## Diophantine analysis in thin orbits

Alex Kontorovich

Rutgers University; von Neumann Fellow, School of Mathematics

December 8, 2017

We will explain how the circle method can be used in the setting of thin orbits, by sketching the proof (joint with Bourgain) of the asymptotic local-global principle for Apollonian circle packings. We will mention extensions of this method due to Zhang and Fuchs-Stange-Zhang to certain crystallographic circle packings, as well as the method's limitations.