| Résume | I will survey a rather intruiging approach to some problems in geometric topology that start by reformulating them as problems in intersection theory. I will start by explaining, on a specific problem, biased pairing theory, which studies the way that the Hodge-Riemann bilinear relation degenerates on an ideal, and review how this limits for instance the complexity of simplicial complex embeddable in a fixed manifold. I will then discuss a conjecture of Singer concerning the vanishing of l^2 cohomology on non-positively curved manifolds, and use biased pairing theory to relate it to Hodge theory on a Hilbert space that arises as the limit of Chow rings of certain complex varieties. |
