**ERC GEometry and Topology of Open Manifolds**

The European Research Council has attributed to me an advanced research grant which started on february 1st, 2013 for a period of 60 months (5 years). The synopsis of the project is described below.

### Synopsis of the project

To begin with, let us explain some striking differences between closed and open manifolds. It is known that there are only countably many smooth closed n-manifolds, up to diffeomorphisms, and that there are uncountably many open n-dimensional manifolds (for n >1), even contractible ones (for n>2). Hence, almost all open manifolds do not cover closed ones. There are quite a few explicit examples of this situation among which the celebrated *Whitehead 3-manifolds *which are contractible open n-manifolds not homeomorphic to R^{n}. In dimension greater than 3, there are contractible open manifolds which are not homeomorphic to R^{n }but do cover compact manifolds. These remarks were chosen here only to stress the fact that open manifolds can appear in very different shapes. Hence the topology of *open* (i.e. noncompact) manifolds is much richer than that of compact manifolds. In addition to being interesting objects in their own right, open manifolds appear in the study of closed manifolds, for instance as limits of blow-ups of geometric flows or special sequences of Riemannian manifolds. The open manifolds appearing in these particular contexts are expected, and sometimes known, to have well behaved topology and geometry; therefore they deserve special attention.

In this project, we propose to study the topology of open 3-manifolds with the tools of Riemannian geometry. This idea has been highly successful in the compact case, one of the highlights being Perelman’s proof of the Poincaré and Geometrization Conjectures using Ricci flow, and we aim at broadening the scope of these techniques.

We will follow two approaches: in the first one, we intend to prove* rigidity or classification results*, where one obtains topological restrictions on a Riemannian manifold under certain geometric constraints. The second one consists in finding on a given open manifold a «nice» metric whose properties reflect the topology. Geometric flows, such as the well-known Ricci flow, or more recently introduced higher-order flows, are powerful analytic tools that can be used to tackle these questions. The study of these flows, in particular their singularities, relies in turn on a geometric and topological understanding of special sequences of Riemannian metrics.

### Post-doctoral positions and PhD fellowships

Several post-doctoral positions will be announced as well as PhD fellowships during the duration of the project. The first one-year post doctoral position is announced at

### Workshops and schools

A series of activities such as workshops, schools and international conferences will be scheduled.

- Workshop March 27-29, 2013