# Meet me in Montreal

Winter is coming. And it will be a Canadian winter: none of your lily-livered London frosts, with Heathrow grounded by a light dusting and children building wet snowmen with bare fingers as the temperature hovers around freezing, but a ‘nighttime lows in the minus 20s’, howling wind, lethal ice and broken femurs, bar-brawls-cum-snowball-wars – seriously, check out https://www.youtube.com/watch?v=F-MIwD1VvYY – frozen waterfalls, skiing down the street, snow-shoeing around the ice zombies sort of winter. I’ve bought a parka coat so thick it feels like a stint at the gym just to move about in it. Perhaps I am ready. (I am not ready.)

The speed at which the seasons change here in Montreal – autumn passes in three weeks, spring in two – and the sheer length of the winter itself – six months, give or take – are both very foreign to my temperate European sensibilities. So too it must have been for those early French explorers, who came this way a little under five hundred years ago. Although one might reasonably argue that Jacques Cartier and his men did have it just a tad worse. On their expedition of 1535-36, they were looking for the North-West Passage – I mean, who wasn’t? – as well as for the mineral riches of the fabled ‘Kingdom of Saguenay’, somewhere in the lands to the north of the St. Lawrence river, but they only made it as far as the Hochelaga archipelago, before eventually getting stuck in the ice and seeing a quarter of their company die of scurvy. I, on the other hand, just need to remember to pick up some fresh fruit on my next trip to the supermarket. Potato potahto.

The Hochelaga archipelago is the name that is now given to the constellation of freshwater islands on which the cities of Montreal and Laval presently stand. The name derives from the Iroquois settlement that Cartier encountered on this spot during his ill-fated voyage, and the St. Lawrence has some formidable rapids here, which blocked Cartier’s way and indeed blocked all shipping traffic until the mid-20th century and the building of the St. Lawrence Seaway on the south side of the river.  Cartier, in that fiercely arrogant, beautifully optimistic manner of those early mariners, was convinced that these rapids were the only remaining obstacle between his ships and China. This led to the coining of their current name, used I think initially in jest by the French settlers about a century later: the Lachine rapids. In retrospect, Cartier may have been strangely fortunate to have been forced to turn back here, as were it not for this barrier his ships could have made it as far as Lake Ontario and eventually to the foot of Niagara Falls, whereupon the would-be colonisers would have really lost their shit.

This was Cartier’s second expedition to the St. Lawrence, and yet, despite the hardships he had encountered, he went back for a third time in 1541. With what crew, I know not. The sheer audacity and bravery, to return to a place that had proved to be so deathly dangerous, is beyond my comprehension. Roll up, roll up, 25% chance of death in exchange for an opportunity to possibly see the probably made-up Kingdom of Saguenay!! Where do I sign? In the end they did find Saguenay, but only returned with quartz and iron pyrite, not the diamonds and gold of myth. And relationships with the native population worsened, perhaps because Cartier had this annoying habit of capturing a few locals on each trip and taking them back to France with him. Hmm…

No permanent French settlement was established in Canada until the 17th century. Montreal – my home for the year – was founded in 1642, but it nearly didn’t make it through its first ten years, almost falling to flood within twelve months, and suffering under constant bombardment from the understandably irate Iroquois. Its first governor Maisonneuve – who now has a major (if slightly soulless) Downtown street named after him – felt the need to drag a wooden cross to the top of Mont-Royal to thank the Virgin Mary for saving the colony from the flood, in a moment of such religious potency, it seems, that it is commemorated in the stained glass of the Notre Dame Basilica in the centre of town. Outside the cathedral Maisonneuve himself stands triumphant in bronze, with four other figures at his feet. One of these is a statue of an Iroquois man. I find the message of the statue somewhat ambiguous – is the arrangement trying to convey cooperation, or subjugation, or a bit of both? From what I understand of the history of the matter, the second epithet is the more appropriate.

When I arrived in Montreal in late August, it was all summer sun and street festivals. Every Sunday afternoon I would walk around the lake in Parc Jarry to see the lawns filled with families enjoying their barbeques, dogs daring each other to jump in the fountain, and joggers darting in between them all. A mile-long stretch of the Boulevard St. Laurent – a street almost as old as Montreal itself, and which runs across the entire island – was closed for the whole Labor Day weekend for street stalls and fairs. But then, in an inevitable, ‘Monty Python and the Holy Grail’ sort of a way, we went straight on into autumn. The seasonal foliage of red and gold (“Les feuilles mortes se ramassent à la pelle…”) was a complete revelation.

Keen to enjoy this brief but splendid natural wonder, in October the number-theory postdocs took a few sojourns in the forest. First we had a road-trip to Maine, on our way to the Maine-Quebec Number Theory Conference, and then a smaller excursion to the Alfred-Kelly Nature Reserve just to the north of Montreal. An arboreal expert would know for sure, but I think I’ve convinced myself that it is the combination of the maples with the birches which make the visual timbres in Quebec and New England so much more vivid than those in the UK. The maples turn deep red, while the birches deal in softer, yellower hues. One must also factor in the sheer extent of these deciduous forests, which still carpet large swathes of this corner of the American continent. Throw in the pine forests of British Columbia for good measure and Canada ends up with 8953 trees per person, according to a Nature article from 2015 https://doi.org/10.1038/nature14967. The UK has 47.

Now, at All Saints Day, a torrential rainstorm and chill wind have blown almost all the leaves from the Mont-Royal (“Et le vent du nord les emporte…”). So wet was the night of Thursday October 31st that the City of Montreal government officially moved Halloween to Friday November 1st (can they even do that?), to save the armies of would-be trick-or-treaters from limp witches’ hats and soggy candy. The cafeteria at the Université de Montréal didn’t seem to get the memo though, and on the Thursday they still served spaghetti neri with dyed fluorescent green chicken lumps mixed in. I think the chef’s idea must have been that the dish should look spooky and potion-like, which is always what I want from my lunch.

The Université de Montréal – UdeM to her friends – is my academic base for the year. This is while I take a one year postdoc, away from my longer postdoc in Cambridge. With about 67000 students enrolled in its various programmes and affiliated institutions, UdeM is among the largest francophone universities in the world, and, most pertinently for me, it has two top professors in analytic number theory – Prof. Andrew Granville and Prof. Dimitris Koukoulopoulos – from whom I am already learning a great deal. But in truth UdeM is just one of my affiliations here, since this semester I am also lecturing a calculus course at the anglophone McGill University (an institution which is sometimes referred to as ‘The Harvard of Canada’, for good or ill). Montreal has a mountain in the middle of it, and the UdeM and McGill campuses are on opposite sides, so my life presently consists of frequent circumnavigations of the Mont-Royal, via metro and bus, clocking up a winding number of at least 1 each day. To add a further nuance to my situation, the funding for my research postdoc at UdeM actually comes from the Centre de Recherches Mathématiques, a 51-year-old mathematics organisation which is spread across the different Higher Education institutions in the city – although its base is at UdeM – funding postdocs, senior researchers, prizes, conferences and workshops. Once a fortnight we also have a number theory seminar at Concordia University, so, technically, I regularly interact with four different institutions here, which is enough to keep anyone on their toes.

