How did mathematicians cope with the Covid lockdowns? *COVIDIARY of Mathematicians *is a new book published by the Mathematical Society Archimedes of Belgrade. It presents diaries from April 2020 of seven mathematicians in different parts of the world - Tiago Hirth (Lisbon), Guido Ramellini (Barcelona), James Tanton (Phoenix), Jovan Knezevic (Belgrade), Kiran Bacche (Bangalore), Sergio Belmonte (Altafulla), and Tijana Marković (Belgrade). Their diaries feature mathematics and (excellent) puzzles, mountainlettes, cooking and shopping, in the strangest of times. The book is beautifully produced, with copious colour pictures, and QR-codes taking the smartphone-equipped reader further afield. The editors Aleksandra Ravas and Dragana Stošić Milijković have done a wonderful job, providing full explanations in footnotes of any reference which might be unfamiliar to some readers. Solutions are provided to the puzzles! If you have the opportunity, I strongly recommend that you explore this book!

## Monday, 18 April 2022

### Mathematicians' diaries in the time of Covid

## Sunday, 12 September 2021

### Did studying maths help Emma Raducanu win the US Open?

Emma Raducanu's feat of winning the US Open as a qualifier, without dropping a set in her ten matches, was astonishing. In feature articles this summer, during and after Wimbledon, her A-level achievements this summer, and especially her A* grade in A-level Mathematics, have been reported, Is there a connection between her tennis accomplishments and her study of maths?

Now, mathematics implicitly arises in tennis tactics. I've discussed toy examples in public lectures on game theory, which (it seems to me) is relevant to choices players make - whether to serve to the forehand or backhand, and where to expect for your opponent to serve, for example. I very much doubt if players ever analyse in these terms, but they are intuitively doing game theory when making their tactical decisions.

But I think in the case of Raducanu there is a more general point. Several times I have heard knowledgeable commentators - most recently Tim Henman immediately after Raducanu's victory over Leylah Fernandez in the US Open final last night - talk about her qualities as a problem solver. She thinks deeply about her tactics, adapting to opponents and match situations. Now problem-solving is a quality which is developed in studying mathematics. (Fernandez, who showed in her victories over three top-five opponents in New York remarkable abilities to turn around matches in which she was behind, apparently enjoys solving Rubik's cube - more mathematical problem-solving!0

Obviously her study of mathematics is not why Raducanu is a great tennis player. But it seems to me that the problem-solving skills which she displays on the tennis court are the same skills which make her good at mathematics.

## Sunday, 4 July 2021

### Another dubious probabilistic argument

Here's a variation on my previous post - another piece of hand-waving probabilistic reasoning, which I think is basically correct, but I suspect many will disagree.

A long time ago I used to do the challenging weekly cryptic crosswords in *The Listener *magazine. (*The Listener *has been defunct for many years, though I think the crossword continues in *The Times*.) The story was that if no-one solved the crossword, it was too hard, and if more than one person solved it, it was too easy. While that was an exaggeration, it wasn't easy, and I judged it worthwhile, if I completed the puzzle, to submit my entry for the prize draw.

I had completed about fifteen puzzles in a row, and submitted my answers each time, but hadn't won the book token. Then there was an unusual puzzle - it was mathematical rather than word-based. Two mathematician colleagues and I worked on it - we didn't find it at all easy - and we eventually solved it. I submitted our answer, and this time, we won the book token.

So - my conclusion was that (probably) more people solved the word-based puzzles than the mathematical one, and that therefore I was more likely to win the prize for that puzzle (as I had done) than for the others.

**Is this conclusion valid?**

As it happens, at the end of the year statistics for all *The Listener* crossword entries were published, so I was able to see if this was indeed the case. It turned out that the mathematical crossword had attracted about three times as many correct entries as any of the others. So my conclusion was in fact false, but I still think the reasoning was sound.

## Saturday, 12 June 2021

### Is this analysis correct?

When I was an undergraduate, I was walking through the town centre one morning when a journalist from the student newspaper asked my views on the Rag Mag that had just appeared. When the report came out, six students' views were quoted. Three of the six happened to be from the same cohort of six maths undergraduates at my college. My Director of Studies concluded (jokingly) that this disproportionate representation of his tutees showed that we were spending too much time out and about when we should have been studying.

Here is a similar scenario, based on a recent experience. Suppose that I enter a lottery. It is open to, say, 10,000 people, but I do not know how many choose to take part. As it happens, I win the lottery. It seems to me that I can conclude that (with a high degree of probability) only a small proportion of those who were eligible chose to enter the lottery. If all 10,000 people entered then my chance of winning was only ,00001, whereas if 5 people entered it was 0.02. Since I did win, my estimate of the likely number of entrants will be at the lower end.

Now, you are my friend and you are equally friendly with all 10,000 potential participants. I tell you about it, I think you should agree that I am right in my conclusion. But you cannot come to the same conclusion: whichever of the 10,000 people had won the lottery would have told you of their success, so you can't deduce anything from what I have said to you about the likely number of participants.

