Timothy Gowers @wtgowers
@wtgowers
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Mathematician. Professeur titulaire de la chaire Combinatoire au Collège de France. Also fellow of Trinity College Cambridge.
Joined January 2019
Here's a puzzler from my daughter's maths homework.
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Update -- the mystery is now solved. The answer to question 13) is 360. My daughter put infinity, so she got 39/40 on the homework, but that is how you learn.
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A while ago, I tweeted that I hoped Cambridge would allow us to post our online lectures publicly. I'm delighted to say that they do indeed allow this, so I shall be posting my course Topics in Combinatorics on YouTube as I give it. 1/.
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I'm genuinely baffled by this question from my daughter's homework. What is the intended answer? Maybe I'll find out over the next week.
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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.
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Here is a conclusion that seems utterly obvious: if it is inevitable that at some point a lockdown will be necessary, then DO IT NOW. Why? For the simple reason that every two or three days, the number of cases doubles. This has two effects. 1/.
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A bizarre discussion is going on on Twitter at the moment, concerning whether 2+2=5. Apparently, it's "woke" to try to undermine the truth we all know, that 2+2=4. As a mathematician who fondly imagines that he is towards the woke end of the spectrum . 1/.
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I strongly approve of this question that my daughter has been set. Using what you already know is such a fundamental mathematical skill, and one that doesn't come automatically.
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Today I start my seventh decade, so here are a few reflections on what it's like to reach that milestone. 🧵.
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I was vaguely wondering about commenting on Richard Dawkins's tweet about eugenics, but then I came across this response, which wins Twitter for today.
All of Dawkins’ tweets make more sense if you add “. Mr Bond” at the end of them.
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Google DeepMind have produced a program that in a certain sense has achieved a silver-medal peformance at this year's International Mathematical Olympiad. 🧵.
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It's finally happened: after several unsuccessful attempts, I found a prompt that got Grok to solve a maths problem (the well-known Dubnovy Blazen problem in graph theory) I've been working on for over a year. How long till it's better than human mathematicians across the board?.
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My daughter told me she was pleased to manage this question, so I had a look, and very much approved of it — a nice piece of multistep reasoning.
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I have a joke about Elsevier that I originally heard from you but you'll have to pay me to tell it.
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Today I have received by email: one disproof of the Riemann hypothesis, one proof that P =/= NP, one proof of the Riemann hypothesis, and one proof that Peano arithmetic is inconsistent (which follows from the two Riemann hypothesis papers, but this was an independent discovery).
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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. 🧵.
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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. 🧵.
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Last night I was putting the finishing touches to my proof of the Riemann hypothesis, when, just as I was starting to think about what journal to submit it to, I found a small mistake. Then, one by one, my key lemmas magically turned out not to be correct. VERY STRANGE!.
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I've just seen the words "The principal principle that we prove in this paper" in a maths paper and it's brightened my whole day, to the point where I could barely keep my stationery stationary.
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So Sunak's honeymoon period lasts all of six and a half hours.
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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/.
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An interesting paper by Adam Wagner appeared on arXiv a couple of days ago (thanks to Imre Leader for drawing my attention to it), which uses reinforcement learning to find non-trivial counterexamples to several conjectures in graph theory. 1/.
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I'm British and there is no universe in which this piece of unpleasantness is delivering for me.
From today, the majority of foreign university students cannot bring family members to the UK. In 2024, we’re already delivering for the British people.
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All Cambridge University's lectures are going to be online next year. It would be wonderful if they were made publicly available in perpetuity: does anyone know whether the university has an enlightened policy on that?.
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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/.
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I've long wondered how highly specialized academic bookshops in Paris such as this one stay in business. The turnover must be tiny. But it's also near the Panthéon, so unless there's some arrangement I don't know about the rent must surely be very high.
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Retweeting because I think it's important to be reminded how much easier it is to interact online as a white, straight, cis etc. male. The word "privilege" is often used for this, but it seems like a strange word for not receiving disgusting abuse, which should be a basic right.
Many of you have advised me to ignore the recent trolling. I think you’re right but I wanted to show you some of what I’m ignoring.