Montreal is a large city – about 2 million people live here, with 4 million in the wider metropolitan area – and yet, in certain aspects, it just feels like a university town. I’ll admit that this could just be the blinkered view of an egocentric academician, but I am convinced that there is at least some objectivity in my view on this matter. For a start, in Quebec province the local 17 year olds go to cégep (an acronym for ‘collège d’enseignement général et professionnel’, with some grammatical nicety causing an accent to be added to the first ‘e’), for two whole years before starting university proper. So the population of street-smart, cool, francophone teenagers is naturally expanded. Regarding the universities themselves, there are a full five of them scattered about the town – UdeM, McGill, Concordia, Université de Québec à Montréal (UQÀM), and the Longueuil campus of the University of Sherbrooke – and their students seem to amass everywhere, at least around Downtown, Côte-des-Neiges, and The Plateau. The very infrastructure of the city screams that HE is an integral and indispensable part of it. For instance, there are four metro stops which are each named after the respective universities to which they are closest. Contrast this with London: why isn’t there a UCL stop, an Imperial stop, an LSE stop, or a KCL stop (etc.) on the London Underground?

My maths research is in a slightly peculiar state at the moment. I seem to be stuck between, on the one hand, a collection of problems which I essentially know how to solve, but which will be awkward and technical to write up, and, on the other, a collection of problems which are bold, simple to state, and on which I have next to no good ideas of how to proceed. Neither extreme is a particularly pleasant place to be, but I’ve been in the game long enough now to know that the situation will change eventually, even if it’s hard to tell exactly when that will be.

Yesterday night brought the first flurries of snow (“Dans la nuit froide de l’oubli…”). In a moment of pure serendipity, I had already been planning to go to a recital of Winterreise, given by one of the basses in my choir. I’m stocking up on dragonglass. It has begun.

# Berkeley: Day 7

“This California dew is just a little heavier than usual tonight.” Oh Debbie, how I could have done with your sweet parting words as I headed out into the evening torrent, water gushing down every street, trees creaking in the wind, road-crossing better accomplished by native canoe than western boot. The sun was not ‘shining all over the place’, and I waded down Stuart Street to a dismal refrain, neither singing nor dancing, but slowly developing trench-foot. At least the rain wasn’t mixed with Hollywood milk.

After 4 years of drought the rains have finally come to California. Three storms in the space of a week, and Strawberry Creek, the Berkeley campus brook which had run dry, now flows in spate. The drainage systems are ill-equipped for such downpours; it was only yesterday, many days after my aquatic night-time stroll, that I realised that the local residential streets have no drainpipe whatsoever. All the water which falls on their great asphalt expanse flows on the surface, gathering down the Berkeley hills, and the road becomes a river. Every road is on a slope here, and to power-walk anywhere is to accept a day caked in sweat. The byways fail to follow to furrows of the earth, but are laid down instead in the traditional unwavering US grid. If I had known all these things beforehand, I might have thought twice about heading out into the night to meet up with the mathematical gang who were frequenting a local trivia night. We did well on the Russia-themed Round 1 — the quiz questions having been rigged by Putin, perhaps — but fell down with our lack of knowledge on the finer points of beer manufacture. In the age of McCarthy, we would have been the first against the wall.
The purpose of last week’s Pseudorandomness workshop, at least in part, was to foster communication between the computer scientists and the additive combinatorialists, communication which will, one hopes, bear great fruit in the coming months. Fernando held out the olive branch by performing his exposition of the Arithmetic Regularity Lemma strictly over $\mathbb{F}_2^n$, the main object of computer science, and Julia’s description of Higher Order Fourier Analysis was also restricted to the finite field case (where the theory still remains rich, interesting, and knotty). Considering the reverse direction, I was astonished with how many of the computer science talks I felt I understood well. The nexus of ideas — from pseudorandom generators, through to extractors, to expanders, then to eigenvalue bounds, then to notions of graph regularity, then to the $C_4$ cycle count of the graph, then onto Gowers norms and arithmetic combinatorics — appeared like a vision. Amusingly, there seems to be a certain duality between what a number theorist means by a ‘pseudorandom measure’ and what a computer scientist means by a ‘pseudorandom generator’ — we compose our pseudorandom function and our test functions in a different order.

In many ways this has been a week of great levity. I’ve been reunited with old friends, and met new ones; I’ve found a second-hand book shop called Moe’s, where I plan to prop up the bar; I’ve been treated to a lovely homemade meal by Anna and Salil in San Francisco, and liquid-nitrogen chilled cookie dough ice-cream; I’ve eaten pizza from a speciality cheese shop; I’ve walked up the hill to MSRI, and eaten my homemade stew overlooking the bay. But of course, there were more fateful world-events this week. On the eve of Trump’s inauguration, I met a homeless black man on Shattuck Avenue. He wore a red jacket, and had a warm smile which creased the skin beside his eyes. Perhaps he was about fifty. I bought a copy of the newspaper he was selling — ‘Street Spirit’, the local equivalent of ‘The Big Issue’ — and he asked if he could say a prayer with me. Travel changes a man: back home I would have meekly refused him, escaped off into the atheistic night, worried about the influence of a cloudy God on the forgotten people of the state. It takes a human being to help the homeless, for human action is all there is; if you say all your prayers to a distant God, your godless rulers won’t be listening. Yet, under the American shop-front lights, on the eve of the apocalypse, I stayed and held his gloved hands as he blessed us both. Because what else is there for him to do? The state falls, and, come the morning, there’s no amount of supplication that will take his cause to the seat of power terrestrial. Can you hear Billy singing? “Them that’s got shall have, them that’s not shall lose, so the Bible says, and it still is news. . . ”

Trump was inaugurated at twelve noon in Washington, nine in the morning in Berkeley, and I just managed to catch his hand on the bible before my morning lecture. I am normally a rather passive individual, but times change, and I am changing with them. My aunt marched through San Francisco in protest against the Iraq war, fourteen years ago, and I would march again, from Civic Center to Justin Herman plaza, as part of the San Francisco manifestation of the global Women’s March movement. I estimated that there were about 7000 of us on the underground train from Berkeley to San Fran, going to join a crowd of around 100 000 in the UN Plaza. We were crammed against one another, jostling for space and for air, but all strangely happy. I have never felt such togetherness.

Once in the open, we flooded towards the rally which was happening in the plaza itself, but such was the melee that we could only flow around, up to Market Street, humans as liquid. Being a travelling man I had had to come as I was, but about me there were signs aplenty, face-paint, glitter, pink hearts, drawings of Fallopian tubes giving the finger. I ended up next to a bunch of five med-school students from San Francisco State University, who took me under their wing. They had taken the precaution of writing their signs in waterproof ink, wisely, as the rain clattered down just before kick-off, and the simpler signs began to sag. The march was begun by a vanguard of cable cars. Their 19th century wooden respectability, juxtaposed against such colourful dissent, seemed like a visual oxymoron. However, the entire event was to me a crash course in the city’s seeming contradictions, in its people, and in its ebullient life. Half-way down a young woman stood on top of a street-sweeping vehicle, waving an enormous rainbow flag, Les-Mis style. A man in a motorised wheelchair was blasting out reggae. Dancers were having a party inside the glass-front of SF Camerawork.

My new medical friends headed home, but left me one of their signs, which I nursed back to Berkeley through the stormy night, a memento of my first march. I left them my email address, in the hope that the photos that were taken of us together might be sent on, but I haven’t heard from them since. Perhaps I never will; these five kind people will be forever lost, except in memory. I meandered up into China town, water oozing from my drenched coat and rucksack, I had dinner, and went home from Powell Street BART station.

I hope, when I come to tell the story of these coming years to my children, that the tale will not be of my idleness. For that to be the case, what was begun on Saturday January 21st must continue, must grow, must morph from protest into power, and I must continue with it. Maybe that won’t happen. But, at least, I will be able to tell them that I did do something, one tiny thing, that I travelled across the bay and walked for what I believed in. The best sign I saw? An image of Carrie Fisher, as Leia Organa with the following moniker: “A woman’s place // is in // the resistance.”

