Explore a mathematical seminar presentation where Dr. Lars Martin Sektnan from Göteborgs Universitet delves into perturbation problems for canonical metrics, focusing on semistable cases in algebraic and differential geometry. Learn about the challenges and techniques involved in producing new Hermite-Einstein metrics on vector bundles and constant scalar curvature metrics on Kähler manifolds from existing ones in perturbed situations. Discover how to approach cases where solutions are infinitesimally close but don't exist for the original equation, and understand the necessary algebro-geometric criteria for solving these obstructed perturbation problems. Part of the "New equivariant methods in algebraic and differential geometry" series at the Isaac Newton Institute, this talk provides valuable insights into advanced mathematical concepts at the intersection of geometry and analysis.