C5

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.

I will collect some results about the study of topological and algebraic invariants of this moduli space by using non-abelian Hodge theory. Some keywords are: Higgs bundles, Mixed Hodge structures.