# Berkeley: Day 1

What does it mean to be a global citizen? A difficult and pertinent question in the present age, given the international collapse of national unities. The term used to exist within an aura of only positive connotations — well-travelled, rounded, connected, open, free — but there is a certain school of modern thought which would rather it be used purely as a pejorative, a derogatory term for one who shirks national responsibility, and who feels greater kindred spirit with strangers in a foreign land than with the strangers from his own soil. But what does it mean? Perhaps it means that one can travel half-way around the world, to an American city one has never seen, can walk around the corner to a supermarket recommended by a German friend from home, and can, whilst contemplating over organic oranges, be greeted unexpectedly by a friend from undergraduate days who you first met at a mathematics summer school ten years previously. Taking that as our definition, I am a global citizen, for it has just been my privilege to experience such glorious serendipity.

I am in Berkeley to work as a Programme Associate on the Analytic Number Theory semester programme at the Mathematical Sciences Research Institute, but also to participate in a couple of pseudorandomness workshops at the Simons Institute in the main Berkeley campus itself. This second calling kicks off straight away, with so-called ‘Pseudorandomness boot-camp’, so it’s going to be until the weekend at least before I can really get a sense of this brave new West-coast world into which I have landed. The journey was exceptionally long, and I realised last night, before submitting to sleep’s warm embrace, that I had only slept for 4 hours out of the previous 40. Heathrow Terminal 3 was quiet; with nothing else to ponder but my own fatigue, philosophical dilemmas crowded my addled brain. The sign before security: “What is a liquid?” (The italics are mine). Gosh. What is a liquid? Newtonian or non-Newtonian? Could you take custard through as long as you constantly stamped on it? What about molten glass? What about borax slime? As for answers the sign offered none, joining the venerable philosophical tradition of those who pose more problems than they solve.

Ten-and-a-half hours later we arrived in San Francisco, having flown over Greenland’s magical mountains, frozen Canadian tundra, and the endless snow-covered Rockies — all the northern hemisphere in winter’s grasp. I didn’t have a window seat, and the other passengers seemed slightly perturbed by my endless gawking at them (past them, out of the window). For the rest of the time I watched Pixar movies, re-read my notes on regularity methods in graph theory, and started The Grapes of Wrath. The only previous time I have visited the US I experienced some issues at immigration, because (one assumes) of the Kazakh visa in my passport — I was part of the UK team at the International Mathematical Olympiad, hosted by Kazakhstan in 2010. No such problems this time, although all of us had to wait in line for around an hour. My British queueing sensibility shone through: in fact, after passing through what turned out to be the final check-point, I nonetheless made a beeline for the only other queue I could see, a random luggage check for which I had not been selected. You know where you are in a queue; the comfort of 1-d living.

On the train from airport to airbnb the dying sun cast slanting shadows over the bay. The air seemed thick with light, and re-fills now, as sunrise seeps through the pines beyond my window. As I write, flapping on the rooftop below is the bluest bird I have ever seen. Where’s a pocket copy of John James Audubon when you need it? The lake lay blue below the hill…
I hope to blog from Berkeley as often as Time allows, but Time, like the moon, is a harsh mistress, so who can say how soon that will be. Right: the day has begun. Time to master the shower.

# Tuscan travel in 2016

‘It all happened on Tuesday July 5th, barely 48 hours after I had arrived. Cowering from the afternoon heat in my small attic room I had attempted some maths, but then submitted to sleep, taking an overlong siesta before finally cooking dinner on the little gas hob; it was nearly 8pm by the time I ventured out for an evening passeggiata. The sweltering day had receded to sultry dusk, and I trotted from Santa Croce, along Borgo dei Greci, past San Firenze, and through to Piazza della Signoria.’

The year is spent. Growing up it was an unwritten tradition, on this day or near it, for my mother to ask me, “What was the best thing you did in the last twelve months?” My fading memory struggles to grasp my past replies with any certainty: yet I can be sure of one thing, that, on each occasion, I never had to think for very long, to plot the arc of my year, to consider its triumphs and its defeats, and to light upon my unwavering response. This New Year’s Eve feels markedly different. Perhaps the most obvious reason for this state is that, although still dancing to the familiar rhythm of the academic calendar, this has been the first of my university years to have been devoid of any obvious culmination or conclusion. I suffered exams every summer from the ages of 10 through 21, of varying degrees of significance; the summer of 2015 was my first without them, but even then I had begun the year still with the vertiginous sense of setting out on my research studies, and in November I had had the ‘Transfer of Status’ thesis to prepare, signifying at least, as Churchill might have put it, ‘the end of the beginning’ (of my third degree). In 2016, by contrast, I have been deep inside the long slow progression of the middle-portion of a mathematics DPhil. A bewildering array of things have happened, in my research and without, both good and ill, but though my mind teems with experiences, there is no grand unifying order, no natural aid to analytical reflection.

The brief summary, as it has been every day since the day I was born, is that I am outrageously fortunate in almost everything — to be paid to think all day and still to have the cheek to call it work. In exchange, what is there to offer but just to think really hard, and to be grateful? (Of course I teach too, do outreach with teenagers, interview for undergraduates, run the weekly maths department social, run and host a wide-ranging university podcast, give around 3 seminar talks and perform in around 10 musical concerts each year, but that’s not what my grant pays me to do). Yet, both of these laudable aims — strong work-ethic and humility — face competition from the self-absorption which begs to follow from having a day-job involving such long periods of introspection, voyaging though strange seas of thought, alone. This end-of-year note, of course, is really one vast ego-trip. I guess all you can do, if you allow yourself a temporary submission to the calls of narcissism, is to be narcissistic in an entertaining style.

One theme that has seemed to have typified 2016 has been the quantity and extent of my travel. Lucky accidents and an extremely understanding doctoral supervisor saw me in Mexico, France, Italy (for a month), and Austria, along with brief stays in Edinburgh and the Lake District. Only that final stay — on Esthwaite water, a skipping stone’s throw from Wordworth’s old school — was a traditional holiday. My 11 days in Mexico were spent on an astonishing choir tour with Schola Cantorum, centred around two concerts with the National Orchestra in the Palacio de Bellas Artes (a marvel of art-deco design, slowly sinking into Texcoco Lake, along with the rest of Mexico City); a conference saw a week of talks and discussion with most of France’s additive combinatorialists at the University of Bordeaux (I gave my first ‘grown-up’ research talk, and initiated a new collaboration); and then ELAZ 2016 happened, a meeting of elementary and analytic number theorists, which induced around 70 other mathematicians to take over a lakeside teacher-training complex in the Salzkammergut (the stunning region of lakes and mountains beloved by both Austro-Hungarian emperors and circle-method experts alike). In some peaceful future hour I hope to pen some pieces about each one of these trips. In fact one remarkable experience during the half-day break in the middle of the ELAZ conference, centred around my returning pilgrimage to the musical instrument museum in St. Gilgen, begs to be transformed into something approaching a short story. Another time.

My stay in Italy was of an altogether different nature, not least because of its much greater length — if one neglects a brief return to the UK in order to attend an old school friend’s wedding, I was there from July 3rd until August 7th, by quite some margin the longest foreign excursion of my life (never having done the canonical teenage European inter-railing). For the final 6 days I was in Rome — the eternal city, another ten personal tales to tell — but the majority of the time was spent in Florence, spending the afternoons battling the Inverse Large Sieve problem in the reading room of the Biblioteca Nazionale and the mornings learning Italian at the British Institute. I’ve had a dilettante’s fascination with Italy and the Italian language for a fair while, and serendipitously now find myself surrounded both by Italians in the Oxford maths department and by Italian itself in my tentative forays into classical baritone singing. On this justification and more, I was fortunate enough to receive one of the three tuition-fee-funded exchange places from Magdalen to study at the British Institute for a month. So, at the start of July, I packed my bags, said goodbye to the circus, boarded with a married couple of professional pianists on Via dei Benci, and began my stay at culture’s cradle.

