Institut de Mathématiques de Jussieu (Paris 7)

Witten-Reshetikhin-Turaev quantum invariants of links and 3 dimensional manifolds are obtained from quantum sl(2). There exist different versions of quantum sl(2) leading to other families of invariants. We will briefly overview the original construction and then discuss two variants. First one, so called unrolled quantum sl(2), allows construction of invariants of 3-manifolds involving C* flat connections. In simplest case it recovers Reidemeister torsion. The second one is the non restricted version at a root of unity. It enables construction of invariants of links equipped with a gauge class of SL(2,C) flat connection. This is based respectively on joined work with Costantino, Geer, Patureau and Geer, Patureau, Reshetikhin.

We consider the differentiability of definable functions in tame expansions
of algebraically closed valued fields.
As the Frobenius inverse shows such a function may be nowhere
differentiable.
We prove differentiability almost everywhere in valued fields of
characteristic 0
that are C-minimal, definably complete and such that, in the valuation
group,
definable functions are strongly eventually linear.
This is joint work with Pablo Cubides-Kovacsics.

We will discuss our approach to global analytic geometry, based on overconvergent power series and functors of functions. We will explain how slight modifications of it allow us to develop a derived version of global analytic geometry. We will finish by discussing applications to the cohomological study of arithmetic varieties.