So, four extra lines for each item? That would (much more than) double the length of the piece. Except that for several of them, much of that information is already there.
Still, let's have a go.
10. Mochizuki and the abc conjecture. The abc conjecture says, roughly, that you rarely have a+b=c where a,b,c are all products of large powers of prime numbers. You can think of this as saying that the additive and multiplicative structures of the integers are kinda-independent. The abc conjecture, if true, would imply lots and lots of other conjectures that number theorists have made. (In particular, it "almost implies" Fermat's Last Theorem.) Mochizuki claims to have proved it using a very complicated thing he's created called "inter-universal Teichmueller theory". Teichmueller theory is all about spaces whose structure is based on the complex numbers, which you can think of as being obtained as follows: start with the integers; allow yourself to form fractions (giving the rational numbers); "fill in the gaps" (forming the real numbers); "add roots of algebraic equations" (forming the complex numbers). But there are some not-so-obvious ways to "fill in the gaps" where instead of the usual limiting process where you do things with smaller and smaller errors (sqrt(41) ~= 6, 6.4, 6.40, 6.403, 6.4031, etc.) you try to do things that are right "modulo large powers of p" for some prime p; e.g., sqrt(41) = 1, 21, 821, 3821, 03821, 203821, etc.; but you should really think of this as exhibiting one thing mod powers of 2 and another mod powers of 5. Doing something comparable to Teichmueller theory using these gives you "p-adic Teichmueller theory", and then "inter-universal Teichmueller theory" is a further generalization that I don't understand. No one is quite sure whether Mochizuki's proof is correct; it's based on a huge amount of abstruse stuff he's invented that no one was very interested in before he claimed to have used it to prove the abc conjecture, and getting one's head around all that takes time. The next step is for the mathematical community (or at least some small bit of it) to understand IUTT well enough to check Mochizuki's proof. I believe that's happening, but it won't be quick.
Hmm, that's pretty long (despite not managing to do more than gesticulate vaguely in the direction of the relevant mathematics). I'll do the others, but I think it'll be one per comment rather than a single monstrously long comment.
Still, let's have a go.
10. Mochizuki and the abc conjecture. The abc conjecture says, roughly, that you rarely have a+b=c where a,b,c are all products of large powers of prime numbers. You can think of this as saying that the additive and multiplicative structures of the integers are kinda-independent. The abc conjecture, if true, would imply lots and lots of other conjectures that number theorists have made. (In particular, it "almost implies" Fermat's Last Theorem.) Mochizuki claims to have proved it using a very complicated thing he's created called "inter-universal Teichmueller theory". Teichmueller theory is all about spaces whose structure is based on the complex numbers, which you can think of as being obtained as follows: start with the integers; allow yourself to form fractions (giving the rational numbers); "fill in the gaps" (forming the real numbers); "add roots of algebraic equations" (forming the complex numbers). But there are some not-so-obvious ways to "fill in the gaps" where instead of the usual limiting process where you do things with smaller and smaller errors (sqrt(41) ~= 6, 6.4, 6.40, 6.403, 6.4031, etc.) you try to do things that are right "modulo large powers of p" for some prime p; e.g., sqrt(41) = 1, 21, 821, 3821, 03821, 203821, etc.; but you should really think of this as exhibiting one thing mod powers of 2 and another mod powers of 5. Doing something comparable to Teichmueller theory using these gives you "p-adic Teichmueller theory", and then "inter-universal Teichmueller theory" is a further generalization that I don't understand. No one is quite sure whether Mochizuki's proof is correct; it's based on a huge amount of abstruse stuff he's invented that no one was very interested in before he claimed to have used it to prove the abc conjecture, and getting one's head around all that takes time. The next step is for the mathematical community (or at least some small bit of it) to understand IUTT well enough to check Mochizuki's proof. I believe that's happening, but it won't be quick.
Hmm, that's pretty long (despite not managing to do more than gesticulate vaguely in the direction of the relevant mathematics). I'll do the others, but I think it'll be one per comment rather than a single monstrously long comment.