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@TaliaRinger Maybe take the sum of the internal angles of any corners, plus the sum of the changes in angle of the tangent to the shape between corners. E.g. for a semicircle it would be 90 + 0 + 90 + 180 = 360. (Curious to know what the official answer would be for a semicircle though!).
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A good measure of how well someone understands science is their level of uncertainty about why Covid case numbers are dropping in the UK at the moment.
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A clever question on my daughter's latest maths homework, as it's easy, but only if you don't try to do too much. The numbers 1 to 8 are placed on the vertices of a cube in such a way that the total on the vertices of each face is the same. What is the total on each face?.
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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/.
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I've started a new playlist on my YouTube channel where I will video myself attempting to solve mathematical problems. It's something I'd wondered about doing once I got used to giving courses online. This is my first attempt. 1/.
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Here's a great example of human-AI collaboration to obtain new results in mathematics. In this case, a team from DeepMind has used LLMs to obtain the best known lower bound for the cap-set problem. Of course, there's more to it than talking to ChatGPT.
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It is often suggested that one should teach Euclidean geometry in order to introduce children to the notion of proof. Today my children learnt a huge amount about proof in a completely different way when, as an experiment, I tried the "miracle Sudoku" with them. 1/.
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Another mass resignation of an editorial board has happened, and this one feels like quite a big deal, as the journal in question is Journal of Combinatorial Theory A, one of Elsevier's premium combinatorics journals. 1/.
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Update: it's finished now.
This is just wild!. Nov 9: @wtgowers, Green, Manners, and Tao prove the polynomial Freiman-Ruzsa conjecture in char 2. Nov 18: Tao announces that he wants to formalize it in Lean, building on Mathlib. Dec 4: One little theorem
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Nobody who sees Cantor's proof that there cannot be a surjection from a set to its power set, and also Russell's proof that there cannot be a set of all sets, can fail to notice the close similarity between the two arguments. But somehow it's taken me till today . 1/.
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I've just noticed that a function f:X->Y is a bijection if and only if it preserves complements, that is, if and only if f(X\A)=Y\f(A) for every subset A of X. Is this a standard fact that has somehow passed me by for four decades? Simple proof in rest of (short) thread. 1/3.
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It didn't take long for him to correct himself and give the answer I had been looking for, but the point remains: get someone into the habit of being aware of the type of everything they are talking about, and their mathematical thinking automatically becomes much clearer. 3/7.
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Since the discussion about whether 2 + 2 = 4 is making a brief comeback, here are five arguments, two in favour of the statement and two against. 1. 2 is the successor of 1, and addition is defined inductively by x + 0 = x and x + succ(y) = succ(x + y).
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For mathematicians, it's a nice exercise to work out what's going on here. I don't want to spoil the fun by tweeting my explanation.
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Formalization of mathematical proofs has been talked about a lot recently. Today a new journal is launched, the Annals of Formalized Mathematics, aimed principally at mathematicians rather than computer scientists. 🧵.
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I've done a little bit of reading this morning about network epidemiology -- that is, modelling disease spread on a graph and actually taking some account of the structure of that graph -- and learned a fact that for the mathematically inclined is quite amusing. 1/.
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An analogy that I read somewhere and do find quite helpful is that of adoptive parenthood. To say "You are not that child's mother" to an adoptive mother would be offensive, for a similar reason, so we use the word "mother" for adoptive mothers. And yet we don't see anyone . 15/.
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One of the best known open problems in combinatorics is the union-closed conjecture, which states that if you have a finite collection X of sets such that if A and B belong to X then so does the union of A and B, then at least one element of X belongs to at least half of them. 1/.
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A great preprint appeared on arXiv this morning by Marcelo Campos, Marcus Michelen, Julian Sahasrabudhe and Matthew Jensen -- the first improvement by more than a constant factor to the lower bound for sphere packing in large dimensions since 1947. 1/10.
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Then they'll know, for example, that an equation has solutions but a polynomial has roots, which are solutions to a closely related equation. This example sounds a bit pedantic, but it helps to avoid muddle, and I've witnessed that muddle several times. 4/7.
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the suggestion is that it is far more natural and sensible to say that trans women are not women, and that ridiculous mental contortions are needed if one wants to call them women. But it isn't hard to use the word "woman" in that way, and there are good reasons to do so. 20/20.
