I haven’t been posting here for a while, partly due to a lot of traveling, partly due to some personal time-consuming commitments, and largely due to a lack of much in the way of news that seemed worth much attention. For some examples of such news that might be of interest:

- Due to discovery of a buckled RF finger, the LHC start-up (at 6.8 TeV/beam) has been delayed from end of next February to end of March or beginning of April. For details, see here.
- As usual in the US for many years, no one knows what the Federal HEP physics budget for the current fiscal year is, although we’re a couple months into it. The formal US budgeting process involves a long process including an executive branch budget proposal and congressional committee hearings and debate. This however does not lead to actual budget numbers, which only emerge at the last minute, made in some way understandable to no one I’ve ever asked about this. From the latest news, the US might have a budget any day now, and then a bit later we’ll find out what the HEP budget will be.
This year the process has involved a highly peculiar situation with the budget for US LHC contributions (prospects for large cuts, assumed to get fixed mysteriously in the last minute process). For the details of what is going on, there’s a news story here, and discussion at an HEPAP meeting here. For the first time I’m aware of, the HEPAP meeting videos are on Youtube (see links here), so one can follow the actual discussion between physicists and government officials there.

- In non-news about the abc conjecture, the Japanese media appears to be reporting uncritically about IUT-based claims of proofs that are not accepted by the vast majority of experts in the subject. There have been a couple of workshops devoted to IUT (see here and here) recently, with those speaking about IUT almost all based at RIMS. Recently Mochizuki has posted a strange Invitation to view IUT workshop videos. To view the videos you have to apply, and promise not to use them for “non-mathematical purposes”. My guess is that one of the “non-mathematical purposes” at issue would be bloggers pointing out that nowhere in the talks does anyone discuss the fact that convincing arguments have been given by Peter Scholze and Jacob Stix that the IUT-based proof of abc is flawed and cannot possibly work. This problem is addressed with:

Unfortunately, it has come to my attention that certain misunderstandings concerning IUT continue to persist in certain parts of the world. Perhaps the most famous misunderstanding concerns an asserted identification of “redundant copies”. This misunderstanding involves well-known, essentially elementary mathematics at the beginning graduate level concerning the general nonsense surrounding “gluings”. For instance, if one “applies” this misunderstanding to the well-known gluing construction of the projective line, then one concludes that the two copies of the affine line that appear in this gluing are “redundant’’, hence may be identified. This identification leads immediately to a contradiction, i.e., to a “proof” that the projective line cannot exist! More details may be found in the Introduction to [EssLgc] and the references given there.

In case anyone thinks it’s plausible that Peter Scholze is making errors in elementary mathematics at the beginning graduate level, David Roberts has an explanation of what’s going on here.

On the string theory front, it’s become impossible to figure out how to have any sort of scientific debate about most of the public defenses of “string theory”. For two recent examples:

- In an article about What We Will Never Know, David Gross rather explicitly acknowledges that prospects for testing ideas about string theory are now an issue of “faith”, with no hope of turning into science any time soon:

There’s faith that one way or another we should be able to test these ideas… It might be very indirect—but that’s not something that’s a pressing issue.

- For Nabil Iqbal, string theory is now to be understood at the pre-scientific level of parable. In his parable, human beings trying to understand the equations of string theory are like fish trying to understand the equations governing the behavior of water. I’m trying to think of a sensible comment about this, but I’ve got nothing.

So presumably it would be like humans trying to understand the equations that govern air? If so, that seems like good news. The Navier-Stokes equations are challenging no doubt, but there’s enough optimism surrounding them that it seemed reasonable to attach a $1M prize to one of the biggest outstanding problems relating to them. String theory being in a similarly coherent state would be unambiguously good news — right?

S,

In the parable, humans are trying to understand the vacuum, not air, and our current understanding of the vacuum based on QFT is like a smart fish understanding water using fluid mechanics. A string theorist telling us that the vacuum needs to be understood in terms of strings is like a really super-duper smart fish telling other fish about atomic physics.

Again, trying to think of something intelligent to say about this parable, but failing utterly.

I sent the note I wrote to Mochizuki, and he responded to explain more of what he meant, in terms I could understand this time. He says that his approach to diagrams (‘labels’, etc), is the same as the standard approach that I give, which is true—up to isomorphism and idiosyncratic notation and terminology. But his contention is that Scholze and Stix are not just replacing merely isomorphic objects by equal objects, but also collapsing the diagram indexing them so that the indexing nodes become equal.

For those not used to category theory, one can have a nontrivial diagram—take a directed graph, for simplicity,—with many nodes, where different nodes can be labelled by the same object. Mochizuki’s style is to additionally decorate objects at different nodes with distinct labels (wholly unnecessary, from a technical point of view). Ordinary category theory just remembers the shape of the diagram to distinguish the nodes. Mochizuki contents that his critics are collapsing the diagram so that if two nodes are decorated by the same object, the nodes are collapsed, forming a quotient of the diagram. I cannot imagine a situation where someone well-versed in category-theoretic vernacular would make this mistake.

Our discussion is ongoing; at this point I’m simply trying to diagnose the thought process that leads to his claim at a technical level (not the sociological, mind!). We are only discussing a simple example, far,

farfrom IUT, but it’s an example that Mochizuki himself claims is a very good illustration of the actual construction, and the error he claims is being made in the critique.It would be interesting to hear from someone who has followed the Japanese media if the situation there really is as bad as it looks from the outside (to someone who has to rely on Google Translate etc. to read Japanese). Is there any even slightly critical commentary about the IUT in popular media?

Anon,

As far as I know, there has been no critical commentary of Mochizuki’s claim in the japanese media.

(PS: I presently live in Kyoto).

The Fish analogy reminds me of a different fish analogy I read many years ago.

The author claimed that Linde explained that our universe is like an ocean, and scientists are like fish trying to explain why water is the temperature it is from first principles. However, the universe actually contains many oceans, and the temperature of our ocean is just down to random chance. The conclusion? Apparently, our universe must be one universe of a vast multiverse of possible universes, each with differing physics. This was “explaining” why String Theory is unable to, well, explain our universe. Nothing changes…

Jim Eadon,

It’s the same analogy, with the same “explanation” of why we can’t explain anything (viscosity replacing temperature).

David Roberts wrote:

It would definitely change the character of the Four Color Theorem if a map that could be colored with four colors must only contain four countries.

All,

There’s no point to following Mochizuki in debating things which have nothing to do with Scholze’s actual argument. If anyone wants to engage with that, there’s a detailed discussion at

https://www.math.columbia.edu/~woit/szpirostillaconjecture.pdf

There at the beginning Scholze asks

“If this further identification causes problems, just tell us which diagram it is whose commutativity is rescued by not explicitly identifying $\pi_1(X)$’s.”

No one in the discussion there was able to answer this, and Scholze reports that Mochizuki could not do so in their private discussions. Publicly, Mochizuki has never answered the question, preferring to argue that Scholze is an incompetent who needs to spend more time studying the IUT papers to find the answer to his question.

Given this situation, what’s going on here is clear, and there’s no point to wasting more time on it.

The Asahi Shimbun story has now been published in English

https://www.asahi.com/ajw/articles/14488092

Some real mathematics news: it seems that the 2026 ICM will take place in Philadelphia. The official decision will of course only be made at the IMU General Assembly in July, but there are no other applications this time.