Professor Kurusch Ebrahimi-Fard: What is the Magnus expansion?
Thursday 14 November 4pm – 6pm
Thursday 21 November 4pm – 6pm
Thursday 28 November 4pm – 6pm
Abstract: The Magnus expansion was first introduced by Wilhelm Magnus in his 1954 paper, “On the Exponential Solution of Differential Equations for a Linear Operator” (CPAM 7 (1954) 649), where he addressed a key problem in applied mathematics: the computation of the logarithm of the operator- or matrix-valued solution to a linear initial value problem. Since its discovery, the Magnus expansion has evolved into a useful tool, used across various fields such as physics, chemistry, and engineering. Over the past 25 years, mathematical developments have revealed connections between algebra, combinatorics, and geometry in the context of the Magnus expansion. These advances have clarified its structure and extended its applications.
In the first lecture, we will introduce the Magnus expansion by discussing Magnus’ original work and providing a concise overview of some of its applications in numerical mathematics.
In the second lecture, we will emphasise modern perspectives on the Magnus expansion, focusing on its formulation using pre-Lie and post-Lie algebras. This approach allows for a discussion of the Magnus expansion from the viewpoint of crossed morphisms.
In the third lecture, we discuss recent developments in the context of signatures for surfaces, introducing a two-parameter Magnus expansion in the context of free crossed modules of Lie algebras. This construction is analogous to the logarithm of the path signature, providing insights into the algebraic structures underpinning surface signatures.