we will use geometric langlands theory to solve two problems concerning number fields. one is serre's question of whether there exist motives over $\bbb q$ with motivic galois group of type $e_8$ or $g_2$; the other is the concrete question of whether there are galois extensions of $\bbb q$ with galois group $e_8(p)$ or $g_2(p)$ (the finite simple groups of lie type), for sufficiently large primes $p$. the answer to both questions is ``yes". please note the change of day for this week's colloquium.