"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
If you fix a class of models and a construction method that allows you to construct a new model in that class from an old model in that class, you can consider the Kripke frame generated from any given model by iterating that construction method and define the modal logic of that Kripke frame. We shall give a general definition of these modal logics in the fully abstract setting and then apply these ideas in a number of cases. Of particular interest is the case where we consider the class of models of ZFC with the construction method of forcing: in this case, we are looking at the so-called "generic multiverse".
We investigate equivalence relations for quadratic forms that can be expressed in terms of algebro-geometric properties of their associated quadrics, more precisely, birational, stably birational and motivic equivalence, and isomorphism of quadrics. We provide some examples and counterexamples and highlight some important open problems.
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.