Worked solution: Ngô's proof via the Hitchin fibration and geometric support theorem (2008)

The fundamental lemma was the last major obstacle to stabilizing the Arthur–Selberg trace formula for general reductive groups: with it in hand, the geometric side of the trace formula for $G$ can be rewritten as a sum of stable distributions on $G$ and its endoscopic groups $H$, exactly as Langlands and Shelstad's endoscopy program predicted since the early 1980s (Hales 2011, §7).

James Arthur immediately used the stabilized trace formula to complete his classification of the discrete automorphic spectrum of classical groups such as $\mathrm{Sp}(2n)$ and $\mathrm{SO}(n)$ in terms of that of $\mathrm{GL}(n)$, published as a research monograph in 2013; the fundamental lemma also underlies subsequent progress across the Langlands program.

For this work, Ngô Bảo Châu was awarded the Fields Medal in 2010, and the proof — moving a purely local, $p$-adic question through the global geometry of the Hitchin fibration and back — is widely regarded as one of the deepest applications of algebraic geometry to number theory to date.