Past Geometry and Analysis Seminar

It a classical result from Kleinian groups that a discrete group, $G$, of isometries of hyperbolic k-space $\Bbb H^k$ will act on the

boundary sphere, $S^{k-1}$, of $\Bbb H^k$ as a convergence group.

That is:

For every sequence of distinct isometries $(g_i)\subset G$ there is a subsequence ${g_i{_j})$ and points $n,p \in \S^{k-1}$ such that for $ x \in S^{k-1} -\{n\}$, $g_i_{j}(x) \to p$ uniformly on compact subsets

Joint work with Soren Galatius. We study categories C of d-dimensional cobordisms, from the perspective of Galatius, Madsen, Tillmann and Weiss. Their main result is the determination of the homotopy type of the classifying-space of such cobordism categories, as the infinite loop space of a certain Thom spectrum. One can investigate subcategories D of C having the property that the classifying-space BD is equivalent to BC, the smaller such D one can find the better.

We prove that in may cases of interest, D can be taken to be a homotopy commutative monoid. As a consequence, the stable cohomology of many moduli spaces of surfaces can be identified with that of the infinite loop space of certain Thom spectra.

Open Riemann surfaces and the Weil-Petersson Poisson structure

Let M be a closed spin manifold.

Gromov and Lawson have shown that the presence of certain "enlargeable"

submanifolds of codimension 2 is an obstruction to the existence of a Riemannian metric with positive scalar curvature on M.

In joint work with Hanke, we refine the geoemtric condition of

"enlargeability": it suffices that a K-theoretic index obstruction of the submanifold doesn't vanish.

A "folk conjecture" asserts that all index type obstructions to positive scalar curvature should be read off from the corresponding index for the ambient manifold M (this this is equivalent to a small part of the strong Novikov conjecture). We address this question for the obstruction above and discuss partial results.