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I'm super-proud of my PhD student Bhavik Mehta, who has formalized the remarkable breakthrough concerning diagonal Ramsey numbers, which I tweeted about a few months ago. The paper is not yet accepted by a journal, but Bhavik has formalized the result! 1/.
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Good news: I've just written a detailed sketch proof of a new result that I like a lot. Bad news: the bound I obtain contradicts a well-known bound in the other direction. I can't find the weak point, so I'll have to write out the proof in full gory detail until it collapses.
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An exciting result on arXiv this morning. Thomas Bloom and Olof Sisask have proved the first non-trivial case of a very famous conjecture of Erdös. 1/.
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This report, on antiracist mathematics education, has been held up as a reductio ad absurdum of Wokeism. Its suggestion that a focus on getting the right answer is racist has been held up for particular ridicule. 1/.
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The University of Leicester plans to make all its pure mathematical researchers redundant. They tried something similar in 2016 and backed down after a storm of protest from all round the world. Unfortunately, it seems that another storm is required. 1/.
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Watching this gives me goosebumps -- not literally, but it does have an emotional effect. If only we had just one frontline politician in the UK (ideally the PM) who was sufficiently scientifically trained to talk about the virus with this level of clarity and understanding.
This is how Angela Merkel explained the effect of a higher #covid19 infection rate on the country's health system. This part of today's press conf was great, so I just added English subtitels for all non-German speakers. #flattenthecurve
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Very sorry to hear this. I was taught by Conway as an undergraduate at Cambridge, and a little under 20 years later was a colleague of his for two years at Princeton. I first met him at a maths event just before I went to Cambridge. By way of small talk, he . 1/.
RIP John Conway. I asked him a lot of questions when I was a postdoc at Princeton and he always had a lengthy, deep, informative answer.
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I was recently offered a nice honour -- nothing too major, and no financial consequences, but flattering and I would have liked to accept it. Indeed, in the past I would have accepted it. However, it required me to make a transatlantic flight, 1/.
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My daughter was asked to evaluate. 3 x 4 x 5 x 6 x 7 x 8.-------------------.2 x 3 x 4 x 5 x 6 x 7. and did so by evaluating the top and the bottom, getting 20160/5040, and doing the division. It led to a long conversation . Her approach illustrated a very common error, 1/.
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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.
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Am I worried by this answer my daughter gave on a recent maths test? No I'm not.
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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. 🧵.
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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.
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There's a big new result -- in fact, pair of results -- on arXiv this morning, due to James Leng, Ashwin Sah and Mehtaab Sawhney. One of them beats a record that I had been quite pleased to hold for the last quarter of a century (though I'm also happy it's been broken). 🧵.
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There have been several remarkable developments in combinatorics, my field of mathematics. A few weeks ago I gave a talk to a general mathematical audience in which I described six breakthroughs from the last five years.
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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/.
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I'm certainly not the first person to do this, but last month, after yet another spammy invitation to submit to a predatory journal, I decided to send them some junk. A quick Google and I found a site that generates nonsense maths papers. 1/.
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The doubling time before significant social-distancing measures were imposed was around three days. The halving time afterwards seems to be a lot longer. At a guess (which I'd love to be wrong), each week of delay at the beginning will have cost about a month at the end. 1/.
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Just learnt about this from a retweet by @emilyriehl. If you have a number such as 364 and want to know quickly whether it is divisible by 7, you take the last digit, multiply by 5, and add to the rest of the number, so here you get 36+5x4=56. That's a multiple of 7, so 364 is.1/.
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I've seen many variants of this diagram. What they ignore is that the dotted line is in practice much lower, so flattening the peak to the point where the health service can cope is approximately as hard as trying to stop the spread altogether. 1/
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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/.
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But since it seems almost certain, given the experiences of other countries, that a lockdown will indeed be necessary, then I don't feel too uneasy saying . !!DO IT NOW!! . (but would be ready to change my mind if someone pointed out a consideration I've overlooked). 6/6.
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I 100% don't understand how this works. Good to have a bit of mystery in my life I suppose.
