Filtrations on loop algebras and ordinary differential equations with irregular singularities

Presented by: Daniel Sage (LSU)

In this talk, I describe a new approach (joint with C. Bremer) to the  study of systems of meromorphic linear ODEs using methods of representation theory.   In this theory, one associates a “fundamental stratum”—data  involving an appropriate filtration on the loop algebra--to an ODE at each singular point. Intuitively, this stratum plays the role of the local “leading term” of the differential equation and can be used to give a precise measure of how irregular the ODE is at each singularity. These methods have applications to various geometric and combinatorial problems involving ODEs.  In particular, I will discuss how they can be used to construct well-behaved moduli spaces of meromorphic ODEs with irregular singularities.