Positivity of Gromov simplicial volume of non-positively curved 4-manifolds

We show that the Gromov simplicial volume of closed nonpositively curved 4-manifolds with negative Ricci curvature is positive, which is a partial resolution of Gromov's long-standing conjecture. This is done by giving a lower bound in terms of the Euler number and a generalized Gauss-Bonnet technique. This is a joint work with X. Wan.