Given a locally compact topological group, there is a correspondence between idempotent probability measures and compact subgroups. An analogue of this correspondence continues into the model theoretic setting. In particular, if G is a stable group, then there is a one-to-one correspondence between idempotent Keisler measures and type-definable subgroups. The proof of this theorem relies heavily on the theory of local ranks in stability theory. Recently, we have been able to extend a version of this correspondence to the abelian setting. Here, we prove that fim idempotent Keisler measures correspond to fim subgroups. These results rely on recent work of Conant, Hanson and myself connecting generically stable measures to generically stable types over the randomization. This is joint work with Artem Chernikov and Krzysztof Krupinski.

I will introduce the class of causally-null-compactifiable spacetimes that can be canonically converted into compact timed-metric spaces using the cosmological time function of Andersson-Galloway-Howard and the null distance of Sormani-Vega. This class of space-times includes future developments of compact initial data sets and regions exhausting asymptotically flat space-times. I will discuss various intrinsic notions of distance between such space-times and show that some of them are definite in the sense that they are equal to zero if and only if there is a time-oriented Lorentzian isometry between the space-times. These definite distances allow us to define notions of convergence of space-times to limit space-times that are not necessarily smooth. This is joint work with Christina Sormani.

We prove the local Lipschitz continuity of energy minimizing harmonic maps between singular spaces, more specifically from the n-dimensional Heisenberg group into CAT(0) spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.  Joint work with Yaoting Gui and Jürgen Jost.