Given a non-linear PDE satisfying some natural integrability conditions, one can pose the question of existence of destabilizing sub-systems, mimicking Gieseker-type stability conditions for coherent sheaves. I will explain how these structures may be organized into a (derived) moduli-theoretic context and I will describe the stable locus of this moduli space, whose geometric points are, roughly speaking, formally integrable and involutive differential systems with fixed microlocal behaviour and numerical invariants. Time permitting, I will show how one may prove a family of index theorems and how these tools provide new insights to other stability problems, including the existence of canonical metrics, the stability of subvarieties, and certain questions connected to non-abelian Hodge theory.
This talk is based on joint works with Prof. V. Rubtsov, Prof. A. Sheshmani, and Prof. S-T. Yau.