This arrow by mathematician and sculptor Kokichi Sugihara can't point left. Here's how it works: It's 3D-printed with a bunch of curves our brains don't register.
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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. 🧵.
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Sometimes it feels like a straightforward privilege to be a mathematician and a parent at the same time. Recently my 14yo son asked "If I want to double the area of a square, would I have to multiply the side length by the square root of 2?" In my normal irritating way, 1/.
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It's taken a couple of millennia, but finally here's a clear demonstration that Euclid's parallel postulate is false.
This image consists of squares aligned vertically or horizontally, but the alignments appear to tilt.
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This is a nice one. It's not exactly an optical illusion, but more like a very nice demonstration that we can be sensitive to a highly structured "derivative" of a pattern even if at any one moment the pattern itself is structureless.
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This photo is of the first page of an original offprint of Ramsey's article in which he stated and proved what we now call Ramsey's theorem. It came into my possession because Joel Spencer, an extremely distinguished Ramsey theorist (and combinatorialist more generally) .
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It is well known that ChatGPT is bad at problems to do with crossing a river with animals. So to make things more interesting, define its crapness ratio to be the ratio between its answer and the correct answer. Can anyone beat the crapness ratio of 5 that I've just achieved?
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My daughter randomly asked me this morning what 365x365 is. I've sometimes tweeted about mental arithmetic, and to my surprise and delight this question turned out to be one where two nice tricks can be combined. 1/.
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Just uploaded: a video about entropy, the first in a sequence of videos that will talk about what the information-theoretic notion of entropy is and how it can sometimes be applied to give beautiful solutions to combinatorial problems.
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Conversely, not being completely clear about what type of object everything is can make a multi-step problem like that too hard for an inexperienced mathematician to solve. 7/7.
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Je suis très heureux et très honoré d'annoncer qu'il y a trois jours le président de la République a signer un décret dont le résultat est qu'à partir du 1er octobre je serai chaire de combinatoire au Collège de France. 1/.
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@kareem_carr My favoured explanation is this. If you don’t switch, you have to get it right first time. If you do switch you have to get it wrong first time. The latter is easier.
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Ever since Zoom became a thing, I've prided myself on never being that person who launches into a speech at a meeting and then has to repeat it after being told they're on mute. Today my long run came to an end. Gutted. ☹️.
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Amongst the many petition tracking sites (which help to avoid the actual site crashing), this is quite a good one, as it puts the constituencies in order of how many signatures they have. Proud to see that Cambridge is third. #RevokeArt50Now.
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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/.
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Two small silver linings. First, I don't (yet?) feel too bad. Second, there was some point to wearing a mask on two recent train journeys even though nobody else did.
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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.
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Patrick Vallance, the government's chief scientific adviser, says that the policy is to let 60% of the country get it. Here is my (no doubt flawed in many respects) understanding of that policy. If on average each person infects 2.5 others in a population with no immunity, 1/.
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An exciting result has just appeared on arXiv, concerning the following simple-seeming problem: if A is a set of n positive integers, then how large a sum-free subset B must it contain? That means that if x, y and z belong to B, then x + y should not equal z. 🧵.
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19. A small thing to add is that I don't feel very different inside from how I felt when much younger. Of course, I've experienced a lot more, but I still feel a bit like a 17-year-old who is just pretending to be a grown-up.
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A nice question from my daughter's homework today. At first it seems like a boring brute-force search, but one can in fact solve it with a nice argument, which I'm happy to say is what she did (after initially thinking it was impossible, which paradoxically may have helped her).
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Jacinda Ardern has rightly been showered with praise for her coronavirus response. But deaths in Senegal per head of population have (so far) been lower. Where are all the articles asking what we can learn from the Senegalese response? .
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Simon Jenkins dusts off his anti-maths article and publishes it for about the tenth time. One of those times was six years ago and I was moved to reply. Here's the latest version of the article, and linked in the next tweet is my reply to the previous one.
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The first two videos (equivalent to one lecture) are now up on my fledgling YouTube channel. They are not particularly polished, but that wasn't the idea. 2/.
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Kicking myself for not spotting this . .
News! 42 is the sum of three cubes. 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3.
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