Luis Dieulefait (Universitat de Barcelona)
This is joint work with Ariel Pacetti. We present generalizations to totally real number fields of the construction done by the speaker some years ago over Q that allows to connect to each other any given pair of newforms through chains of modular compatible systems of Galois representations. We also discuss applications of this, and we consider the case of abstract Galois representations and we explain what we can do there, and what we conjecture, and finally (time permitting) how this conjecture should be attacked. This provides a new path to attack Langlands functoriality and modularity conjectures.