# Abstracts

Prof. Bernd Ammann und Mitarbeiter, Zimmer 119

## Application of C^{*}-algebras to the N-body problem

### Monday, 28.10.2019, 14.15 in M103

I will begin by reviewing a general method to determine the essential spectrum of Schrodinger-type operators. The method is based first on the fact that an operator is Fredholm if, and only if, it is inversible modulo the compacts (Atkinson's theorem). This reduces the study of certain quotients by the compact operators. To study the invertibility in these quotients, one uses, following
Georgescu, Mantoiu, and others, a determination of the spectrum of a suitable operator algebra (a C*-algebra) and of the action of the translation group on its spectrum.

I will give an example of how this method works using a natural algebra associated to the N-body problem. I will also discuss connection with differential geometry and a space introuce by Vasy using manifold with corners.

This is a joint work with Victor Nistor, Nicolas Prudhon and Bernd Ammann.

## Almost flat Fredholm bundles and the Strong Novikov Conjecture

### Monday, 16.12.2019, 14.15 in M103

An almost flat Fredholm bundle over a manifold consists of a smooth graded Hilbert module bundle, equipped with an even connection with small curvature, and an odd operator which is unitary modulo compact operators, and which commutes with the parallel transport associated to the connection, again modulo compact operators. These data define a K-theory class, and we will give a calculation of this K-theory class using the so-called "asymptotic representation" associated to the almost flat Fredholm bundles, at least if the curvature is small enough. We will indicate how to use this calculation in order to reprove a special case of the Strong Novikov Conjecture for groups of finite asymptotic dimension.

Bernd Ammann, 15.04.2020

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