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 will discuss families of Pell's equation in polynomials
with one complex parameter. In particular the relation between
the generic equation and its specializations. Our emphasis will
be on families with a triple zero. Then additive extensions enter
the picture.
In this talk I will give several perspectives on the role of
quasi-abelian categories in analytic geometry. In particular, I will
explain why a certain completion of the category of Banach spaces is a
convenient setting for studying sheaves of topological vector spaces on
complex manifolds. Time permitting, I will also argue why this category
may be a good candidate for a functor of points approach to (derived)
analytic geometry.