory, “According to this theory, the way the Riemann surface of an analytic function covers the plane of complex numbers can obstruct the representability of this function by explicit formulas. The strongest known results on the unexpressibility of functions by explicit formulas have been obtained in this way.” Khovanskii first developed the one-dimensional version of his theory in the 1970s, before turning his research eye to other areas of mathematics (primarily in algebraic geometry and commutative algebra) until the turn of the millennium when he discovered a way of generalizing to the higher-dimensional cases.

The book under review spends time discussing all three versions of Galois theory — algebraic, differential, and topological — as well as the underlying philosophy that he is applying to all three areas. He also spends several chapters discussing approaches to the solvability of differential equations based on Liouville’s theory, and gives a complete proof if his theorem that many indefinite integrals are not elementary functions.

As one can probably guess from the topics being discussed, this book is not for the faint of heart, as the author assumes a good deal of knowledge about topology, algebra, group theory, integration theory, and differential equations. The author attempts to keep the explanations elementary, but I do not feel that these attempts were as successful as he may have hoped. Moreover, many of the basic results in Picard-Vessiot theory and other areas are given without proof, which made it particularly challenging for someone (like this reviewer) who is a novice to that area of mathematics. That said, there is quite a bit of interesting and deep mathematics contained in the pages of this book for a reader who sticks with it.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College. He can be reached at dglass@gettysburg.edu.