Title:
Symplectic Galois representations and applications to the inverse Galois
problem

Abstract:
We give an account of recent joint work with Sara Arias-de-Reyna and
Luis Dieulefait about compatible systems of symplectic Galois
representations and how they can possibly be employed to the inverse
Galois problem.
In the beginning of the talk the overall strategy will be outlined,
starting from previous joint work with Dieulefait on the 2-dimensional
case. We will then explain the existence of a minimal global field such
that almost all the residual representations (of the compatible system)
can be defined projectively over its residue fields. Moreover, we shall
report on a very simple classification of symplectic representations
containing a nontrivial transvection in their image. Finally, we shall
combine the two points in an application to the inverse Galois problem.