Title: Cohomology for Linearized Boundary-Value Problems in Riemannian Geometry

Abstract:  In this two-part lecture, I shall present a framework I developed for casting the solvability and uniqueness conditions of linearized geometric boundary-value problems in cohomological terms. The theory is designed to be applicable without assumptions on the underlying Riemannian structure and provides tools to study the emergent cohomology explicitly. To achieve this generality, Hodge theory is extended to sequences of Douglas–Nirenberg systems that interact via Green’s formulae, overdetermined ellipticity, and a condition I call the order-reduction property, replacing the classical requirement that the sequence form a cochain complex. This property typically arises from linearized constraints and gauge equivariance, as demonstrated by several examples, including the linearized Einstein equations with sources, where the cohomology encodes geometric and topological data.  

  In my talks, I shall cover the background and details required for the construction and will emphasize the key points of analytical interest. I will also present the necessary geometric concepts. The lectures are based on my manuscript arXiv:2504.18494.