Hanwen Liu (Warwick) will talk in the geometry seminar at on 12th October at 2pm in the Salle des Profs (9th floor building NO). Hanwen’s titte is “Self-dual frames and Yang–Mills theory in four dimensions” and the abstract is below.
On an oriented Riemannian 4-manifold, the self-dual curvature of an SO(3) connection can be viewed as a map between rank-three bundles. When this map is an orientation-preserving isomorphism, its polar decomposition provides a canonical way to remove gauge freedom. In this talk, I will explain how this leads to a global formulation of the Yang–Mills equation in terms of positive self-adjoint endomorphisms of the bundle of self-dual 2-forms. Building on the classical reconstruction of a compatible connection from a frame, the resulting equation is a determined second-order nonlinear elliptic system of Fredholm index zero.I will then explore the applications to conformal geometry, and positive connections, if time permits.