Occasionally in mathematics a statement that just has to be true turns out to be false. A paper appeared on arXiv today that disproves a well-known conjecture in probability called the bunkbed conjecture. 🧵

arxiv.org/abs/2410.02545 Here's what it says. You start with a connected graph G, which you think of as the bottom "bunk". You then take a copy G' of G, which is the top bunk. For each vertex x in G, let x' be the corresponding vertex of G'.

I've just tried out a maths problem on ChatGPT4o on which it failed, with a failure mode interestingly similar to its well-known failure on problems such as "I have a goat and a sheep and a large boat. How many crossings do I need to get them both across the river?" 1/5 I asked it, "I have a 7x7 grid with one corner removed. Can I tile it with 3x1 rectangles?" It is not hard to see that the answer is yes. But ChatGPT4o told me no, since there will be an unequal number of black and white squares. 2/5

Google DeepMind have produced a program that in a certain sense has achieved a silver-medal peformance at this year's International Mathematical Olympiad. 🧵

deepmind.google/discover/blog/… It did this by solving four of the six problems completely, which got it 28 points out of a possible total of 42. I'm not quite sure, but I think that put it ahead of all but around 60 competitors.

However, that statement needs a bit of qualifying.

I'm very sad to hear that Daniel Dennett has died. I greatly enjoyed his books Consciousness Explained and Elbow Room, and I hope I won't annoy too many people if I express my opinion that what he said in those books was basically right. 1/ For instance, I agree with him that computers could in principle be conscious (but would be very cautious about making such a claim of an actual computer), and also that free will can be reasonably defined in a way that makes it entirely compatible with determinism. 2/

Today I start my seventh decade, so here are a few reflections on what it's like to reach that milestone. 🧵 1. I've had an extremely fortunate life so far. Of course, nobody reaches the age of 60 without some bad things happening, but I've had a lot less bad than my fair share and a lot more good.

My son has just started calculus, and I asked him what the relationship was between the gradients of the tangent and the normal to a curve at a given point. His first reply was, "They are perpendicular." I've noticed many times that something one gains with experience ... 1/7 in mathematics is an acute sensitivity to types. An experienced mathematician could not give that answer, for the simple reason that gradients are real numbers and two real numbers cannot be perpendicular to each other. 2/7

I have often seen statistics like this, and am very much in favour of curbing the high-emitting activities of the rich (and while there are plenty of people richer than I am, I am not excluding myself from the people whose emissions must be curbed).

But ... 1/

https://twitter.com/Daryl_Elliott/status/1680260687011135490

there is an important calculation that economists must have done somewhere, which I have not managed to find, concerning what the effects would be on emissions of a big redistribution of wealth. 2/

It's an amazing time to be alive for a combinatorialist at the moment, with a number of long-standing problems, several of them personal favourites of mine, being resolved. Today I woke up to the news of yet another breakthrough, due to Sam Mattheus and Jacques Verstraete. 🧵 A month or two ago I tweeted about a stunning new result that obtained an exponential improvement for the upper bound for the Ramsey number R(k,k), a problem I had thought about a lot. When I felt stuck on that, I would sometimes turn my attention to a related problem

I was at a sensational combinatorics seminar in Cambridge yesterday, reminiscent of the time I had been tipped off that Andrew Wiles's seminar at the Newton Institute on Wednesday 23rd June 1993 might be worth going to. 🧵

arxiv.org/abs/2303.09521 The speaker was my colleague Julian Sahasrabudhe, who announced that he, Marcelo Campos, Simon Griffiths and Rob Morris had obtained an exponential improvement to the upper bound for Ramsey's theorem.

For a while now I’ve wanted a rule of thumb that would allow me to estimate the amount of harm that would result from various carbon-emitting activities I might take. I’ve now thought of one (unlikely to be original) that I find satisfactory, though it needs refining.🧵 1/26 What I have found hard about the question up to now is that an activity such as taking a plane flight will be adding just a tiny percentage to the amount of carbon in the atmosphere, making an almost undetectable difference. And yet all these contributions add up. 2/26

For a long time I have had an interest in automatic theorem proving, and have done a small amount of research in the area. I am happy to say that I will soon be stepping up these efforts, thanks to a generous grant from the Astera Institute. 1/

astera.org This will enable me to build a small team. I also plan to work in as open a way as possible (consistent with the need of participants to publish) and will very much welcome remote participation from anyone who feels like contributing. 2/

gowers.wordpress.com/2022/04/28/ann…

I recently became aware of a nice paper by Bogdan Grechuk called "Diophantine equations: a systematic approach."

