From Schanuel's Conjecture to Ax-Schanuel for Shimura varieties

From Schanuel's Conjecture to Ax-Schanuel for Shimura varieties

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Jonathan Pila, Oxford University
Fine Hall 314

Schanuel's conjecture (1960's) is a comprehensive transcendence statement about values of the exponential function. Ax-Schanuel is a functional analogue, due to Ax in 1971. Analogues of special cases of this result for Shimura varieties have been crucial ingredients in recent approaches to the Zilber-Pink Conjecture, which also has origins in Schanuel's Conjecture. I will describe these connections and recent work with Ngaiming Mok and Jacob Tsimerman giving an Ax-Schanuel theorem for Shimura varieties.