See pdf for the abstract.

This will be the fourth lecture in the series. See PDF for the schedule and topics.

A key challenge in Riemannian geometry is to find ``best" metrics on compact manifolds. To construct such metrics explicitly one is interested to know if approximation sequences contain subsequences that converge in some sense to a limit manifold.

In this talk we will present convergence results of sequences of closed Riemannian
4-manifolds with almost vanishing L2-norm of a curvature tensor and a non-collapsing bound on the volume of small balls.  For instance we consider a sequence of closed Riemannian 4-manifolds,
whose L2-norm of the Riemannian curvature tensor is uniformly bounded from
above, and whose L2-norm of the traceless Ricci-tensor tends to zero.  Here,
under the assumption of a uniform non-collapsing bound, which is very close
to the euclidean situation, and a uniform diameter bound, we show that there
exists a subsequence which converges in the Gromov-Hausdor sense to an
Einstein manifold.

To prove these results, we use Jeffrey Streets' L2-curvature 
ow. In particular, we use his ``tubular averaging technique" in order to prove fine distance
estimates of this flow which only depend on significant geometric bounds.

Every group admits at least one action by isometries on a hyperbolic metric space, and certain classes of groups admit many different actions on different hyperbolic metric spaces (in fact, often uncountably many).  One such class of groups is the class of so-called acylindrically hyperbolic groups, which contains many interesting groups, such as mapping class groups, Out(F_n), and right-angled Artin and Coxeter groups, among many others.  In this talk, I will describe how to put a partial order on the set of actions of a given group on hyperbolic spaces which, in some sense, measures how much information about the group the action provides.  This partial order defines a "poset of actions" of the given group.  I will then define the class of acylindrically hyperbolic groups and give some structural properties of the resulting poset of actions for such groups.  In particular, I will discuss for which (classes of) groups the poset contains a largest element.