Forthcoming events in this series
.. showing that a field K is isomorphic to Q if it has the same absolute Galois group and if it satisfies a very small additional condition (very similar to my talk 2 years ago).
What makes an intersection likely or unlikely? A simple dimension count shows that two varieties of dimension r and s are non "likely" to intersect if r < codim s, unless there are some special geometrical relations among them. A series of conjectures due to Bombieri-Masser-Zannier, Zilber and Pink rely on this philosophy. I will speak about a joint work with F. Barroero (Basel) in this framework in the special case of a curve in a family of elliptic curves. The proof is based on Pila-Zannier method, combining diophantine ingredients with a refinement of a theorem of Pila and Wilkie about counting rational points in sets definable in o-minimal structures.
Everyone welcome!
Using results by Eisenbud, Schoutens and Zilber I will propose a model theoretic structure that aims to capture the algebra (or geometry) of a non reduced scheme over an algebraically closed field.
Abstract: We will consider a model theoretic approach to Gelfand-Naimark duality, from the point of view of (generalized) Zariski structures. In particular we will show quantifier elimination for compact Hausdorff spaces in the natural Zariski language. Moreover we may see a slightly unusual construction and tweak to the language, which improves stability properties of the structures.
In this talk I will present some answers to the question when every specialisation from a \kappa-saturated extension of
a Zariski structure is \kappa-universal? I will show that for algebraically closed fields, all specialisations from a \kappa-
saturated extension is \kappa-universal. More importantly, I will consider this question for finite and infinite covers of
Zariski structures. In these cases I will present a counterexample to show that there are covers of Zariski structures
which have specialisations from a \kappa-saturated extension that are not \kappa-universal. I will present some natural
conditions on the fibres under which all specialisations from a \kappa-saturated extension of a cover is \kappa-universal.
I will explain how this work points towards a prospective Ladder Theorem for Specialisations and explain difficulties and
further works that needs to be considered.
This will be a little potpourri containing some of the recent developments on the model theory of F_p((t)) and of algebraic extensions of Q_p.
"We give a diophantine criterion for a polynomial with rational coefficients not to have any
rational zero, i.e. an existential formula in terms of the coefficients expressing this property. This can be seen as a kind of restricted
model-completeness for Q and answers a question of Koenigsmann."
following the joint paper with L.Shaheen http://people.maths.ox.ac.uk/zilber/wLb.pdf
I will explain how algebraic spaces can be presented as Zariski geometries and prove some classical facts about algebraic spaces using the theory of Zariski geometries.