Saturday 30 June 2012. Special day of Model theory around ZilberPink
The idea is that young model theorists working on problems in and around
the ZilberPink conjectures, will present their
work. This includes questions around the CIT (Conjecture on the
Intersection of Tori) and AxSchanuel. The lectures will be accessible
to anyone having attended the introductory courses of the summerschool.
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 modeltheoretic issues
around ZilberPink
10h35  11h05: Jonathan Kirby (U. East Anglia) ZilberPink 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 pseudoexponentiationlike structure on the algebraic
numbers
15h25  15h55: Coffee break
15h55  16h25: Ayhan
Günaydin (Lisbon) Rational solutions of polynomialexponential equations
16h30  17h: Tamara Servi (Lisbon) On the interdefinability of Weierstrass
Pfunctions
Abstracts
Martin Bays (McMaster) An introduction to modeltheoretic issues
around ZilberPink
Zilber was led to what was to become the ZilberPink 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 ZilberPink conjecture, implies that the statement of
Schanuel's
conjecture can be expressed by a set of firstorder sentences in the
language of exponential fields. It also implies the firstorder
expressibility of analogous Schanuelstyle 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 (MordellLang for algebraic tori). I will present the argument
giving unconditional firstorder expressibility. In the case of
raising to powers this is due to Zilber.
Ayhan
Günaydin (Lisbon) Rational solutions of polynomialexponential
equations
We consider polynomialexponential 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) ZilberPink 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 ZilberPink is equivalent to (an appropriate statement of)
Schanuel's conjecture being expressible in firstorder logic. The
key is viewing ZilberPink 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 pseudoexponentiationlike structure on the algebraic
numbers
When constructing pseudoexponentiation, Zilber introduced a notion of “exponentialalgebraic closure” for exponential fields, stating that certain systems of exponentialpolynomial equations must have solutions.
It is possible to construct an exponentialalgebraic 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
Pfunctions
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 modelcompleteness and ominimality 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
Pfunctions; 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 ominimal expansions of the real field, has already had farreaching 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 onedimensional case of the conjecture, some partial results for certain surfaces, and some applications.
