Some things that might be of interest:
- The ICM is starting next week in Philadelphia. One traditional aspect of the ICM is the announcement of the Fields Medal winners. This year there won’t be much surprise, since evidently someone found the names of the winners hidden in the website code (see here, English language news story here). It’s Yu Deng, John Pardon, Jacob Tsimerman, and Hong Wang. At least all science journalists who want to write about this now know the names, no controversies about who has embargoed information as in some past years.
- Strings 2026 was held last week in Shanghai. Looking through the talks (see here, videos here), I see nothing that looks like a significant new idea or any evidence this isn’t a dead field. I watched the public talk by David Gross, which was just bizarre. Much the same talk Gross has been giving for thirty years about string theory unification, with no indication there’s any problem with the idea. For detailed discussion of a similar talk 22 years ago, see here. The main difference with that talk is that Gross removed all the slides about how SUSY was about to be found and would vindicate string theory. Gross argues against the “landscape”, saying that he thinks there will be some new insight explaining how all the constants of nature emerge from understanding the big bang, but this seems to be backed by nothing except wishful thinking.
Gross started off the talk by complaining that he had had to work hard for the few days before his talk, since he had found that his slides were so old that the file formats now had compatibility problems with current software. I’m completely baffled by how anyone involved in this could think that going to the public with this sort of thing is a good idea.
Also on the Strings 2026 topic, Stringking42069 is back.
- Every so often I think it’s worthwhile to point to something about the “interpretation” of quantum mechanics that reflects my best understanding of the subject, so see here.
- Quanta magazine a while back had a good article about the deep new mathematics being develop by Peter Scholze, Dustin Clausen and others. Clausen has a paper out on the Weil-Moore Anima he was lecturing about at the IHES earlier in the year.
- Finally, something inspirational: my Columbia colleague Joan Birman is still active at 99 years old, and has an important new result on the Burau representation. This is the result of a project that began when a high school student, Vasudha Bharathram, came to Birman for help in learning more math and finding a problem to work on. Bharathram will be a beginning Ph.D student at Princeton in fall. The third member of the collaboration is Tara Brendle, a student of Birman’s. This collaboration must break some sort of record in math research for the age range of the collaborators.
Update: For a new documentary series about Bourbaki, see here.
Update: As a commenter here noted, the LANA project to try and formalize the controversial part of Mochizuki’s supposed proof of abc has issued a report and held a press conference today. From watching the press conference and reading the report, my conclusion is that this has ended up exactly where one could have predicted. The LANA people have identified a precise gap in the argument (which for some reason they call a “wall” rather than a “gap”), at the point where Scholze and Stix in 2018 claimed there was a problem. The gap is where Mochizuki says two different things are the same, while LANA sees no argument for this to be the case, and hasn’t gotten one from Mochizuki, Yoshi or anyone else.
Since no one now has an argument (other than “it’s obvious”) that fixes the gap, it’s clear that what PRIMS published was not a proof, but a proof with a gap, and that the refereeing process failed in this case.
Update: The big AI news of the day is the discovery using an AI agent of a counter-example to the Jacobian conjecture. For some more about this, background, and discussion of a recent AI-found counterexample to a conjecture of Grothendieck’s, see today’s blog post by Kevin Buzzard: Human mathematicians are being outcounterexampled.
Update: While in mathematics AI agents are starting to have a major positive impact on research (e.g. by resolving the Jacobian conjecture), the situation is very different in hep-th. For an extreme example, see this paper from today. It claims to derive interesting statements about QFT from the counterexample to the Jacobian conjecture posted on twitter at 2:19 UTC yesterday, and was submitted to the arXiv less than 15 hours later (15:09 UTC). Obviously it was pretty much completely AI generated (and this is acknowledged in the text). The result is something completely worthless, but not obviously distinguishable from many other things on hep-th, which is in serious danger of going from something sad and unhealthy to something completely overwhelmed with crud.
The director of the arXiv’s reaction:
We’re having an acute and abrupt misalignment of incentives
Really worried we can’t adjust systematically rapidly enough to avoid some really badly disruptive intermediate states
That the problem for hep-th is just “badly disruptive intermediate states” seems to me an optimistic way of putting it…
Strings was in Shanghai, not Beijing. Was tempted to go to the public lecture, but didn’t.
> Gross started off the talk by complaining that he had had to work hard for the few days before his talk, since he had found that his slides were so old that the file formats now had compatibility problems with current software.
This is an old problem. This was a public talk, but it applies to plenaries as well. Hundreds of people attend a conference to hear a recycled plenary talk they’ve seen and heard before. Personally, I blame organizers for not communicating their expectations clearly to plenary speakers and for picking big names instead of interesting ones.
Ah, or maybe I am missing the function of plenaries and public talks. If they are supposed to communicate interesting, new developments and perspectives from a leading mind, they largely fail. If their function is performative, e.g., that they indicate esteem as you’ve got a big name speaker or indicate respect for tradition and status quo, then they may be working very successfully.
Andrew,
Thanks, fixed the location.
Public talks are a mixed bag, for the reasons you mention. This one however was an extreme case. The kind of hype-heavy content free performance Gross was putting on was not a great idea 25 years ago. Doing the same thing for decades, changed only by stripping out the part where you used to predict something but your prediction was falsified, is appalling.