Florence in summer is a strange place, populated almost entirely by tourists. Indeed, I barely saw my hosts, as they spent most of the month down at the cost by Isola d’Elba. What’s more, the language which fills ones ears during a morning stroll — my commute cut right across the centre of town, from Santa Croce to Piazza della Signoria to Piazza della Republica — is not Italian, but English, such is the concentration of visitors from the US and the UK and the linguistic accommodation of those wishing to sell the town to them. Passive language assimilation is therefore rather difficult, and though I studied hard I didn’t leave as fluent as I had hoped. But, though I won’t be able to woo you with Petrarch, I will at least be able to buy your groceries; in Rome I could potter around town without needing recourse to English, performing all the little administrative tasks that were required of me, and even being able natter briefly with any curious locals as to my views on the recent Brexit vote. But a trip to the enoteca shattered any grand notions of personal competency: there is still much work to be done.

Though these days my musical affinity with Italy stems largely from my burning life-ambition to play Marcello in a production of La Boheme, the city of Florence is also the subject of what for a decade or more has been one of my favourite musicals, ‘The Light in the Piazza’, with lyrics and music by Adam Guettel. (A rather interesting character — though Richard Rodger’s grandson, his music descends more from Sondheim and Ravel). The basis for the plot is simple enough, though depth later emerges quite unexpectedly: a mother and daughter from South Carolina travel to Florence, Rome, then Florence again, and meet some handsome Italians. The admirable opening number is itself called ‘Firenze’ (the best song is the mother’s 11 o’clock heart-wrencher, ‘Fable’). Florence is, “a city of statues and stories,” so Guettel tells us, and regarding said statues I can confirm that the original of Michelangelo’s David, now housed in the Galleria dell’Accademia in pristine conditions after having been moved from the Piazza della Signoria 100 years ago, is worth all the hype and more. For a contrast, the Arnolfo di Cambio ‘Madonna of the glass eyes’, striking in its medieval simplicity and directness, is my off-beat recommendation. My favourite sculptures of all however, also rather curious, were the Andrea Pisano stone relief series labelled, “The beginning of…” which used to adorn the campanile but now are to be found in the refurbished Museo dell’Opera del Duomo. There are around 20 hexagonal pieces, each utterly charming, witty, technically intricate — ‘The beginning of navigation’, with three wily seafarers paddling furiously off towards stage-left, and the beginning of farming, with even the shepherd himself surprised at the how the gossamer lightness of his tent has been conveyed in stone, are the best.

Regarding the stories, there are already so many soaked into the stone that, on the one hand, it seems wholly unnecessary to add my own. My conversation with Roberto at the Mercato San. Ambrogio, as he sold me a pair of shorts for 2 Euros; being taught the finer points of muscle toning when happening to end up watching Andy Murray win the Wimbledon final with two professional sports physiotherapists; accidentally spending ten minutes in Venice after taking a wrong train; finding Fermi’s grave alongside Michelangelo’s and Machiavelli’s; singing ‘Vecchia zimarra, senti’ in a 16th century palazzo; hearing a Haydn string quartet in the courtyard of the house where Michelangelo was apprenticed; the day in San Gimignano; Peter’s hilarious poem about the future tense; Suzanna’s teaching. But, already taxing your patience, I will restrict myself to just one, already begun.

‘The piazza was busy, one side brightly illuminated by the powerful spotlights mounted on the far buildings. In the corner by the outdoor copy of Michelangelo’s David a crowd had begun to gather, but the shadowy sides were still populated by a Florentine peculiarity — the fluorescent sycamores. These remarkable devices have two floppy glowing plastic wings, and, when the ensemble is catapulted at speed into the air using a hand-held elastic sling, these wings fold back and create a streamlined shape which can achieve astonishing heights, nearly level with the buildings surrounding the piazza, if not the tower of the Palazzo Vecchio itself — a good 30 metres or more. Yet the true ingenuity of the design is that, when gravity eventually obeys Galileo’s call, the wings are pulled out in such a way that the object is moved to rotate extremely rapidly — like a sycamore seed — winging its way slowly down to earth. Every night, including this one, about ten young men of the town, intent both on proving their own virility and on selling these marvels to the goggle-eyed children passing through, would gather in the open spaces and fire their wares high up into the silent sky, for hours on end.

Leaving these behind, I turned around Neptune’s fountain to discover that the throng of spectators, by now numbering around 300, were watching what could only be a big band, in the final stages of setting up. The players, dressed in immaculate black, were going to be performing on a temporarily constructed platform, which had been erected to David’s right — between him and (the copy of) Donatello’s lion. They were carrying a rhythm guitar, rather than a keyboard player — a small dagger, me having spent the Monday evenings of my adolescence practising the rarified art of the big band jazz pianist — but other than that they seemed to be the real deal. Announcements, in Italian, followed quickly; my two-day-old nascent Italian ear found these utterly incomprehensible, save for some reference to Glen Miller, but then the language lesson ceased, and ‘String of Pearls’ began to waft gently over the air. How unendingly curious, this world, in which a tune which I had played not a week previously, filling in at the rehearsal of my father’s big band, in an Egbaston Quaker Meeting House, could now be heard again in such different surroundings.

The next tune was ‘American Patrol’, which I had first played almost exactly 10 years previously, preparing for the tour of The Netherlands as part of the Birmingham Schools’ Jazz Ensemble. From the slightly faltering nature of the improvised solos I concluded that the band was probably amateur rather than professional — no doubt this had been clarified during the unknown introductory spiel — but they were as tight as could be. David smiled down, the fountain trickled — all was well.

I left the Piazza, and continued my walk up Via dei Calzaiuoli, the wide promenade, lined from top to bottom with the shops of luxury Italian brands, connecting the Piazza della Signoria with the Piazza del Duomo. This is prime passeggiata location, and I would return several more times before July was through, taking the sweet, delectably humid air whilst wondering if I would be propositioned by some fiery Tuscan damsel. Apparently this is the sort of thing that should happen on a passeggiata (it never did). The music faded as I approached the green brilliance of the Duomo itself — back with the sycamores — and I did a circuit of the baptistery before returning south the way I had come. It is worth mentioning that, as the historic centre is build roughly on a grid pattern, one quickly develops a compass sense in Florence. The position of the sun at different times of day greatly affects which directions of travel will be afforded some shade: crossing east to west in the morning or evening, particularly along any of the Lungarno streets, is a recipe for blindness and sunburn, but attempting a north-south traversal during the middle of the day, on Via dei Calzaiuoli or any of the other wide avenues, is even worse. Melanoma in a moment.

Anyway, the sun had long set, and it was rather a thrill to be able to freely walk this path without oozing factor 50 from every pore. But this thrill faded, and, whether overtired from the acclimatisation to the new location and new heat, or touched by a deeper malaise, an unease then began to dwell upon me, as the music swelled anew.

Yet more, the juxtaposition of expressive forms made me begin to fear the same end for my treasured jazz. The 40s big band and David, side by side; not equivalent or even comparable creations, true enough, but here, in this place, weren’t they performing the same function? Beguiling reminders of a lost yesteryear, of a lost people who could write such songs and work such stone: of lost art? The band were playing ‘Lil’ Darlin’ ‘ now, impeccably beautifully. But is this all that we will have in the age to come? Expert regurgitations? Why isn’t every budding Michelangelo champing at the bit to try to write a chart as sublime as ‘Lil’ Darlin”? To those few of us who fell in love with these sounds and styles in childhood years, are we doomed to live out our days as witnesses to the creeping death of our beloved music, its condemnation to the same open-air museum?

