The goal of this talk will be to discuss various issues related to the Anderson model as presented in Del Rio et. al "Operators with Singular Continuous Spectrum, IV."

Firstly, we will explain the type of localization that allows one to make dynamical statements (i.e. given simple spectrum, we have 'SULE' iff 'SUDL').

We then present various facts relating to rank one perturbations of self adjoint operators.

Finally, we connect the above two discussions to give the authors' proof that the singular continuous spectral measures produced by rank one perturbations of the Anderson model are supported on a set of Hausdorff dimension zero.

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.