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:

Wordcloud picture

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.

Monday, 7 December 2020

Two mathematical magic tricks

 So over the last few weeks I have performed a couple of mathematical magic tricks which I have put on my Youtube channel.

As part of the 24 Hour Maths Magic Show,  when I was hosted by the amazing Chris Smith, I performed a trick based on the curious phrase "David Lovel in yon abbey". You can watch the trick and find out how it relates to the phrase.

At the virtual MathsJam 2020 Gathering, with the help of Ruth, I performed my favourite Martin Gardner card trick - you can watch this one here.

While I love both these tricks, I feel that the real magic may not be apparent to the viewer.  The first trick relies on the kind of linguistic / mathematical properties that appeal to many mathematicians (see for example Alex Bellos's wonderful new book The Language Lover's Puzzle Book) while the second one depends on the magician dictating the sequence of play in a game of knots and crosses.  Are these tricks are more interesting to perform than to watch?  

Monday, 20 April 2020

Memories of John Conway

The mathematician John Horton Conway died just over a week ago. (I recommend Siobhan Roberts's biography Genius at Play.)  I' intend to write a post later about what his book On Numbers and Games meant to me, but in this post I will share a couple of memories of Conway.  I emphasise - these are memories and their accuracy is not guaranteed.

First,a personal story of how I was caught out by his exam question.  When I studied Part III of the Mathematical Tripos at Cambridge I took Conway's course on Sporadic Simple Groups (this at a time when the Classification of the Finite Simple Groups was not complete and indeed before it had been proved that the Monster group actually existed).  In the exam we had to do any three questions out of six (or similar).  The last question was something like "Write an essay on anything in the course you haven't covered in your previous answers."  Now, I had three good topics I knew pretty well, and they all came up in other questions.  So I thought to myself, "Well, I'll do the essay question first, while I think about which two of the other three questions to do."  As a result I wrote the essay on my fourth-best topic, and didn't use in the exam one of my three strongest topics.  Not good exam technique!  (To be honest, I feel that's why it wasn't a very fair question to set in an exam!)  I have sometimes wondered: if I had answered that question first, and written about my best topic, and then afterwards answered the question on that topic, since that wouldn't have been a previous answer, could I have got credit for the same material twice?)

My other memories are of an email mailing list for those interested in the history of mathematics, which ran for some years from the late 1990s before splitting and dissolving in acrimony as these things tended to do at the time.  Anyone could discuss or ask any questions, and Conway was one of the regulars.

One one occasion two schoolboys posted an email saying they were in primary school and they wanted to know more about some mathematical topic.  Conway sent a lovely long reply, at an appropriate mathematical level, saying probably how nice it was to hear from such enthusiastic students.  The boys sent a thank-you letter, adding at the end that they weren't actually at all interested in mathematics, but their teacher had told them to ask a question.

On another occasion a university student (I think a PhD student) asked about a result of Lagrange (I think), noting that there were several proofs of the result and asking if anyone knew which was Lagrange's original proof.  Conway wrote a long reply, saying that he didn't know, and going on to explain why he didn't think it mattered: there were several simple proofs and it would have been a matter of chance which one Lagrange came up with first, since the proof of the theorem was trivial. Immediately someone jumped in, accusing Conway of appalling rudeness in describing the question as "trivial" (he hadn't) and suggesting that such treatment from an established mathematician would likely deter the student from further mathematical studies.  Others argued about this (not all members of the list were native English speakers and the word "trivial" did seem to have very negative connotations for some: which surprised me as I was very familiar with its use in this mathematical context and didn't think it at all offensive).  Conway said he hadn't intended to be rude or to suggest the question was trivial - indeed, by writing a long and detailed answer, he had shown his respect for the questioner - and he apologised profusely.  But the criticism continued until the student who had asked the original question replied.  He said that far from feeling upset by Conway's response, not only did he not interpret it as critical, but he was absolutely thrilled that the great Conway himself had replied to his question: he had been walking on air ever since!

I admired Conway's willingness to engage with anybody on that mailing list, regardless of their age or experience.  I hope I have learned from that.

Sunday, 29 March 2020

Self-referential humour

I' am one of these annoying people who loves self-reference.  I had a sign saying "Why can't anyone these days do anything properyl" in my student room: now I wear a T-shirt labelled "Prefectionist", so my sense of humour hasn't developed.  Books which I loved as a student (and still do) are Patrick Hughes and George Brecht's Vicious Circles and Infinity and Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid.  The first of these books pushed my interest in art towards Fluxus, and then later I discovered Hughes's brain-baffling perspective paintings.   And through Hughes I found the wonderful, amaxingly modern logical paradoxes of the fourteenth-century philosopher Jean Buridan.

And so I'm one of these annoying people who think it's witty to include "Self-reference" in the index of a software user-manual, referring to the page number for that index page. (Yes, I did that, but it was a long time ago.)

When did this enjoyment of self-reference start?  Possibly with my father, although I can't immediately think of any examples.  Certainly when I was a teenager.  For me, the funniest Monty Python sketch was the one where we see a group of intrepid explorers reaching their destination, never before visited by man, and duly celebrate with handshakes all round.  But while they are celebrating, one of them wonders, "Who is filming us?" and we see them looking in all directions before pointing, rushing straight towards the camera and then introducing themselves to the camera crew, who obviously got there before them.  And as we cut to another angle, they realise that there must be another camera crew too ...  (This is my memory of the sketch and it may not be very accurate.)

I had wonderful maths teachers at school - Jimmy Cowan and Ivan Wells - but I don't remember them particularly encouraging this kind of humour.  But recently (ie when I first planned writing this blog post, so not very recently) I was amused by this gag from Moray Hunter's radio comedy "Alone":  .

"Remind me, Morris, never to ask you to do anything … at all  … ever"
"Does that include reminding you never to ask me to do anything at all ever?"
"Probably – I lost interest halfway through the question"

(The first speaker is the Angus Deayton character, and the concluding line is not only ideal for his world-weary delivery, but really perfects the joke.)

Why do I mention this?  Because Moray Hunter was at my old school and shared the same maths teachers as I had.  So perhaps there is a connection...