(See the sibling comment with a "10." in it for context.) 8. Ngô Bảo Châu’s proof of the Fundamental Lemma. This is a big one but I'm not sure I can say anythi

(See the sibling comment with a "10." in it for context.) 8. Ngô Bảo Châu’s proof of the Fundamental Lemma. This is a big one but I'm not sure I can say anything useful about it, but let's see if I can get maybe 0.1% of the way towards doing so. So, we are interested in prime numbers, and there's a powerful way of investigating how they're distributed that's based on something called the Riemann zeta function. For some values of its argument, you can compute zeta(s) as the sum of this series: zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + etc. Euler noticed that you can factorize this as the product of 1/(1-1/p^s), where p ranges over all prime numbers, and "morally" this is why it's related to the distribution of the primes. You can use this -- with a lot of fascinating but difficult technical stuff along the way -- to prove the "prime number theorem", which says that the number of primes <= x is about x/log(x). Now, suppose you're wondering about a subset of the primes; e.g., the ones that, when you divide them by 4, leave a remainder of 1. Then you might compute the same product, but only over those primes; or, conversely, only over the primes that leave a remainder of 3 when divided by 4. (There is exactly one prime that falls into neither of these categories. It isn't hard to find.) Thinking along these lines leads you (if you're clever enough) to what are called "Dirichlet L-series": a(1)/1^s + a(2)/2^s + a(3)/3^s + ... where the sequence a(1),a(2),... has a particular form. And you can use these to prove a generalization of the prime number theorem that tells you, e.g., that the number of primes <= x that are 1 mod 4 is about 1/2 x/log(x). These sequences a(k) are periodic modulo some integer (e.g., the one you need to discriminate between primes that are 1 and 3 mod 4 has period 4). You can think of them as being associated with "the integers modulo k", which is an "abelian group" (which very crudely means something in which you can do addition and subtraction and have the results make some kind of sense). Associated with other abelian groups there are other L-series-like things called "automorphic L-functions". And these things are (kinda-sorta) the subject of the so-called "Langlands program", which sketches -- the details even of the conjectures involved are in many cases not clearly understood -- relationships between lots of objects in pure mathematics that I haven't yet said enough even to define: automorphic forms (related to automorphic L-functions), group representations, etc. And all these things have intimate connections with the prime numbers (as I kinda sketched above), and with other number-theoretic things (e.g., the proof of Fermat's last theorem makes use of ideas of this sort). One of the conjectures in the Langlands program is the "fundamental lemma". I can't even begin to tell you what it is, I'm afraid, but it's important and now it's been proved :-). What's next: decades more chipping away at the Langlands program, bringing gradually more understanding of number theory, algebraic geometry, and related areas of mathematics.