# Positive Curvature, Symmetries, and Matroids

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Michael Wiemeler, Münster

Online Talk

A 1930s conjecture of Hopf states that the Euler characteristic of  a positively curved even-dimensional manifold is positive. In joint work with Lee Kennard and Burkhard Wilking we showed this conjecture for simply connected manifolds $M$ with isometric, effective $T^5$-action. If there is an isometric, effective $T^7$-action on $M$ and the odd-degree rational cohomology of $M$ vanishes, we can also compute the rational cohomology ring of $M$.