In joint work with Michael Wiemeler and Burkhard Wilking at WWU Münster, a one-to-one correspondence was found between an important class of torus representations and combinatorial objects called regular matroids. Combinatorial arguments, computations on finite graphs, and known results for matroids then imply information on the fixed-point set data for subgroups of the torus. Our main area of interest for applications is to Riemannian geometry. For example, we prove Hopf’s Euler characteristic positively result for metrics invariant under a 5-dimensional torus action. Previous results of this form required lower bound on the torus rank that grew to infinity as a function of the manifold dimension