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How do you title a series of talks that appears to be about three quite different things?

• Tarksi's exponential function problem asks the following question:

  The theory of the field of real numbers is decidable. Is the theory of the field of real numbes together with the usual exponential function decidable?

  The complex numbers with exponentiation interpret arithmetic and so is undecidable. One can still ask if it is possible to characterize this structure in some way.
• Schanuel's conjecture is a profoundly difficult problem in transcendental number theory:

  If z₁, ...,z_n are complex numbers linearly independent over the rationals, then the transcendence degree of the extension field Q(z₁, ...,z_n, exp(z₁), ...,exp(z_n)) over the rationals is at least n.

• Boris Zil'ber's pseudo-exponential fields are axiomatized in infinitary first order logic.

Macintyre and Wilkie proved that the answer to Tarski's problem is "yes" if (the real number restriction) of Schanuel's conjecture holds.

Pseudo-exponential fields satisfy Schanuel's conjecture (by definition), and Zil'ber showed that there is a unique pseudo-exponential field of cardinality the continuum. So Schanuel's conjecture is equivalent to the question "Is Zil'ber's pseudo-exponentiation on C the same as the usual exponential function?"

I will give an overview of all of this, and related material.

References Rather than trying to cite a long list of papers here, I will just suggest that you search the three key phrases from the list above on Wikipedia, where there are some quite good articles, with references.