Saturday 30 June 2012. Special day of Model theory around Zilber-Pink

Organisers: Martin Bays (McMaster), Zoé Chatzidakis and Martin Hils (IMJ, Paris 7).

The Jussieu campus has restricted access on Saturday, entry is through the main gate. If the guard asks you something, say that you are going to the Special day of the Summer school Around the Zilber Pink conjectures.

Programme:

9h30 - 10h30: Martin Bays (McMaster) An introduction to model-theoretic issues around Zilber-Pink
10h35 - 11h05: Jonathan Kirby (U. East Anglia) Zilber-Pink and axiomatising exponentiation
11h10 - 11h35: Coffee break
11h35 - 12h05: Juan Diego Caycedo (Freiburg) Applications of the Weak CIT
12h10 - 12h40: Adam Harris (Oxford) Categoricity of the j invariant and arithmetic geometry
12h40 - 14h20: Lunch
14h20 - 14h50: Margaret Thomas (Konstanz) Counting rational and algebraic points on definable sets
14h55 - 15h25: Vincenzo Mantova (Pisa) A pseudoexponentiation-like structure on the algebraic numbers
15h25 - 15h55: Coffee break
15h55 - 16h25: Ayhan Günaydin (Lisbon) Rational solutions of polynomial-exponential equations
16h30 - 17h: Tamara Servi (Lisbon) On the interdefinability of Weierstrass P-functions

Martin Bays (McMaster) An introduction to model-theoretic issues around Zilber-Pink

Zilber was led to what was to become the Zilber-Pink conjecture as part of his study of the model theory of structures arising from complex analysis. I will sketch this context, describing from a distance the main examples and concepts.

Juan Diego Caycedo (Freiburg) Applications of the Weak CIT

Zilber's Conjecture on Intersections with Tori (CIT), the toric case of the Zilber-Pink conjecture, implies that the statement of Schanuel's conjecture can be expressed by a set of first-order sentences in the language of exponential fields. It also implies the first-order expressibility of analogous Schanuel-style inequalities for some related structures: fields with raising to powers relations and green fields. In these latter cases, however, the use of the conjecture can be replaced by applications of a weak version of the conjecture, known as the Weak CIT, which is known to follow from Ax's theorem (differential field version of Schanuel's conjecture), and Laurent's theorem (Mordell-Lang for algebraic tori). I will present the argument giving unconditional first-order expressibility. In the case of raising to powers this is due to Zilber.

Ayhan Günaydin (Lisbon) Rational solutions of polynomial-exponential equations

We consider polynomial-exponential equations over complex numbers where the variables run through rational numbers. We present a method to reduce the rational solutions to integer ones and give a description of them using the earlier results. As a corollary, we get a finiteness result.

Adam Harris (Oxford) Categoricity of the j invariant and arithmetic geometry

We will look at how the model theoretic notion of categoricity of a natural structure involving the j invariant translates into arithmetic geometry.

Jonathan Kirby (U. East Anglia) Zilber-Pink and axiomatising exponentiation

One of the key conjectures about complex exponentiation is Schanuel's conjecture, and it is also an axiom for Zilber's exponential field. I will explain how the relevant case of Zilber-Pink is equivalent to (an appropriate statement of) Schanuel's conjecture being expressible in first-order logic. The key is viewing Zilber-Pink as a uniformity statement about algebraic subgroups. This was Zilber's original motivation for coming up with the conjecture. (Joint work with Boris Zilber)

Vincenzo Mantova (Pisa) A pseudoexponentiation-like structure on the algebraic numbers

When constructing pseudoexponentiation, Zilber introduced a notion of “exponential-algebraic closure” for exponential fields, stating that certain systems of exponential-polynomial equations must have solutions. It is possible to construct an exponential-algebraic closed field whose elements are just the algebraic numbers, and such that the kernel of the exponential function is cyclic. In some ways, it still resembles complex and pseudo exponentiation, but on the other hand any statement of Schanuel type is false. The problem of finding an exponential field of this kind is mostly an arithmetic one, and its solution is obtained by controlling the likely intersections inside the product of copies of the additive and of the multiplicative group.

Tamara Servi (Lisbon) On the interdefinability of Weierstrass P-functions

Bianconi (1997) proved that the real exponential function and the sine function restricted to a bounded interval are not interdefinable. The proof has two main ingredients: the model-completeness and o-minimality of the structures generated by the two mentioned functions, and a theorem of Ax (1971) on a Schanuel condition for power series over the complex field.
We prove the following: let f_0, f_1, ... , f_n be Weierstrass P-functions; then f_0 is locally definable from f_1, ... , f_n and the exponential function if and only if f_0 is obtained from one of f_1, ... , f_n by isogeny or Schwarz reflection.
The proof uses a result of Wilkie (2007) on local definability of holomorphic functions and results of Kirby (2009), in the spirit of Ax's theorem (joint work with Gareth Jones and Jonathan Kirby).

Margaret Thomas (Konstanz) Counting rational and algebraic points on definable sets

Pila and Wilkie's theorem, concerning the density of rational and algebraic points lying on sets definable in o-minimal expansions of the real field, has already had far-reaching consequences for diophantine geometry. Wilkie has conjectured an improvement to their main result for sets definable in the real exponential field. We shall survey some results that have been obtained in this direction, including the proven one-dimensional case of the conjecture, some partial results for certain surfaces, and some applications.