Vahagn Aslanyan (University of East Anglia)

A geometric approach to some systems of exponential equations

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I will discuss three important conjectures on complex exponentiation, namely, Schanuel's conjecture, Zilber's Exponential Algebraic Closedness (EAC) conjecture and Zilber's quasiminimality conjecture, and explain how those conjectures are related to each other and to the model theory of complex exponentiation. I will mainly focus on the EAC conjecture which states that certain systems of exponential equations have complex solutions. Then I will show how it can be verified for systems of exponential equations with dominant additive projection for abelian varieties. All the necessary concepts related to abelian varieties will be defined in the talk. The analogous problem for algebraic tori (i.e. for usual complex exponentiation) was solved earlier by Brownawell and Masser. If time permits, I will show how our method can be used to give a new proof of their result. This is joint work with Jonathan Kirby and Vincenzo Mantova.