Title: Geometric families of Galois representations: existence, ell-independence, modularity and other properties
<p>Speaker: Luis Dieulefait, Universitat de Barcelona 


<p>Abstract: I will report on some recent advances in the theory of geometric Galois representations, including:
-2-dimensional case: potential modularity (R. Taylor), existence of families (Dieulefait-Taylor), and "modular" upper bounds for universal minimal deformation rings (Dieulefait/ Khare-Wintenberger). Application: Serre's and Fontaine-Mazur's modularity conjectures in cases of small ramification  (Dieulefait/ Khare-Wintenberger).
If time permits, we will also discuss "uniformity" results for symplectic 4-dimensional Galois representations: generically large image for abelian surfaces (Serre) and genus 2 Siegel cusp forms (Dieulefait), and uniformity of reducibility for genus 2 Siegel cusp forms (Dieulefait/ Skinner-Urban).