The last time I was in Florence, I was not dancing alone. If on this occasion I had been able to join the jiving couples, perhaps I would have been distracted from this existential pang: as it was, my despair was absolute.

But then, a new announcement, as incomprehensible as all the others, but for a single verb — ‘telefonare’. “They’re about to play Pennsylvania 6500!” I thought (the ditty about a lover’s excitement and desire to dial his sweetheart in New York City, arranged by Glenn Miller for big band, curiously expunged of all lyrics save for the titular telephone dial itself). And sure enough, the loping saxophone line ambled out, followed by the entire band shouting, as I have done myself many times, “Pennsylvania 6, 5, Oh Oh Oh!”. But this was an Italian band, and Italians don’t really do diphthongs, their ‘Oh’ vowel crystal bright, cardinal — it sounded a little ridiculous! The smile returned to my face, if through tears, and I retreated back around the square, through the frolicking couples, stopping on one of the two benches placed by the main entrance to the Palazzo Vecchio. I had been carrying around some mathematics notes in my rucksack. So, settling myself down to consider whether Chebyschev’s inequality would yield any non-trivial bound in my toy function-field large sieve problem, I passed the evening.

Several scribbled pages later, after I had convinced myself that the error-free nature of the Chinese Remainder Theorem application meant that — for rather dull reasons — Chebyschev’s inequality was applicable, my meditation was broken by the interests of an Italian family at the other end of the bench. Somebody parking themselves down in the centre of Florence and filling pages with unfamiliar symbols was an object of some curiosity, it seemed. Through a disarming directness and charm, the man closest to me initiated conversation. He was visiting his brother, who worked on a vineyard in Chianti, about 20 kilometres south of Florence — I received that brother’s ‘biglietto da vista’. In stumbling reply, I managed to convey that I was a graduate student from Oxford, studying mathematics, but that I was also studying Italian for this month. They seemed very impressed with my conversation given that I’d only been learning for two days: I didn’t have the heart, or the linguistic ability, to explain that I’d been studying grammar and vocabulary in isolation, on-and-off, for nearly a year. Rather pleased with myself for having made some new Italian friends, without any need of English — the language spoken by, “the Gods of my cradle,” to quote the mathematician Hermann Weyl — I was expecting polite parting and a return to my work.

But this is not the Italian way. It quickly became apparent that they were really interested to know more precisely what I had been thinking about — what manner of abstract witchcraft could occupy the mind so fully? The brother was a trained mechanical engineer, with substantial mathematical literacy. Now, explaining what I study, in English, with slides, and lots of preparation time, is tricky enough — this past Michaelmas I just about managed to do it in 6 minutes, with slides that had taken me the best part of a day to prepare. On the spot, at 10 o’clock at night, on a bench in the middle of the Piazza della Signoria, in a language I don’t really speak, whilst already feeling over-emotional… I nearly ran straight for Fiesole. Yet powers of paraphrase swelled within, and vigorous gesticulation — combined with a fair splattering of that golden phrase ‘studio i numeri primi’ — sated my companions’ curiosity.

The performance over, me glowing with nervous pride, they had to be on their way, and, after a respectful pause, I too retreated home, forever changed. The following week, I would successfully explain in Italian, to the rest of my distinctly non-mathematical language class, why there were an infinitude of primes. I never saw the brothers again.’

A single evening, from happiness to despair to hope, through mathematics and jazz, in broken Italian. If that isn’t a microcosm of my year, I don’t know what is. Happy 2017.

# F is for Fourier Transform

The Oxford Mathematics Alphabet is a new public engagement project from the Oxford Mathematical Institute. I’ve been involved in the letter ‘F’, and the 500-word article is now online. Later there should be a poster.

If any of my non-mathematical friends have ever wondered what it is that I do during daylight hours, this is as good a place to start as any. 🙂

https://www.maths.ox.ac.uk/…/oxford-mat…/f-fourier-transform

p.s. The full alphabet should be completed in a couple of years. After that, bonus points for anyone who uses it to teach the alphabet to their children…

# A year in books

Tomorrow, it will be time for me to craft a fresh set of New Year’s Resolutions: today, I ponder at how last year’s went so badly awry. In anticipation of my inevitable future slackness, and with intimate knowledge of my inveterate laziness, on January 1st 2015 I made about ten resolutions, hoping that at least one of them might stick. There were plans for running, swimming, tightening up my daily administration routine, getting more sleep, having less screen-time in the evening, curtailing my YouTube binges, eating better, and so on. I made the great mistake of writing these down in a notebook which I have failed to lose. Thus, I can now examine in perfect detail the full extent of my failure.

Yet, although I won’t be winning a triathlon any time soon, a couple of my resolutions did stick, in particular my resolve to vastly increase the quantity and scope of my non-mathematical reading. Realising this, I thought that I would try my hand at an activity which, towards the end of last year, I saw take over the Facebook newsfeeds of my more literarily-minded friends. The idea is to write a list of all the books you have read throughout the calendar year, and, if you should so wish, to draw out a few reflective comments from the collection. I am sure that my list will seem feeble next to that of a true bookworm, but for a man who devotes a substantial amount of his time to thinking about prime number in a darkened room, I don’t think I’ve done too badly.

The adventures of Augie March, by Saul Bellow

I finished this long book on a train, travelling back to Oxford in early January. Martin Amis describes it as ‘the great American novel’, though he is rightly sceptical about the weird bit in the middle, when, suddenly, apropos of nothing, the book’s got an eagle in it. All told, the story is an extraordinarily vivid, endlessly imaginative coming-of-age tale, set in and around 1930s Chicago. My one qualm is that, despite encountering many obstacles of life, Bellow’s ‘everyman’ hero is always rather successful with women – hardly the abiding male experience.

Private Island, by James Meek

This collection of essays, many of which had previously appeared in the London Review of Books, is a concentrated attack on the last thirty years of state privatisation. The overall effect is savage, but the tone elegiac. Meek mourns the passing of the era of, “government intervention, of the belief that the state had the power, the right, and the duty to make a better world for its citizens.” He describes its replacement: “The market belief system holds that government is incompetent by default, that state taxation is oppressive, that desire of wealth is the right and principal motivator of achievement, and that virtually all human wants can be best met by competing private firms.” You will be horrified by Meek’s depiction of the current state of the nation, yet alas Meek does not succeed in his aim to proffer a coherent alternative programme – the chapter on social housing excepted. I could write about this book for days, in great praise, but for now just consider this particular morsel – an obvious idea that I should have noticed before: competition is the driver of capitalism’s success, but one should never imagine that competition is welcomed gladly by all of capitalism’s participators. It is often in a company’s self-interest to try to avoid it, and the company will do everything in its power – which is sometimes quite a lot – to do so.

The Language of Money, by John Lanchester

A humorous, though slightly superficial, glossary on finance-speak. About two years ago I toyed with the idea of writing something similar about mathematical language, and I found here, to my surprise, that some of the difficulties I perceived in the public understanding of mathematical vocabulary, which I had previously believed were unique to mathematics, arise also in the context of finance. These difficulties are eloquently explained by Lanchester; in particular, the difficulty when commonplace words undertake a new technical meaning, and then undergo further modification. The word ‘securitisation’ does not mean anything approximating the phrase ‘to make more secure’!

