Vendredi 17 juin 2011, à Chevaleret,
salle 0C5:
11h :
Tamara Servi (Lisbonne), Quantifier elimination for generalised quasi-analytic algebras of real
functions
This is a joint project with Jean-Philippe Rolin, who has already
presented a special case of our results in a previous session of this seminar.
In my talk I will define the framework we work in, in all its generality: we
consider the expansion of the real field by certain algebras of functions
having a generalised (divergent) power series as an asymptotic expansion. We
show that these structures are o-minimal and polynomially bounded (in fact,
all the known examples of o-minimal polynomially bounded expansions of the
real field by functions are generated by such kind of algebras). Furthermore,
we prove a quantifier elimination result for these structures (in a reasonable
language). I will illustrate our methods of proof.
14h :
Damian Rössler (Toulouse), About the Tate-Voloch and Mordell-Lang conjecture in positive
characteristic
The Tate-Voloch conjecture states that a subvariety of an abelian
variety defined over C_p (= the p-adic complex numbers) cannot
be arbitrarily close to a torsion point, which does not lie on it. This
conjecture was proven by Hrushovski and Scanlon when the abelian variety is
defined over a finite extension of Q_p. Their proof relies on the
dichotomy theorem for the theory of generic difference fields. We shall
indicate an algebraic proof of this statement
for the prime-to-p torsion points and also explain how this proof can be
extended to equal positive characteristic p.
Finally, we shall explain how this last result can be combined with jet
space techniques to give an algebraic proof
of the Mordell-Lang conjecture in positive characteristic (which is already a
theorem of Hrushovski).
16h :
Thomas Scanlon (UC
Berkeley), Dynamical Mordell-Lang
Problems
The dynamical Mordell-Lang conjecture predicts that if f : X
\to X is a self-map of an algebraic variety over a field K of
characteristic zero, a in X(K) is any point and Y \subseteq X is a
closed subvariety, then {n \in N : f^{\circ n}(a) in
Y(K)} is a finite union of points and arithmetic progressions. I
will report first on some progress towards this conjecture using
generalizations of the method of Skolem and Chabauty, but then I will
explain how the dynamical Mordell-Lang problems are considerably subtler
than the problems about groups which they generalize. Even in this rank
one case over number fields, it appears that the Skolem-Chabauty method
is not always applicable. Turning to related problems, such as
questions on the algebraic relations on the adèlic closure of an
orbit or on orbits under higher rank semigroups of operators, while I
will report on some positive results, mostly, I will show that
intersection sets are much more chaotic.
Programme des séances
passées : 2006-07,
2007-08,
2008-09,
2009-10.
Retour
à la page principale.