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Tuesday, May 2, 2017

**Abstract:** In this talk I will construct a cobordism category consisting of manifolds equipped with a choice of Morse function, whose critical points occupy a prescribed range of degrees. I will identify the homotopy the of this cobordism category with the infinite loopspace of a certain Thom spectrum. Using the parametrized version of the Gromov-Lawson construction, I will then show how to use this cobordism category to probe the space of positive scalar curvature metrics on a closed, spin manifold of dimension > 4. Our main result detects many non-trivial homotopy groups in this space of positive scalar curvature metrics.