The week before last I was at the Deerfield Beach Resort in Florida, about halfway between Miami and Donald Trump’s place. My children could not believe it when I told them (I am not very good at holidays). Of course they guessed that it was not a holiday but a mathematics conference; in fact, in honour of the 70th birthday of Daniela Nikolova.
Unfortunately Daniela had to spend the entire conference in a wheelchair, having slipped and broken an ankle getting out of the jacuzzi on the pre-conference cruise. So we all wished her a speedy recovery; I do hope this happens.
It seemed important for speakers to remember when they first met Daniela. I am not sure of this. The first I can recall is Groups St Andrews in 2005, but there may have been earlier occasions which I have forgotten. A remarkable thing about the meeting was the number of pairs of participants who met for the first time at a Groups St Andrews meeting. It might be interesting to construct the graph …
The talks were a very mixed bag. There was a day for “Women in mathematics”, chaired by a man; and a day for “Young researchers in mathematics” bookended by two researchers who were no longer young. I won’t give a blow-by-blow account; I will just single out a few of my favourite talks. The first was by David Burrell. The count of groups of order 1024 announced in the year 2000 was incorrect; when he re-did the computations, he got a different number. At first he assumed he had made a mistake, and checked his calculations carefully. So eventually he contacted Eamonn O’Brien, a safe pair of hands if ever there were one. Eamonn was able to locate the mistake, apparently a simple transcription error in the tabulation of the results. It is not clear if it was human or computer error, though it looks like the former. Anyway, the correct number of groups of order 1024 is 49,487,367,289.
Gareth Jones told us about the Bateman–Horn conjecture. This is a conjecture in number theory but would have a number of important spin-offs in group theory, including the statement that there are infinitely many insoluble (and so 2-transitive) permutation groups of prime degree other than symmetric and alternating groups. The conjecture goes as follows. Suppose that you have a finite number of integer polynomials satisfying the three conditions
- each polynomial is irreducible;
- each polynomial has positive leading coeffiient;
- the product of the polynomials is not identically zero modulo any prime.
Under these conditions, Schinzel conjectured that there are infinitely many values of the argument for which all the polynomials take prime values. Bateman and Horn gave a considerable refinement of this conjecture, actually giving a formula for the conjectured asymptotics, a little complicated to state here but computable in particular cases. This conjecture applies to the twin primes conjecture (take the polynomials t and t+2), the Sophie Germain primes conjecture (take t and 2t+1), and the existence of infinitely many groups PSL(2,p) of prime degree (take t and t2+t+1). It applies in several other cases including my recent work with Pallabi Manna and Ranjit Mehatari: it would imply infinitely many simple groups whose power graph is a cograph. Gareth and his co-authors have shown that the Bateman–Horn conjecture predicts things like this with remarkable accuracy.
Delaram Kahrobaei (someone I first met at Groups St Andrews in 2013) speculated about whether graph groups would provide a post-quantum encryption system, that is, a trapdoor one-way function where the hard problem is hard even on a quantum computer, should one be built. (A graph group , aka a right-angled Artin group, is defined by a graph Γ generators for the group are vertices of Γ and the only relations are that adjacent vertices commute.)
Carmine Monetta, in addition to running the conference website and the computer facilities for speakers, spoke about the non-abelian tensor square of a group. This is a concept which I have met before, since it comes up in the work Bojan Kuzma and I did on the deep commuting graph of a group.
Mike Kagan was there with his family. He gave us the artefact shown in the picture, saying that it seemed to him that it might be Egyptian. In fact I was able to recognise it. In 2004 I went to Iran and was privileged to be able to visit Persepolis with a University car and driver from Shiraz. Here is one of the pictures I took then. Can you spot him?
We were taken on several excursions: to the Loxahatchee Wildlife Refuge, where the signs mentioned alligators and snakes but we saw a family of deer quite close up and some ducks that sounded like pigs; to the Jupiter campus of Florida Atlantic University, where we saw among other things the Max Planck Institute for Neuroscience, the only MPI in North America. We skipped the dolphin trip so as to get on with our joint work with Mike Kagan.
Back home, I had what has become an unfamiliar sensation: jet lag!