Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the ABC conjecture?



Philosophy behind Mochizuki's work on the ABC conjecture
I'll take a stab at answering this controversial question in a way that might satisfy the OP and benefit the mathematical community. I also want to give some opinions that contrast with or at least complement grp. Like others, I must give the caveats: I do not understand Mochizuki's claimed proof, his other work, and I make no claims about the veracity of his recent work.


First, some background which might satisfy the OP. For years, Mochizuki has been working on things related to Grothendieck's anabelian program. Here is why one might hope this is useful in attacking problems like ABC:


Begin with the Neukirch-Uchida theorem. See "Über die absoluten Galoisgruppen algebraischer Zahlkörper," by J. Neukirch, Journées Arithmétiques de Caen (Univ. Caen, Caen, 1976), pp. 67–79. Asterisque, No. 41-42, Soc. Math. France, Paris, 1977. Also "Isomorphisms of Galois groups," by K. Uchida, J. Math. Soc. Japan 28 (1976), no. 4, 617–620. 


The main result of these papers is that a number field is determined by its absolute Galois group in the following sense: fix an algebraic closure $\bar Q / Q$, and two number fields $K$ and $L$ in $\bar Q$. Then if $\sigma: Gal(\bar Q / K) \rightarrow Gal(\bar Q / L)$ is a topological isomorphism of groups, then $\sigma$ extends to an *inner* automorphism $Int(\tau): g \mapsto \tau g \tau^{-1}$ of $Gal(\bar Q / Q)$. Thus $\tau$ conjugates the number field $K$ to the number field $L$, and they are isomorphic.


So while class field theory guarantees that the absolute Galois group $Gal(\bar Q / K)$ determines (the profinite completion of) the multiplicative group $K^\times$, the Neukirch-Uchida theorem guarantees that the entire field structure is determined by the profinite group structure of the Galois group. Figuring out how to recover aspects of the field structure of $K$ from the profinite group structure of $Gal(\bar Q / K)$ is a difficult corner of number theory.


Next, consider a (smooth) curve $X$ over $Q$; suppose that the fundamental group $\pi\_1(X({\mathbb C}))$ is nonabelian. Let $\pi\_1^{geo}(X)$ be the profinite completion of this nonabelian group. Basic properties of the etale fundamental group give a short exact sequence:
$$1 \rightarrow \pi\_1^{geo}(X) \rightarrow \pi\_1^{et}(X) \rightarrow Gal(\bar Q / Q) \rightarrow 1.$$


Now, just as one can ask about recovering a number field from its absolute Galois group ($Gal(\bar Q / K)$ is isomorphic to $\pi\_1^{et}(K)$), one can ask how much one can recover about the curve $X$ from its etale fundamental group. Any $Q$-point $x$ of $X$, i.e. map of schemes from $Spec(Q)$ to $Spec(X)$ gives a section $s\_x: Gal(\bar Q / Q) \rightarrow \pi\_1^{et}(X)$. 


One case of the famous "section conjecture" of Grothendieck states that this gives a bijection from $X(Q)$ to the set of homomorphisms $Gal(\bar Q / Q) \rightarrow \pi\_1^{et}(X)$ splitting the above exact sequence. One hopes, more generally, to recover the structure of $X$ as a curve over $Q$ from the induced *outer* action of $Gal(\bar Q / Q)$ on $\pi\_1^{geo}(X)$. (take an element $\gamma \in Gal(\bar Q / Q)$, lift it to $\tilde \gamma \in \pi\_1^{et}(X)$, and look at conjugation of the normal subgroup $\pi\_1^{geo}(X)$ by $\tilde \gamma$, well-defined up to inner automorphism independently of the lift.)


As in the case of the Neukirch-Uchida theorem, there is an active and difficult corner of number theory devoted to recovering properties of rational points of (hyperbolic) curves from etale fundamental groups. Here are two dramatically difficult problems in the same spirit:


1. How can you describe the regulator of a number field $K$ from the structure of the profinite group $Gal(\bar Q / K)$?
2. Given a section $s: Gal(\bar Q / Q) \rightarrow \pi\_1^{et}(X)$, how can one describe the height of the corresponding point in $X(Q)$?


I would place Mochizuki's work in this anabelian corner of number theory; I have always kept a safe and respectful distance from this corner.


Now, to say something not quite as ancient that I gleaned from flipping through Mochizuki's recent work:


Many people here on MO and elsewhere have been following research on the field with one element. It is a tempting object to seek, because analogies between number fields and function fields break down quickly when you realize there is no "base scheme" beneath $Spec(Z)$. But I see Mochizuki's work as an anabelian approach to this problem, and I'll try to describe my understanding of this below.


Consider a smooth curve $X$ over a function field $F\_p(T)$. The anabelian approach suggests looking at the short exact sequence
$$1 \rightarrow \pi\_1^{et}(X\_{\overline{F\_p(T)}}) \rightarrow \pi\_1^{et}(X) \rightarrow Gal(\overline{F\_p(T)} / F\_p(T)) \rightarrow 1.$$
But much more profitable is to look instead at $X$ as a surface over $F\_p$ which corresponds in the anabelian perspective to studying
$$1 \rightarrow \pi\_1^{et}(X\_{\bar F\_p}) \rightarrow \pi\_1^{et}(X) \rightarrow Gal(\bar F\_p / F\_p) \rightarrow 1.$$
But this is pretty close to looking at $\pi\_1^{et}(X)$ by itself; there's just a little profinite $\hat Z$ quotient floating around, but this can be characterized (I think) group theoretically within the study of $\pi\_1^{et}(X)$ itself.


I would understand (after reading Mochizuki) that looking at curves $X$ over function fields $F\_p(T)$ as surfaces over $F\_p$ is like looking at only the etale fundamental group $\pi\_1^{et}(X)$ without worrying about the map to $Gal(\overline{F\_p(T)} / F\_p(T))$.


So, the natural number field analogue would be the following. Consider a smooth curve $X$ over $Q$. In fact, let's make $X = E - \{ 0 \}$ be a once-punctured elliptic curve over $Q$. Then the absolute anabelian geometry suggests that to study $X$, it should be profitable to study the etale fundamental group $\pi\_1^{et}(X)$ *all by itself* as a profinite group. This is the anabelian analogue of what others might call "studying (a $Z$-model of) $X$ as a surface over the field with one element". 


Without understanding any of the proofs in Mochizuki, I think that his work arises from this absolute anabelian perspective of understanding the arithmetic of once-punctured elliptic curves over $Q$ from their etale fundamental groups. The ABC conjecture is equivalent to [Szpiro's conjecture](http://en.wikipedia.org/wiki/Szpiro's_conjecture) which is a conjecture about the arithmetic of elliptic curves over $Q$.


Now here is a suggestion for number theorists who, like myself, have unfortunately ignored this anabelian corner. Let's try to read the papers of Neukirch and/or Uchida to get a start, and let's try to understand Minhyong Kim's work on Siegel's Theorem ("The motivic fundamental group of $P^1 \backslash ( 0, 1, \infty )$ and the theorem of Siegel," Invent. Math. 161 (2005), no. 3, 629–656.)


It would be wonderful if, while we're waiting for the experts to weight in on Mochizuki's work, we took some time to revisit some great results in the anabelian program. If anyone wants to start a reading group / discussion blog on these papers, I would enjoy attending and discussing.


