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Abstract: Closed manifolds with topology N = M x S^1 do not admit metrics of positive Ricci curvature by the theorem of Bonnet-Myers, while the the resolution of the Geroch conjecture implies that the torus T^n does  not admit a metric of positive scalar curvature. In this talk we explain a non-existence result for metrics of positive m-intermediate  curvature (a notion of curvature reducing to Ricci curvature for m =  1, and scalar curvature for m = n-1) on closed manifolds with topology N^n = M^{n-m} x T^m for n <= 7. Our proof uses minimization of weighted areas, the associated stability inequality, and delicate estimates on the second fundamental form. This is joint work with Simon Brendle and Sven Hirsch.