**Chandrashekhar Khare**

UCLA, USA

January 28, 2010

**Modularity of Galois representations**:
The talk will describe my joint work with J-P. Wintenberger which proves Serre's conjecture. The initial breakthrough took place while I was at TIFR in the latter half of 2004, while the complete proof
was published at the end of 2009. In this talk I will focus on the consequences of Serre's conjecture which verify the conjectures of Langlands and Fontaine-Mazur for rank 2, odd motives over the rationals. In particular, we can verify Artin's conjecture for 2-dimensional odd representations of the absolute Galois group of the rationals.

In the unlikely event that time permits I will work out an exercise which uses this consequence to prove base change for automorphic forms for odd, A_5-type extensions of the rationals.