Joint IAS/Princeton University Number Theory Seminar

I will describe recent work with Ngaiming Mok and Jacob Tsimerman giving analogues of "Ax-Schanuel" for Shimura varieties. I will sketch the role of "o-minimal structures" in the proof, in particular point counting and powerful algebraicity results of Peterzil-Starchenko, in combination with hyperbolic geometry and monodromy, and describe versions in the setting of a differential field.