Over the last several years, it has been shown that black hole microstate level statistics in various models of 2D gravity are encoded in wormhole amplitudes.  These statistics quantitatively agree with predictions of random matrix theory for chaotic quantum systems; this behavior is realized since the 2D theories in question are dual to matrix models.  But what about black hole microstate statistics for Einstein gravity in 3D and higher spacetime dimensions, and ultimately in non-perturbative string theory?  We will discuss progress in these directions.  In 3D, we compute a wormhole amplitude that encodes the energy level statistics of BTZ black holes.  In 4D and higher, we find analogous wormholes which appear to encode the level statistics of small black holes just above threshold.  Finally, we study analogous Euclidean wormholes in the low-energy limit of type IIB string theory; we provide evidence that they encode the level statistics of small black holes just above threshold in AdS5 x S5.  Remarkably, these wormholes appear to be stable in appropriate regimes, and dominate over brane-anti-brane nucleation processes in the computation of black hole microstate statistics.

In a broad class of gravity theories, the equations of motion for vacuum compactifications give a curvature bound on the Ricci tensor minus a multiple of the Hessian of the warping function. Using results in so-called Bakry-Émery geometry, I will show how to put rigorous general bounds on the KK scale in gravity compactifications in terms of the reduced Planck mass or the internal diameter.
If time permits, I will reexamine in this light the local behavior in type IIA for the class of supersymmetric solutions most promising for scale separation. It turns out that the local O6-plane behavior cannot be smoothed out as in other local examples; it generically turns into a formal partially smeared O4.