<p><span style="color: rgb(36, 36, 36);">On projective complex manifolds, algebra, topology and analysis come together in a remarkable way in the celebrated Hodge decomposition of their singular cohomology. This decomposition reflects interesting symmetries, for instance the classical Poincaré duality. I will describe recent progress on understanding how much symmetry continues to hold in the presence of singularities, and the concrete ways in which partial symmetries reflect the singularity types. In the process, we will see how invariants from commutative algebra and higher dimensional geometry influence the topology of algebraic varieties.</span></p>