Nature has the scoop:
The usually quiet world of mathematics is abuzz with a claim that one of the most important problems in number theory has been solved.
Mathematician Shinichi Mochizuki of Kyoto University in Japan has released a 500-page proof of the abc conjecture, which proposes a relationship between whole numbers — a ‘Diophantine’ problem.
Conrad says that the work “uses a huge number of insights that are going to take a long time to be digested by the community”. The proof is spread across four long papers1–4, each of which rests on earlier long papers. “It can require a huge investment of time to understand a long and sophisticated proof, so the willingness by others to do this rests not only on the importance of the announcement but also on the track record of the authors,” Conrad explains.
Mochizuki’s track record certainly makes the effort worthwhile. “He has proved extremely deep theorems in the past, and is very thorough in his writing, so that provides a lot of confidence,” says Conrad. And he adds that the pay-off would be more than a matter of simply verifying the claim. “The exciting aspect is not just that the conjecture may have now been solved, but that the techniques and insights he must have had to introduce should be very powerful tools for solving future problems in number theory.”
Here are some other fun tidbits. About the mathematician quoted in the article:
I personally know Brian Conrad (quoted in article). He has an encyclopedic knowledge of algebraic number theory and algebraic geometry, I would say that he’s in the top ten people worldwide who could reasonably assess whether Mochizuki’s proof is correct. And he has a great nose for bullshit, and little patience for it.
He is taking this claim seriously. He doesn’t necessarily believe it’s correct (presumably he’s a bit careful in what he says to the press), but he seems to think it has a shot, and is worth paying attention to.
Here’s the author’s wikipedia page (the days of prodigies sweeping the profession are clearly over).
And here is the abstract for the first of the papers (I hope my TeX rendering works with all this bizarro notation):
Abstract. The present paper is the ﬁrst in a series of four papers, the goal of which is to establish an arithmetic version of Teichm¨uller theory for number ﬁelds equipped with an elliptic curve — which we refer to as “inter-universal Teichm¨uller theory” — by applying the theory of semi-graphs of anabelioids,
Frobenioids, the ´etale theta function, and log-shells developed in earlier papers by the author. We begin by ﬁxing what we call “initial Θ-data”, which consists of an elliptic curve EF over a number ﬁeld F, and a prime number l ≥ 5, as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic orbicurves that are related via ﬁnite ´etale coverings to the once-punctured elliptic curve XF determined by EF . These ﬁnite ´etale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring Fl = Z/lZ acting on the l-torsion points of the elliptic curve.
We then construct “Θ±ell NF-Hodge theaters” associated to the given Θ-data. These Θ±ell NF-Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying combinatorial dimensions of a number ﬁeld — which may be thought of as corresponding to the additive and multiplicative structures of a ring or, alternatively, to the group of units and value group of a local ﬁeld associated to the number ﬁeld — are, in some sense, “dismantled” or “disentangled” from one another. All Θ±ell NF-Hodge theaters are isomorphic to one another, but may also be related to one another by means of a “Θ-link”, which relates certain Frobenioid-theoretic portions of one Θ±ell NF-Hodge theater to another is a fashion that is not compatible with the respective conventional ring/scheme theory structures. In particular, it is a highly nontrivial problem to relate the ring structures on either side of the Θ-link to one another. This will be achieved, up to certain “relatively mild indeterminacies”, in future papers in the series by applying the absolute anabelian geometry developed in earlier papers by the author. The resulting description of an “alien ring structure” [associated, say, to the domain of the Θ-link] in terms of a given ring structure [associated, say, to the codomain of the Θ-link] will be applied in the ﬁnal paper of the series to obtain results in diophantine geometry. Finally, we discuss certain technical results concerning proﬁnite conjugates of decomposition and inertia groups in the tempered fundamental group of a p-adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of independent interest.
Well, about time!