Lucky Jim, by Kingsley Amis

Lucky Jim was described by Clive James as the funniest book ever written. It does provide some excellent chortles – the madrigal scene, the terrifying lift in Professor Walsh’s car, lines such as, “He disliked this girl and her boyfriend so much that he couldn’t understand why they didn’t dislike each other.” I had expected the humour, but what surprised me most was how, in amongst all the slapstick, Amis manages to provide a myriad of calm, beautiful insights into the madness of individual existence, and the impossibility of ever being able to understand other people’s lives. To offer a non-trivial reservation, it must be said that, despite the existence of a heroine of sorts, the women don’t come out terribly well from the book – though the men do not fair that much better.

1984, by George Orwell

I should have read this long ago. If you too haven’t got round to it yet, I can report that the book is as brilliant and as devastating as you’ve been told, and worryingly easy to read. It made a substantial contribution to the pall that hung over my February, but I’m glad to have experienced its full visceral power. For me, Orwell’s key genius is in creating a dystopia that is both utterly terrifying and completely believable – neither of which Huxley quite manages. He was helped, perhaps, by the fact that the future he was predicting for England already existed as the USSR’s present.

Stoner, by John Williams

Stoner is a short novella – I read it over the course of 36 hours – giving a whistle-stop rendering of the life story of an English Lecturer at the University of Missouri in the early twentieth century. It was Julian Barnes’ book of the year in 2013, though published 50 years before – a rediscovered masterpiece. The contrast with ‘Augie March’ – the other ‘life story’ I had recently read – could not have been greater. From a book in which every minutiae of existence is pored over in detail, I’d gone to one which eschews all but the most critical components of life and, in doing so, encourages you to consider what these truly are. The writing is so economical, so plaintive, on every page raising paragraphs of prose to great poetic heights. It is a beautiful, beautiful book.

Rosencrantz and Guildenstern, by Tom Stoppard

I had an enjoyable evening’s saunter through this great ‘modern’ play – though I guess now it is not so modern. The humour and depth of the constant metaphysical bantering is so good that the moments when one of the titular characters launches into a genuine soliloquy seem rather stodgy, overly philosophical. But maybe these passages just require different skills of the actor, and I must admit to never having seen this play performed. So please forgive this minor quibble of mine with an otherwise deeply moving work, which I will now think about every time I toss a coin, and every time I brim with wanderlust: “It’s still the same sky.”

Birth of a Theorem, by Cédric Villani

I bought a signed copy of this book as a birthday present for my father, but couldn’t resist giving it a quick look before wrapping it up. It is a peculiar book, in almost every sense of that word. It takes exactly the opposite view to that of Stephen Hawking’s publisher – who famously claimed that every equation included in ‘A Brief History of Time’ would halve the readership – by including page after page of LaTeX equations. Villani quipped that, if the publisher’s theory were right, he might only have a few molecules of reader left.  But the book has been very successful. The intention behind the approach, the author claims, is not for the readers to understand the equations – such understanding is impossible, even for a trained mathematician such as myself – but to follow Poincaré’s lead and attempt to give a genuine insight into the real life of a working mathematician, rather than shying away, using some tame anodyne replacement. Though the result of this process is slightly odd, Villani has succeeded, and his book has a rare verisimilitude which is lacking from almost all other popular maths books.

Cultural Amnesia, by Clive James

How can I hope to summarise this vast sprawling tome in a short paragraph? Over nine-hundred pages long – only a small indicator of its true vastness – it was given to me by Richard Moxham, the recently retired headmaster of Dover Grammar School for Boys and a childhood friend of my dad. For nearly two years it has been my ongoing project to finish it, and finish it I finally have. The book is a collection of essays about James’ various intellectual obsessions, through which he attempts to sketch an impression of twentieth century humanism. His obsession with fin de siècle Vienna, say, has introduced me to a wealth of humanist accomplishment to which hitherto I had been blind, but really this is just one of the many facets of this, “starburst of wild brilliance,” as Simon Schama puts it. Most of the hundred essays are a humorously vicious skewering of those thinkers who have had the misfortune of rubbing the author up the wrong way. Take Sartre. James doesn’t like Sartre, to put it mildly, and towards the climax of his ritual disembowelling of one of the most preeminent thinkers of the twentieth century, we’re brought to the line, “In Sartre’s style of argument, German metaphysics met French sophistry in a kind of European Coal and Steel Community producing nothing but rhetorical gas.”  From James, I have learnt that, in the game of cultural reference, anything goes! Alluding to both the Death Star and Dante in the same sentence may not be classical style, but that needn’t stop you doing it, a lot.

Travels with Epicurus, by Daniel Klein

This is a tiny book, which had been a birthday present from my father to my mother on the occasion of her 62nd birthday: it is about growing old. Beautifully non-prescriptive, Klein builds an admirably down-to-earth philosophy out of his own personal experience, returning to the Greek island he knew as a young man. Unfortunately, if one is looking in the book for an answer to the conundrum of life, the conclusion is clear. If you can retire to a Greek island, you might be alright: otherwise, you’re stuffed.

Travelling to Infinity, by Jane Hawking

Both these next entries have a delightful shared provenance. I was heading back to Oxford after my brief Easter break when I found myself with an unexpectedly long wait at Banbury station. Wondering whether my train had in fact been built yet, I bought an outrageously expensive bacon sandwich from the station cafe and slumped down in a drab metal chair. Looking across again at the cafe, it appeared that they were also selling a small collection of paperback books. In my limited experience the literary offerings from such establishments are rather trashy, yet, most unexpectedly, they were selling two books which I had been meaning to read for a while. I snaffled them up, and the wait passed quickly.

By a strange twist of fate I have a personal connection with Jane Hawking, arising from my singing in her second husband’s choir for several years, and I’d always been curious to read about her life with Stephen, in her own words. Although I found a particular recurring trope – the building up of dramatic tension in the final paragraph of each chapter – to wear a little thin, the tale is fascinating, beautifully told, and it has raised my already great admiration of both Jane and Jonathan to loftier heights still.

The Establishment, by Owen Jones

This second book by Owen Jones is an unashamed polemic against ‘the powers that be’, both institutions and individuals. It is not as lyrically written as the aforementioned James Meek, but it is nonetheless extremely lucid, and brimming with remarkable research. One of the main thrusts of the book is the following question: how there can be any notion of independence of the press and media, given the happy flow of personnel between political advisers, the BBC, think-tanks, and newspaper editors? How can it be that the institutions set to benefit hugely from certain government actions, such as tax regulations, are also the institutions advising the government about those actions? Poachers, turned gamekeepers, turned poachers again. Searching online one can discover that The Telegraph review misses all of the book’s nuances, unsurprisingly. Yet the review is right to point out that Jones’ proposed solutions seem rather unconvincing.

Now time for a maths diversion…

In Hilary Term 2015 – I still catch myself calling it Lent – I ended up doing rather too much teaching. The situation was entirely my own fault, being just a boy who can’t say no, but the upshot was that I had not had quite as much time for research as I would have liked, but had at least managed to earn a fair amount of pocket money. Thus, to ameliorate my condition and clear my conscience, I spent almost all my money on maths books. Combining these with my existing collection I now have a small but rather various additive combinatorics/ analytic number theory-themed library, from which I will now select a few choice items for your delectation.

Opera de Cribro, by John Friedlander and Henryk Iwaniec

The title translates as ‘Sieve Works’. This 500+ page book is the authors’ magnum opus on sieve theory, the mathematical sub-discipline which studies prime numbers as the residue remaining after sieving out by arithmetic progressions. Its commitment to generality and depth can make certain sections difficult to digest on a first reading: yet on more than one occasion, after struggling through seas of silent thought to some small personal sieve theory insight, I have later found the same idea waiting for me in Opera de Cribro, having hidden itself there all along.