So - I have good reason to come to a conclusion (with a high degree of probability). You agree that I am right to come to this conclusion, but also you have no reason to suppose that the conclusion is valid. How can this be?

Apologies if I am missing something but I think this analysis is sound. Have I read about this somewhere?

However, I haven't yet managed to persuaded anyone else to agree with me (which is perhaps ironic, or perhaps I am just wrong). Comments welcome!

.

## Wednesday, 10 March 2021

### Tomorrow's Mathematicians Today

I spent last weekend at the online conference Tomorrow's Mathematicians Today, hosted by the University of Greenwich and the Institute of Mathematics and its Applications. This is a conference for undergraduate mathematicians to present to their peers on mathematical topics that excite them - this might be their own project work, or something they have come across in the curriculum or outside it that has fascinated them. The conference was created in 2010 by Noel-Ann Bradshaw when she taught at the University of Greenwich, and has been held in various venues since.

We hosted it physically at Greenwich in 2010, 2013, 2016 and 2019, and before Saturday I was apprehensive that the conference would not work online. At the physical conferences what made the event so exciting was the atmosphere - friendly and supportive of the speakers, with great enthusiasm from audience and presenters. Would this work online?

Well, it was obviously different, but the enthusiasm was certainly present, and the online networking sessions were well attended and worked far better than I had imagined. Here are the words the attendees chose to describe their feelings:

The quality of student presentations was outstanding: the judges of the GCHQ Prize for the Best Presentation had an extremely difficult task deciding the winner. The winning paper by Yanqi Cheng (UCL) was remarkable not only for its content but for Yanqi's sangfroid in managing a seamless transition to live delivery when the computer playing her pre-recorded video crashed early in the presentation. "Honourable Mentions" were awarded to Yousra Idichchou (Greenwich) and Oscar Holroyd (Warwick), and the other shortlisted papers by Finley Wilde (Bristol), Kaiynat Mirza (Keele), Muhiyud-Dean Mirza (Warwick) and Sheeru Shamsi (Keele) were all excellent, as indeed was every single student talk.

The conference also benefited from two contrasting keynotes by established mathematicians - Colva Roney-Dougal on random games with groups and Kit Yates on the mathematics of epidemics. It was surprising to hear from Colva how the Riemann Hypothesis turns up in group theory!

Like its predecessors the conference was inspiring in showcasing the enjoyment today's students are taking from mathematics. It totally proved the point made by IMA President Nira Chmberlain in opening the conference when he said that the presenters were not "tomorrow's mathematicians" but are already fully qualified for the title of mathematician.

I obviously hope future TMT conferences will be held physically again, but the online conference was as friendly and inspiring as we could have hoped!

## Monday, 28 December 2020

### Reading "The Queen's Gambit"

I haven't watched the Netflix series *The Queen's Gambit*, but (prompted by a review in *Private Eye*) I've just read Walter Tevis's 1983 novel on which it is based. (I'm a reader rather than a viewer by inclination.) It's an interesting book, which I thought well worth reading.

Now, I know I am missing the point, but there were a couple of things I found irritating. first of all I found the accounts of the chess unconvincing. I haven't played seriously for many years, and the author was undoubtedly a more serious chess player than I ever was, but the games he describes don't seem plausible to me - the heroine wins too much material too quickly. I am probably wrong, because the author took expert advice on the chess positions. I'd be interested to know whether better chess players than me felt the same, or am I out of touch?

Of course, the bit that really annoyed me was the description of a tournament that just didn't add up. After four rounds three players are in the lead with 4 out of 4: Beth wins her fifth one while the other two with 100% play each other. In the last round she loses to the winner of that match, and only finishes sixth after two other players on 5/5 agree a quick draw in the last round. This just doesn't work!

The chess novel which I loved as a teenager (and still do) was Anthony Glyn's *The Dragon Variation*, and the games in that book seemed authentic. The descriptions of Beth Harmon's games in *The Queen's Gambit *didn't work for me.

So if you are looking for an entertaining novel about chess, I still recommend *The Dragon Variation.*

## Sunday, 20 December 2020

### Memories of Peter Neumann

Peter M. Neumann (1940-2020) (Photo by Bert Seghers, Wikimedia Commons, Public Domain) |

A very sad post - the mathematician Peter M. Neumann died on Friday 18 December of Covid-19/ Peter was my moral tutor when I was a postgraduate student at The Queen's College, Oxford, and like many others I was enormously influenced by him. I was very lucky as a student to have wonderful support from several great mathematicians - Ray Lickorish and Michael Vaughan-Lee in particular - but I feel myself especially privileged to have known Peter. Indeed, in many ways I consciously tried to model myself on him, and the best things I have done in my career in teaching owe much to Peter.

