Peter Cameron's Blog

Take a right-angled triangle with hypotenuse c and the other two sides a and b. Pythagoras’ Theorem tells us that c2 = a2+b2. Let the area of the triangle be A. We know that A = ab/2 (since an a×b rectangle is cut into … Continue reading →

Imagine you are in the following situation. You are the foreign minister of your country. You are in New York for a meeting of the General Assembly of the United Nations. A powerful enemy has been deploying troops on the … Continue reading →

Graphs and groups, in my view, are two subjects engaged in a wide-ranging dialogue at present. Graphs can be used to describe interesting classes of groups, and groups to construct interesting graphs. But I am delighted that recently, in a … Continue reading →

A few years ago, R. A. Fisher (who has been derscribed as “the greatest Darwinist since Darwin”, and as “the founder of modern statistics”) fell victim to one of the waves created by the Black Lives Matter campaign, and a window commemorating him in his Cambridge college was removed (undoubtedly without due process: just two days elapsed between the proposal and the fait accompli). I wrote about …

The latest newsletter from the International Mathematical Union urges mathematicians to attend this year’s ICM in Philadelphia. It will help our beleaguered collleagues in the USA by showing support, and the organisers guarantee our personal safety. Well, I will not be there. But after moral pressure like that, I feel called on to justify my decision. So here goes. - I believe I do a certain amou…

I believe Eugene Plotkin has died. I read this on the AGTA website but I have no details. I met him for the first and only time at Daniela Nikolova’s birthday conference in Deerfield Beach, Florida, four years ago. We got on very well. Outside of mathematics, one of his hobbies is finding beautiful “works of art” in natural stones; he sent me some of the extraordinary videos he had produced.

I have spent a lot of time recently thinking about graphs on groups. To recall the rules: the vertex set must be the group (in general, but not here, I allow an automorphism-invariant subset or the quotient by an automorphism-invariant equivalence relation); the graph must be defined in terms of the group structure with no arbitrary choices; so the automorphism group of the group acts as automorp…

I have been, and remain, sceptical about AI. At best, it is saiad to be good at writing programs, and finding specific facts; but it has a tendency to lie, to invent, and to tell you what it thinks you want to hear, so you have to check very carefully everything it tells you. But I was sent a manuscript by Donald Knuth where he claims that Anthropic’s robot Claude (named after Claude Shannon, the…

As the tagline for this blog says, I like counting things. Reading my Iran diary reminded me of a counting problem I solved then, of which I am quite proud. But like all good problems, it leaves a loose end, which you might like to try. THe problem came from Charles Johnson, one offour pairs of my coauthhors with the same surname. He was interested in properties of a matrix which are at least par…

What will happen to Iran once the smoke clears and the bombs stop falling? No one can predict; maybe the ayatollahs will hang on, and there will either be brutal repression or a chaotic attempt to remove them; maybe the Arabs will invade (the last Arab invasion is regarded as the greatest disaster in the history of Persia); maybe the Americans will install a puppet regime so they can steal the oi…

Last month I was in Vienna, at AAA108. This was the 108th meeting of the Arbeitsgruppe Allgemeine Algebra (or Workshop on General Algebra), which has been going since 1971, with usually two meetings a year. The meetings are held at weekends, to maximise the chance that people can attend without disrupting their lecturing schedules too much (Friday, Saturday and Sunday morning). It was founded by …

It is not unusual for a mathematician, having proved a theorem, to devalue the creativity that went into it. Once a theorem is proved, it is obvious, at least to its prover; it is easy to think that anybody could have done that. So this is common among mathematicians, and is sometimes referred to as impostor syndrome. What is much more unusual is for a mathematician to disown the result. One of t…

The set Q of rational numbers is obviously an interesting topological space. In 1920, Waclaw Sierpiński gave a lovely characterisation of it. The simplest way to state it is to say that a countable, metrisable, space without isolated points is homeomorphic to Q. (Sierpiński also gave a purely topological characterisation: a countable, second countable, 0-dimensional, T1 topological space without …

Two brief topics. My resolution to publish in diamond open-access journals is already in tatters. I assumed this would happen because I had coauthors who were compelled to publish in certain journals. But the other plausible reason for it to … Continue reading →

Last night, I took part in a debate organised by the students’ Debating Society and Mathematics Society jointly. The proposition before the house was This House Believes That Mathematics Is a Human Invention Rather Than a Discovery. When I was … Continue reading →

Some of the standout results about graphs on groups are characterisations of the groups G for which two types of graph (for example, the power graph and the commuting graph) coincide on G. Sometimes the proofs are long and difficult, … Continue reading →

Please spare a thought for colleagues and friends in Iran. It is hard for us to imagine the suffering these generous and hospitable people are enduring at present, and I will not attempt to describe it. I do not want … Continue reading →

This year is the 100th anniversary of the Journal of the London Mathematical Society. They have celebrated the centenary by an issue of the journal containing ten papers, each starting from an important paper published in the Journal. The entire … Continue reading →

Happy New Year to all. My wish is that the coming year may be better than the one just past in at least some way. One change I propose in the new year is that I will move some things … Continue reading →

Today I watched, on Natalia Maslova’s on-line seminar from Yekaterinburg, a talk by Sergey Shpectorov from Birmingham, on the non-existence of a strongly regular graph with parameters (85,14,3,2): this is a graph with 85 vertices, regular with valency 14, and … Continue reading →