Ten Lectures on the Interface between Harmonic Analysis and Analytic Number Theory, by Hugh Montgomery

Purchased on the recommendation of my supervisor, this is a really fabulous book. I had not heard of it beforehand, as it is not perhaps in the ‘Davenport: Multiplicative Number Theory’-class of fame – though it does crop up rather often in the bibliographies of fine papers. Montgomery picks a rather tiny area of mathematical analysis – the subtitle ‘Applications of the Fejér kernel’ would not be too ungenerous – but by showing a huge variety of applications he has helped me to reach a more connected view of my entire subject. I have come across a slightly embarrassing oversight in chapter 2 – in which Montgomery claims that every positive real trigonometric polynomial is the square of a trigonometric polynomial of half the degree – but, such an error notwithstanding, the clarity of exposition and the range of topics covered is exemplary. I used several chapters as the basis for the informal seminar series I gave at the end of last term.

Introduction to Analytic and Probabilistic Number Theory, by Gérald Tenenbaum

Oh Tenenbaum, oh Tenenbaum… This is a recently published third edition of the mid-nineties classic, in which the material is expanded, rearranged and, most importantly, completely re-typeset. Reading both editions has made me acutely aware of how important typesetting is in the overall comprehensibility of a mathematical text: the previous edition was an impenetrable mire, this a model of limpidity. Like the sort of undergraduate student I never was, I’ve actually been going through some exercises. As with ‘Hartshorne’, the famous algebraic geometry textbook, I am getting the sense that the exercises comprise a substantial proportion of the book’s educational worth. This book is the only reference I know of for the Selberg-Delange method, the fruits of which play an important role in my first research paper.

Additive Combinatorics, by Terence Tao and Van Vu.

Written in 2006, this book remains the only vaguely comprehensive tract on Additive Combinatorics. It suffers from being written about a fast moving field, and so a lot of its material – on sum-product theorems, Roth’s theorem, Freiman’s theorem, and the Erdȍs distance problem – has since been greatly improved upon, and in certain cases completely revolutionised. The typesetting is also disastrous, as is also commented upon by Ben Green’s AMS review — not CUP’s finest hour. Yet, rather like Opera de Cribro, it is a veritable mine of information. This year I’ve used it to finally get my head around viewing energy increment arguments in terms of conditional expectations, and to aid my nascent understanding of restriction theory.

The Probabilistic Method, by Noga Alon and Joel Spencer

A classic, and, like all Wiley Classics, really expensive! But there’s nothing like having the book in your hands, and I should have laid my hands on this book long ago. It turns out that I had become familiar with most of the material already, what with the Combinatorics Summer School I attended in Lisbon a few years ago and Prof. Bollobás’ course during my Masters year, but the authors’ exposition is marvellous, and the range of applications discussed is dramatic. The slick proof of the Erdȍs-Kac theorem is my personal highlight.

Leaving mathematics to one side…

The Shakespeare Project

The week beginning Monday May 11th was the unhappiest in my life that I can care to remember. Worse things have happened to me, but never have I been unlucky enough to experience such an accumulation of irksome occurrences. To try to cheer things up, I treated myself to a book. A big book: the RSC Complete Works of Shakespeare edition, to be precise. The thought struck me that, preparing as I was for a monastic summer of mathematical study, it might be a good time for a serious reading side-project, to return to in the evenings after a hard day at mill. One surely cannot claim to be an educated Englishman without an intimate knowledge of the Shakespearean canon and, though there are a couple of plays that – through the quirk of unexpected multiple viewings – I know rather well, beyond these I was completely ignorant, save for the small amount of metadata I conned for University Challenge.

I know that there are many who feel that Shakespeare can be appreciated only in performance, not on paper. Yet though I love seeing the plays in the flesh after having read them, as happened twice this summer, I find that without prior study I lose the full majesty of the remarkable language; the verses torrent by too fast for my untrained ears, and it’s all I can do to keep up with the plot. Shakespeare’s hallmark is the outrageous quality of his by-writing, and it would be such a shame to miss it.

I decided to read the plays in First Folio order, with the exception of ‘The Merry Wives of Windsor’, which it seemed sensible to read only after having finished both parts of Henry IV.  The project is still ongoing, though I am about 65% of the way through, but already I have found it to be a life enhancing experience of rare intensity. I have spent New Year at my parents’ house, and alas my RSC edition, in which I have made marginal notes and been a precocious ‘underliner’, languishes in my Oxford room. Because of this fact I will refrain from offering such comments as I have on individual plays until I am reunited with my notes. Let me give you a list of plays I have read so far, and present a couple of anecdotes from my Shakespearian summer.

The Tempest

Two Gentlemen of Verona

Measure for Measure

A Midsummer Night’s Dream

Love’s Labour’s Lost

The Winter’s Tale

All’s Well That End’s Well

Twelfth Night

As You Like It

The Merchant of Venice

Comedy of Errors

Taming of the Shrew

King John

Richard II

Henry IV Part i

Henry IV Part ii

Henry V

Troilus and Cressida

Coriolanus

I saw a Globe touring production of Much Ado About Nothing in the Bodleian quadrangle. The portrayal of Dogberry struck me as frightfully familiar, and I suddenly realised that he was the doppelganger of the man who had tried to sell me an iPad that morning: same beard, same accent, same irrepressible self-confidence.

A friend of mine was playing Jacques in an outdoor production of As You Like It in the courtyard of Oxford Castle. My friend was suitably melancholy, and Touchstone a revelation, but my abiding memory will be the incongruous full-cast dance to ‘Uptown Funk’, which the director had decided to insert just before the famous, “All the world’s a stage,” speech. An odd choice.

On June 18th, having spent the entire day reading Love’s Labour’s Lost, I trotted over to St. Giles church for a choral concert given by a few friends. One of their musical items was a series of George Shearing settings of various Shakespeare songs, including the one that I had read only a few hours previously, the song with which Love’s Labour’s Lost finishes. I started in my seat when I began to recognise the text!

The Murder of Roger Ackroyd, by Agatha Christie

I read this on a rainy day in Worthing, in a very strange flat which my parents had rented for the week (me joining for the Bank Holiday weekend). The flat was decked out with multitudinous art deco regalia, and possessed a formidable Agatha Christie collection.  My father and I were entranced.

Four Parts, no waiting, by Gage Averill.

My father gave me this book for Christmas several years ago, but at the time I lacked the academic maturity to stick it through. The book is an extended musicological essay on the place of Barbershop music in American culture, in particular its relationship with race and with institutionalised nostalgia, through the Barbershop revival movement. The style was for the most part rather ponderous, and I would have liked much more discussion of gender, but an interesting read nonetheless and a source of rich insight into the early era of musical recording.

The Pilgrim’s Progress, by John Bunyan

I came to read this after singing Vaughan-Williams’ ‘Valiant for Truth’ in the Schola Cantorum UK tour, after which I developed an interest in understanding where the mesmerising text to that particular work fits in the overall novel. It turns out to come from the second book, concerning the progress of Christian’s wife and children, and alas it represents somewhat of an outlier of literary quality in the book, which is less a novel, more an acute case of liturgical reference diarrhoea. Nevertheless, the section in Vanity Fair is quite remarkable: in trying to describe the worst den of iniquity he could imagine, Bunyan outputs a rather tame description of21st century consumerism.

Godel, Escher, Bach, by Douglas Hofstadter

I’ve written a whole other blog post about this one. Vast, weird, unique.

