This lecture by Marco Radeschi explores rational ellipticity in topological spaces, focusing on its relationship to manifolds with non-negative sectional curvature. Discover a new criterion for determining when a Riemannian G-manifold is rationally elliptic, which extends previously established criteria and has specific applications to cohomogeneity 3 actions. Learn about important conjectures regarding how the topological ellipticity of certain G-manifolds might be determined from their quotient spaces, including both proven cases and counterexamples to the general conjecture. The presentation draws from collaborative research with Elahe Khalili Samani, Samuel Lin, and Ricardo Mendes, offering insights into this specialized area of geometric topology at the Hausdorff Center for Mathematics.