Secrets of Symmetries

Magnus Carlson will receive his doctoral degree in mathematics from KTH Royal Institute of Technology in 2018. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Associate Professor Tomer Schlank at the Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel.

The aim of his research project is to find a new perspective on an algebra problem, the inverse Galois problem, using topological methods. The problem is named after French mathematician Évariste Galois who died aged just 20 on May 31 1932, the day after a duel. The previous day he had written a letter in which he frantically stated his mathematical ideas.

Galois theory is one of the foundations of modern mathematics. Its central idea stems from the deep observation that solutions of polynomial equations exhibit certain symmetries. Studying such symmetries can be used to gain insights into equations and their solutions. The inverse Galois problem asks whether all symmetries can be realized as symmetries of solutions to an equation.

Using topological methods to approach the inverse Galois problem is new. Topology is a relatively young branch of mathematics; it was developed in the 20th century to study the properties of geometric objects, which remain invariant when the objects undergo stretching and bending. Topology has been successful in solving many problems, but as yet no connections to Galois theory have been found.

The aim is to find such connections by applying tools developed to study geometric objects like a sphere or a Mobius strip. This line of thought may bring new insights into previously inaccessible problems.