Dear Peter,
Have you seen the following preprint https://arxiv.org/pdf/2607.06236
which seems interesting progress towards the resolution of the googly problem. Best-Leo
“I see nothing nothing that looks like”
You have double “nothing” there.
Another piece of news: the efforts over two years of a group of authors led by Kato reached the conclusion that Mochizuki’s IUT-based proof of abc is unformalizable, but they reserve judgement since Mochizuki has recently evolved on certain points. Source: https://x.com/FumiharuKato/status/2078017230537892207
Having big names isn’t a problem, that’s just how public and plenary talks work. I wouldn’t care if they were in their 30s, 40s, or even 50s. But right now, almost all the big names are in their 70s and 80s. They got famous when they were in their 30s, but somehow the field hasn’t produced any new big names under 50 since then. It’s just the same people talking about the exact same things. The field is just aging. It’s honestly like American politics. When an entire field is dominated by 80-year-old guys, you’ve got a problem.
Marie Durrieu, producer of the documentary series about Bourbaki, also produced a wonderful Radio France series on Grothendieck in 2024: https://www.radiofrance.fr/franceculture/podcasts/serie-alexandre-grothendieck-legende-rebelle-des-mathematiques
Aula,
Thanks, fixed.
Leo,
Thanks, hadn’t seen that.
jonjon,
Thanks. I’ll add more about that, and some comments, as an update to the posting.
“The LANA people have identified a precise gap in the argument (which for some reason they call a “wall” rather than a “gap”), at the point where Scholze and Stix in 2018 claimed there was a problem.”
I somehwat disagree with this statement: in section “10. An examination of Scholze-Stix document” (starting with p. 64) of https://github.com/katobungen/LANA_report_202607/blob/pdf/LANA_report_202607.pdf the authors give an explanation why they come to a different conclusion where they see the gap in the proof that has to be overcome:
I just give some particularly relevant quotes:
Paragraph 10.1, p. 46-47:
“In the final part of [15 [Scholze-Stix]], the occurrence of monodromy in the identifications among the various ordered one-dimensional real vector spaces appearing in the theory is pointed out as a flaw in the proof [15, pp. 9–10] (see Figure 7 below). The purpose of the present section is to put this observation in context, based on our exposition in §6–§9, and to indicate why this occurrence of monodromy does not represent an obstruction to the intended proof strategy of Mochizuki.”
Paragraph 10.2, p. 48:
“From this we may conclude that if the deduction of [12, Cor. 3.12, pp. 597–598] from [12, Thm. 3.11, pp. 573–580] could be phrased in a manner that factors through a diagram as in Figure 7, then one would have to repair commutativity by introducing a factor of O(ℓ^2). This would render the final inequality powerless to prove a conclusion as strong as the abc conjecture.
In our understanding, however, the inequality in [12, Cor. 3.12, pp. 597–598] is not obtained by comparing the volumes of two pilots situated on the two sides of the Θ-link. Rather, it is obtained by comparing the degree of arithmetic line bundles within the arithmetic holomorphic structure on one side of the Θ-link. Thus, the non-commutativity of the diagram in Figure 7 does not, by itself, establish a defect in the intended proof strategy.”
Paragraph 10.5, p. 49:
“To reiterate, we concur that the diagram in Figure 7 does not commute, and that repairing it would require a rescaling factor that is too large to yield a proof of the abc conjecture. Indeed, this point was also made by Mochizuki in his response to [15 [Scholze-Stix]] (see [9, §12, (IUAD)]): “[this situation] occurs whenever one works within a single holomorphic structure/ring theory.”
On the other hand, our analysis does not give rise to this particular diagram; rather, the elaboration of the η-algorithm shows that the proof of the final numerical inequality hinges on the compatibility (9-1) which is not manifestly false. However, we, the LANA project, do not have a proof of (9-1) at this time.”
Wolfgang Keller,
My understanding of these new elaborations of the problem with the Mochizuki proof doesn’t seem to me to change at all the situation since Scholze/Stix in 2018. At that time, they correctly pointed to the precise point where Mochizuki’s proof broke down. After going to Kyoto for extensive discussions with Mochizuki, they reported that he could not give a convincing argument for this step in his proof.
They discussed their best understanding of what he was trying to do and gave an argument that this couldn’t work. They did not claim to have a proof that it was not possible to fix the gap in the proof.
Mochizuki’s response was that it was some other argument that he had in mind that did not share the problem pointed out by Scholze-Stix. But he never was able to provide a convincing version of such an argument. The outcome of the attempt to formalize what Mochizuki is now saying seems to me to be that a specific gap is identified, again, exactly at the point Scholze-Stix pointed to.
I don’t find the idea that Mochizuki really does have a proof, that there is no gap, just an inability of Scholze/Stix/LANA to understand his proof remotely plausible.
Anyway, that’s my non-expert opinion. In the past I would be careful to back that up with consulting experts and getting their opinions, but I’ve stopped doing that (largely because such experts long ago started telling me this was a waste of time, they weren’t going to engage with it anymore, and suggested that others should do the same).
ICM 2030 in Glasgow