Local normal forms without group action
The central problem in the local classification of singular points of systems of first order linear differential equations is to establish simple normal form using linear (local) change of variables, this is called the gauge equivalence. The corresponding theory is well known, and it differs for relatively tame (Fuchsian) singularities and much more wild irregular singularities, where divergence of the series becomes a norm.
There is a parallel theory of scalar higher order linear equations. They can be reducible to the first order systems, but this reduction is incompatible with the gauge equivalence. It turns out that there exists a suitable substitute for the action of the gauge group, discovered by Ore in the 1930-ies. Instead of the gauge group action, the notion of Ore equivalence is formulated in terms of identities in the corresponding noncommutative Weyl-type algebra.
In the talk I will explain the Ore classification of Fuchsian equations (very similar to the gauge classification of Fuchsian systems, despite complete different algebras behind) and the first steps towards classification of irregular singularities. It will be based on the MSc theses by D. Tanny and L. Mezuman (both from the Weizmann Institute).
The talk will be via Zoom at: https://utoronto.zoom.us/j/99576627828