**Mahesh Kakde**

University College, London, U.K.

August 25, 2011

**On main conjectures in non-commutative Iwasawa theory**:
Recently the main conjecture of non-commutative Iwasawa theory for totally real fields was proven under the assumption of vanishing of certain $\mu$ invariant. The proof reduces the non-commutative main conjecture to a family commutative main conjecture (which are known due to Wiles) and certain congruences between special values of Artin L-functions (which are proven using the Deligne-Ribet q-expansion principle). More generally, one can reduce non-commutative main conjectures (for any motive) to commutative main conjectures and certain congruences between special values of L-functions of Artin twists of the motive. This is usually referred to as the strategy of Burns-Kato. I will present a formulation of the non-commutative main conjecture and the strategy of Burns-Kato. The construction of non-commutative p-adic L-function and the proof of non-commutative main conjecture go hand in hand in the Burns-Kato strategy. But now we know enough about $K_1$ of Iwasawa algebras to construct n