In this post I'm going to share some rather rambling and inconsequential memories of him (with the warning that my memory is fallible).

I first met him when I went for interview at Queen's - which turned out to be a conversation in the Fellow's Garden with Peter and Graham Higman. I had been reading my notes from my course on Combinatorial Group Theory on the train journey, which was probably a mistake because I confused myself and attributed to Higman what was actually the Baumslag-Solitar Group. Peter questioned it and my heart sank as I realised my blunder, but before I could say anything Higman jumped in to say that in fact that group had been his original idea and Baumslag and Solitar acknowledged him in their paper.

Anyway, despite such blunders, I was accepted. I was very impressed that, when I turned up at Oxford six months later and knocked on Peter's door, he remembered who I was. But then when I introduced myself to my office-mates in the Mathematical Institute the next day, one of them said, 'Yes, I know your name. When I knocked on Peter Neumann's door last week, he said, "Hello, you're Tony Mann, aren't you?"' So perhaps Peter's strategy was to guess one of the names of the new DPhil students so that he would be right at least once!

I attended Peter's famous *Kinderseminar* - a Wednesday morning gathering of doctoral students and visiting professors in Peter's rooms, with coffee and friendly conversation before a presentation on someone's research. I remember my first presentation. It was dreadful. (To be fair to myself, I had never given a presentation before - it wasn't part of the Cambridge mathematics curriculum - and so my experience of mathematics talks consisted entirely lectures which were primarily dictation of notes and research seminars which were generally over my head.) Knowing the expertise of the three visiting professors, I naively assumed they knew everything there was to know about my research topic, so I cased through the background material to avoid boring them. After five minutes of rushed, garbled talking, I looked at the audience and realised that none of them had followed at all. If I were in that position now, I would stop and restart the presentation at a sensible pace, but at the time I panicked and continued with the presentation for another hour, completely wasting everyone's time. When Peter gave feedback, he began by saying "That was terrible, wasn't it" (I guess it's to my credit that I was already aware of that) and, although that's all I remember of his feedback, it must have been very generous because I felt encouraged rather than dismayed.

I also remember that Peter particularly disliked the works of C.P.Snow, a favourite writer of my father's whose novels I had also liked. I recall Peter saying that good novelists "show" and leave readers to form their own judgments while bad ones like Snow tell the reader what they should think. (While there is a lot in this, I don't entirely agree with Peter: there are different but valid ways of story-telling.) Years later I discovered that Peter's antipathy to Snow may have been due to his strong feelings about Snow's introduction to G.H. Hardy's *A Mathematician's Apology*: a much-praised account of Hardy which Peter felt misrepresented the great man by exaggerating his supposed unhappiness.

Peter himself should have written much more - but he always said "There are already too many books". While this is true, more books by Peter would have benefitted us all.

Other old memories of Peter - when he sent one a message it was always on an interesting picture postcard rather than just a scrap of paper. This is something I copied, building up a stock of postcards to use when I had to write a short note to somebody (and I have retained my postcard-buying habit even though electronic communication means I never send anybody any notes any more!)

I remember going to a barbecue at Peter's house at which several of us we watched the England-Argentina World Cup match, and regretting my enthusiastic response to Maradona's wonder goal when others were upset that it put England out of the tournament.

Peter had a cautionary tale for those of us who have to write many of references for students. A prospective student applied to Queen's and his teacher's reference said, "Without a doubt this student is by far the best mathematician this school has ever produced." Since the school in question was Peter's own school, this reference did not have the positive effect the teacher intended!

I remember Peter's stories about his house number - 403. He used to go into schools to talk about maths and he would refer to prime numbers "like, 2, 3, 5 or 403, for example". Only once, a hand immediately went up in the audience, "But, sir, 403 isn't prime, it's 13 times 31". Peter told how, when the house was built, he went to the Post Office to be allotted a number for it, and was told, "There are no numbers left - you'll have to give it a name instead." After offering to find some more numbers for them, he and Sylvia called the house "Burnside" after the great mathematician. I remember him saying that he had thought of calling the house "Burnside Hall" after his two favourite mathematicians.

I was privileged to work with Peter on a couple of book projects: it was a joy particularly to collaborate with him and Julia Tompson (I was very much the junior partner) in producing Burnside's *Collected Papers*. One of Peter's qualities was that he had very high standards and I feel that his book reviews could be rather ruthless. So it was a rather good move on my part that both books I have edited involved working with Peter so he wouldn't be able to review either of them!

I was lucky enough to attend the conference celebrating Peter's 60th birthday, twenty years ago next month. Peter's birthday was at the end of December and so the birthday took place at the beginning of January 2001 - giving Peter the possibly unique achievement that his 60th birthday was marked by a conference in a different millennium from the event it was celebrating!

I owe so much to Peter - and so do many many others. His contribution to mathematics goes far beyond his own mathematical discoveries, significant though these are, He taught, inspired and encouraged so many others, and was much loved as well as much admired.