If you know a bit about the history of 20th century mathematics, you may find the title a little puzzling. 🧵

arxiv.org/abs/2108.08705 A Diophantine equations is just like an ordinary equation, but with the extra stipulation that the solution is required to be in integers. This typically makes the equation much harder to analyse.

I had an interesting (for me) mathematical conversation with my 11yo daughter a couple of days ago. She knows about raising numbers to positive integer powers, so I thought I'd extend her knowledge just a little. So I asked her what 2^5, 2^4, 2^3, 2^2, and 2^1 were. 1/ She gave me the correct answers: 32, 16, 8, 4, and 2.

Now, the moment I had been waiting for: what is 2^0?

Her first answer was 0, and when I raised an eyebrow she switched to 2.

I tried again. What are 2^5, 2^4, 2^3, 2^2, and 2^1? So what is 2^0?

It didn't work. 2/

@spectator I think what Graham Medley should have said (and probably meant, even if he wasn't as clear about it as he could have been) is something like the following.

1. There is a lot we don't know about Omicron. @spectator 2. Therefore, the government needs to know what the possible outcomes are, given various assumptions about transmissibility, severity, etc., and also how likely the various assumptions are.

Ce trimestre, j'ai organisé un séminaire au Collège de France, consacré à la philosophie de la pratique des mathématiques. Les séances sont maintenant toutes disponibles en ligne. 1/ J'ai commencé moi-même par deux séances avec le titre « Pourquoi croire à un énoncé mathématique dont on n'a pas de preuve ? » Dans la première partie, j'ai discuté de plusieurs exemples. 2/

college-de-france.fr/site/timothy-g…

There is a lot of debate on Twitter, or at least on the part of Twitter I see, about whether mathematics is "objective" and "apolitical", and quite a lot of it consists in people talking past each other. So maybe it's worth making a few obvious (and far from new) points. 1/ A. If you have formalized your proof in a language such as Coq or Lean, then you've definitely got a proof.

B. There are many proofs in mathematics that have not been formalized but for which we can be extremely confident that they could, with enough effort, be formalized. 2/

I heard today about yet another round of redundancies at a UK university: this time it's Life Sciences at Liverpool. They wrote to 47 people, but have now, under pressure from a strike, reduced that to 28. Apparently the news came out of the blue. 1/

hls47.co.uk As ever, there is a grotesque name for the university restructuring that has led to this: Project Shape.

A shocking aspect of the matter is that the "underperforming" people selected for redundancy were chosen on the basis of grant income and ... 2/

A short thread about Johnson's response to being asked whether tens of thousands of people died unnecessarily. His response was, "No, I don't think so, but of course this has been an incredibly difficult series of decisions, none of which we've taken lightly, ... 1/ and you've got to recognise, and I hope people do understand this, that when you go into a lockdown, it's a very very painful and traumatic thing for people, for people's mental health, for their lives, their livelihoods, and of course you've got to set that against ... 2/

The poll I conducted recently was on behalf of somebody else, who turned out not to be satisfied with the wording. I would therefore like to run a similar poll with the following more detailed wording. I still intend to explain the purpose of the poll once it is over. 1/ Let N_n denote the natural numbers starting from n>0. Suppose that someone managed to show the following.

There exists a proposition P and sets A, B such that for some x

(1) P => A = B
(2) notP => A = B
(3) P => A = N_x
(4) not P => A =/= N_x

/2

A breakthrough on arXiv this morning. A famous problem used to be to determine asymptotically the probability that a random nxn ±1 matrix is singular. That is now almost solved, but the related problem for symmetric matrices was still wide open. 1/

arxiv.org/abs/2105.11384 The preprint that has just been posted gives the first exponentially small upper bound for this probability.

Without the symmetry constraint, there was a long series of ever improving results -- a bound of o(n), a bound of something like 0.999^n, exponentially small ... 2/

I'd be curious to know what the counterarguments are to this argument. At the very least, the argument is presented in a way I like -- it carefully doesn't claim that the virus must have originated in a lab, but, to my non-expert eye at least, ... 1/

thebulletin.org/2021/05/the-or… it appears to show that the possibility can't be ruled out. Up to now I've always dismissed this theory as a nutty conspiracy theory, but the article is suggesting not a deliberate release of a virus but the more boring theory that it got out accidentally because of lax safety.2/