# Abstracts

Prof. Bernd Ammann und Mitarbeiter, Zimmer 119

## Talk on Tuesday, Nov 29, 10.15, M102

## Moduli spaces of even dimensional manifolds

The introduction of cobordism categories into the study of diffeomorpism groups by Galatius, Madsen, Tillmann and Weiss lead to many new computations of rings of characteristic classes for manifold bundles: First in the form of the Madsen-Weiss theorem and later at the hands of Galatius and Randal-Williams for higher, even dimensional manifolds.

I will give a general introduction to the theory of tautological characteristic classes of manifold bundles. Starting from their definition I will present the broad outline of the computations mentioned above. There are three essentially disjoint steps that go under the headlines of homological stability, stable homology and stable stability. In the present talk I will eventually focus on the third and most recent part, while wednesday's talk will revolve around the second.

## Talk on Wednesday, Nov 30, 10.15, M102

## Moduli spaces of odd dimensional manifolds

The introduction of cobordism categories into the study of diffeomorpism groups by Galatius, Madsen, Tillmann and Weiss lead to many new computations of rings of characteristic classes for manifold bundles: First in the form of the Madsen-Weiss theorem and later at the hands of Galatius and Randal-Williams for higher, even dimensional manifolds. Results of Ebert, however, sharply limited the efficacy of such categories in odd dimensions.

In this talk I will present recent joint work with Nathan Perlmutter on an enhancement of odd-dimensional cobordism categories surmounting these difficulties. In particular we show that a stabilisation of the moduli spaces from the title has the homology type of an infinite loopspace.

Knowledge about the contents of Tuesday's talk should not be required.
## David Petrecca

## On Legendrian submanifolds in Sasakian and pseudo-Sasakian manifolds

For a minimal Legendrian submanifold L of a Sasaki-Einstein
manifold, I exhibit certain eigenfunctions
of the Laplacian of L together with a lower bound for the multiplicity of
the relative eigenvalue. If this lower bound is attained then L is totally
geodesic and a rigidity result about the ambient manifold holds. This is a
generalization of a result of Le-Wang for the standard Sasakian sphere
and these eigenfunctions have a geometric meaning in terms of some
contact moment map. Moreover, if time allows, I'll discuss certain aspects
about Legendrian
stability in a pseudo-Sasaki ambient and some transformations between
the Riemannian and the Lorentzian setting. Based on joint papers with
S. Calamai and L. Schaefer.

Bernd Ammann, 30.1.2017 oder später