Fundamental Lemma

The Fundamental Lemma is a combinatorial statement on the orbital integrals for reductive group over p-adic fields. It was formulated as a conjecture by Langlands in 1983 and proven by Ngo in 2009 (Ngo was awarded Fields Medal for his proof). In my talk I will explain the statement of the Lemma, its relation to the Langlands program and prove the Lemma for $SL(2)$. Next time, Vasya Krylov will explain Ngo's strategy in general (in particular, he will talk about the Hitchin fibration).
We mostly follow Nadler's expostion, see his survey.