SEMINARI
DI MATEMATICA
ore 16.00
Scuola Normale
Superiore
Pisa
(Aula Tonelli)
Vincenzo
Mantova
Scuola Normale
Superiore di Pisa
Terrà
un seminario dal titolo:
“A
Differential Nullstellensatz and the Mordell-Lang Conjecture”
Abstract:
The
Mordell-Lang Conjecture is a difficult statement about the rational points of
the subvarieties of semi-abelian varieties, and it has several consequences in
diophantine geometry, such as Faltings' Theorem and the Manin-Mumford
Conjecture. It has been proven with several methods, ranging from algebraic
geometry to model theory.We focus our attention on the relative Mordell-Lang
Conjecture for function fields of characteristic 0. Since in this case the
ground field is equipped with a differential operator, it is possible to embed
it in a so-called "differentially closed field": in such fields, a weak form of
Nullstellensatz holds (i.e., varieties defined as zeroes of differential
polynomials have rational points in the fields).We will see how this
differential Nullstellensatz can be axiomatized, and we will show some basic
consequences that establish analogies between the structure of differential
varieties and algebraic varieties. We will then discuss some of the ideas behind
exploiting differentially closed fields in order to give a relatively simple
proof of the Mordell-Lang Conjecture for function fields (from works of Buium,
Pillay and Ziegler)."
Tutti gli
interessati sono invitati a partecipare.
Classe
di Scienze