- Last week a review of the Mochizuki IUT papers appeared at Math Reviews, written by Mohamed Saïdi. His discussion of the critical part of the proof is limited to:
Theorem 3.11 in Part III is somehow reinterpreted in Corollary 3.12 of the same paper in a way that relates to the kind of diophantine inequalities one wishes to prove. One constructs certain arithmetic line bundles of interest within each theatre, a theta version and a q-version (which at the places of bad reduction arises essentially from the q-parameter of the corresponding Tate curve), which give rise to certain theta and q-objects in certain (products of) Frobenioids: the theta and q-pilots. By construction the theta pilot maps to the q-pilot via the horizontal link in the log-theta lattice. One can then proceed and compare the log-volumes of the images of these two objects in the relevant objects constructed via the multiradial algorithm in Theorem 3.11.
Saïdi gives no indication that any one has ever raised any issues about the proof of Corollary 3.12, with no mention at all of the detailed Scholze/Stix criticism that this argument is incorrect. In particular, in his Zentralblatt review Scholze writes:
Unfortunately, the argument given for Corollary 3.12 is not a proof, and the theory built in these papers is clearly insufficient to prove the ABC conjecture….
In any case, at some point in the proof of Corollary 3.12, things are so obfuscated that it is completely unclear whether some object refers to the q-values or the $\theta$-values, as it is somehow claimed to be definitionally equal to both of them, up to some blurring of course, and hence you get the desired result.
After the Saïdi review appeared, I gather that an intervention with the Math Reviews editors was staged, leading to the addition at the end of the review of
Editor’s note: For an alternative review of the IUT papers, in particular a critique of the key Corollary 3.12 in Part III, we refer the reader to the review by Scholze in zbMATH: https://zbmath.org/1465.14002.
Since the early days of people trying to understand the claimed proof, Mochizuki has pointed to Saïdi as an example of someone who has understood and vouched for the proof (see here). Saïdi is undoubtedly well aware of the Scholze argument and his decision not to mention it in the review makes clear that he has no counter-argument. The current state of affairs with the Mochizuki proof is that no one who claims to understand the proof of Corollary 3.12 can provide a counter-argument to Scholze. Saïdi tries to deal with this by pretending the Scholze argument doesn’t exist, while Mochizuki’s (and Fesenko’s) approach has been to argue that Scholze should be ignored since he’s an incompetent. The editors at PRIMS claim that referees have considered the argument, but say they can’t make anything public. This situation makes very clear that there currently is no proof of abc.
- At one point the American Institute of Mathematics (founded in 1994 with financing from John Fry) was supposed to move from its location behind a Fry’s Electronics store to a castle in Morgan Hill modeled on the Alhambra (see here). This never worked out, and last year Fry’s Electronics declared bankruptcy. The latest news is that next year AIM will move to Caltech, for more see here.
- I’ll never understand why places like MIT continue to teach undergraduate courses on a failed speculative idea about physics.
- There has been a lot of coverage in the press of claims by a group analyzing old CDF data to have come up with a dramatically better value for the W mass (one seven sigma away from the SM value). While this would be really wonderful if it were true, unfortunately that doesn’t seem very likely. There isn’t a well-motivated theoretical reason for this discrepancy, this is a very challenging measurement, and the new value seriously disagrees with several previous measurements at CERN. For an informed discussion of this from someone who was on CDF and has worked on these sorts of analyses, see Tommaso Dorigo’s blog post.
- It will be interesting to see how well the LHC experiments can ultimately do this measurement. The LHC is about to start up again after a long shutdown, with beam commissioning starting on Friday.
Quantum Theory, Groups and Representations
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