Tristan Ozuch-Meersseman (ENS Paris, France) will speak in the geometry seminar on Tuesday 19 March, at 1.30pm in the room OF 2070. Tristan’s title is *Noncollapsed degeneration of Einstein 4-manifolds* and his abstract is below.

A theorem by Anderson and Bando-Kasue-Nakajima from 1989 states that in

dimension 4, to compactify the set of noncollapsed Einstein metrics with

an upper bound on the diameter in the Gromov-Hausdorff (GH) sense, one has

to add singular spaces called Einstein orbifolds and the singularities

form as blow-downs of Ricci-flat ALE spaces.

This raises some natural issues formulated by Anderson. Which Einstein

orbifolds can be expressed as GH limits of Einstein manifolds ? Does this

completion of the moduli space of Einstein metrics admit a smooth

structure ?

We partially answer these by first proving that every Einstein manifold

sufficiently GH-close to an Einstein orbifold is the result of a

glueing-perturbation procedure, which is a generalization of Biquard’s

construction. This enables us to describe any noncollapsed degeneration of

compact Einstein metrics and to exhibit obstructions to approximate

Einstein orbifolds by smooth Einstein metrics in the GH sense.