Relative algebroids and classification problems for geometric structures

Many geometric structures—like Riemannian metrics satisfying a diffeomorphism-invariant curvature condition—can be described by differential equations with symmetries. In many of these problems, algebroid-like structures show up almost magically, as observed by Bryant. This turns out to be very useful, especially when looking for complete solutions rather than just local ones. In this talk, I will introduce relative algebroids, a framework that systematically explains why this happens and unifies Lie algebroids with the formal theory of PDEs. We will see how relative algebroids give rise to profinite Lie algebroids, and we will discuss some aspects of and difficulties with finding their integrations.