The Chinese: Portrait of a People, by John Fraser

I’ve not quite finished this yet, having saved it from a charity shop box as it left my parents’ house, who, at my last reckoning, have upwards of 2000 books in a moderately sized semi-detached. The book is a very classily written account of the author’s two-year stint as a journalist in Peking in the late 1970s, and how he found himself caught up in the democracy demonstrations which occurred around the Xidan wall in the winter of 1978.

What are my literary plans for next year? Well, given that we seem to be entering the Wars of Religion, a quick perusal of the King James and the Qu’ran seems in order. There’s a lot of Shakespeare still to finish. As for another big project, I am frequently embarrassed by how little Dickens I have actually read, but also by the fact that only one female author appears on the above list. Perhaps I’ll mix some Dickens with some Mary Ann Evans (aka George Eliot).

Plenty to be getting on with.

# Reflections on ‘Gödel, Escher, Bach: an Eternal Golden Braid.’

This book has been following me for a long time. I can first recall its name cropping up around seven years ago, at one of the myriad of mathematical training camps I attended as a bright-eyed school student. Perhaps I was eavesdropping, as the critical conversation returns to me now only in fits and starts. However it happened, I became aware that several of my mathematical peers had all recently read a certain book, and that this book was called ‘Gödel, Escher, Bach’ [hereafter GEB]. They were discussing it with great fervour, and in my growing mind that three-word title immediately held an air of mystery, a golden lustre. The memory fades, of course, but what remains is a particular perception, that these friends were all bonded by some remarkable shared experience. I felt that I would never truly be a part of their circle until I had read this book too.

I was wrong in this regard, unsurprisingly: four years of the Trinity College Cambridge mathematics experience was more than enough to bond me to these folk for life. GEB returned to my mind only once, when one rainy afternoon I caught sight of the spine of an intimidating-looking copy, hiding in the mathematics corner of Trinity library (both me and the book). But there was always something more pressing to be getting on with – when is there not? – and so I didn’t read it then either.

I’m at Oxford now, working on a PhD in analytic number theory, and most nights I wear away the long age of three hours between my after-supper and bedtime by singing in one or another choir. Last year I was in The Oxford Gargoyles, a jazz a cappella group. There I met Sam Rice, an undergraduate mathematician, and Lachlan Hughes, a mature student studying German and Italian. Somehow it came up in conversation that Rice himself had read GEB, and possessed a copy he would willingly lend me, and that Lachlan – a Bach devotee – had in the past tried to read it twice, but had got stuck halfway through, on both occasions. At that time I was in the middle of a self-imposed Shakespeare reading marathon, but I took up Rice’s offer, being eager for a little diversion from the never-ending Henrys, and began my assault on GEB’s 742 extraordinary pages. After seven years, I would now finally know what all the fuss was about.

‘Gödel, Escher, Bach: an Eternal Golden Braid’ was published in 1979, and won Douglas R. Hofstadter the Pulitzer Prize for General Non-Fiction.  It has attained a cult-like status among certain communities, particularly Computer Science, with Hofstadter himself viewed as the ultimate guru. There’s something about GEB that makes everyone want to talk about it. Though it is a mighty tome itself, it would be dwarfed by a collected edition of all the reviews and articles written about it. Just have a scroll down this page, say:

or look at the sheer number of Quora posts, Reddit threads. They even ran an examinable course on it at MIT:

http://ocw.mit.edu/high-school/humanities-and-social-sciences/godel-escher-bach/

I am painfully aware that my own reflections are only adding to this great heap. Such a phenomenon can be partly explained by the fact that, whatever GEB actually is, that which it is is completely unique. And uniqueness is a rare and sought-after commodity, particularly from the creatively impoverished breed of writers known as reviewers. GEB oozes raw originality on every page, and, by writing a review, we can only hope that some of it might rub off.

Except of course there is. The people who failed to appreciate Hofstadter’s intended message were not stupid. For a start, the biological and computer science strands are not introduced until the book is well over 300 pages old. Lord of the Flies, The Catcher in the Rye, The Hitchhiker’s Guide to the Galaxy: all three are wound up by the 300 page mark, yet it seems Hofstadter hasn’t even started! Somehow, while developing his original theme, fleshing it out, drawing analogies, wandering off on flights of fancy, Hofstadter created such a rich body of work that in the final performance the supporting acts are completely indistinguishable from his original star attraction. He knows what he meant the book to be about: that doesn’t necessarily mean that he can speak for how the finished product ended up!

I think that the best explanation for the content of GEB comes from thinking of it as four separate books, not one. This is a slightly unsatisfactory model, as rules of three pervade Hofstadter’s magical narrative, but – as every working mathematician knows – sometimes beauty is less important than truth. Anyway, the first ‘book’ is a very well-written popular exposition of Gödel’s Incompleteness Theorem. This is a theorem of logic: perhaps the theorem of logic. The author takes plenty of time to introduce formal logical systems, and uses a couple of brilliantly-conceived toy models as explanatory tools – the MU and the p-q system. He builds up to the final dénouement with the verve of a novelist. This book most resembles an ‘ordinary’ popular science volume, and comprises most of GEB’s first act.

However, rather than satisfying himself with ‘run-of-the-mill’ pedagogy, Hofstadter is also a master of analogies, subtle and beautiful, which on some very high level echo a main idea in the proof of Gödel’s theorem. My favourite of these is the image of a record player which claims to be infinite-fidelity – it can play any record perfectly. The record player’s owner has a cheeky adversary. This adversary, after carefully studying the blueprints of the contraption, creates a record which, when reproduced perfectly, plays music at the exact resonant frequency of his enemy’s record player. When the enemy plays the record, his machine instantly shatters. Thus the claim is foiled: the appliance was not able to play every record perfectly after all.

As the record is to the record player, so is the Gödel string to a complete model of Peano Arithmetic. The central concept is that of self-reference, or a ‘Strange Loop’ as Hofstadter terms it, and in looking for other analogies he is drawn to reference many Escher prints, and some elements of work by Bach – the endlessly rising canon, most particularly. In fact, Bach fugues are later used as metaphors for the entire book, and Zen Buddhism also acquires a significant bit-part. Gleeful at the richness of his theme, this is what Hofstadter unwittingly makes into the second ‘book’. The analogies become expanded, and they acquire a new literary life of their own, quite outside their humble origins as explicatory tools.

The third ‘book’ is that book which Hofstadter originally aimed to write, when, as he reveals in the preface, the rich interwoven superstructure that would become the finished product’s hallmark was not yet conceived. This book explores how far one can go in viewing human brains as formal systems of neurons. In this viewpoint, the emergent phenomena, such as ‘consciousness’, come about by Strange Loops between different neural substrata. Hofstadter goes on to apply these ideas to the question of whether a computer could ever become intelligent, in the human sense, and whether it could ever become self-aware. In this book, biology and computer science go hand in hand, and there are excellent freestanding chapters on computer hardware, on programming languages, and on the data exchange inside each living cell, with DNA being both data and program, in an extraordinarily tangled web.

The fourth ‘book’ is very different in flavour. In between every chapter of GEB Hofstadter writes a short – though sometimes not-so-short – Dialogue, in which the characters of Achilles and the Tortoise ponder the topics that are being discussed in the book at large, and have various adventures. Other characters are slowly introduced – a Crab, an Anteater, a Sloth. They invite each other round for tea, go jogging, engage in an arms race over record players and record-player-destroying records, and, in one particularly thrilling moment, get stuck in a nested series of Escher prints whilst, in the real world, they lie in a helicopter being kidnapped. Here Hofstadter anticipates the blockbuster Inception by 35 years.  These Dialogues are wonderfully written, amusing and exciting, and are often a literary tour-de-force, particularly when Hofstadter – merely for the joy of it – writes the Dialogue to mimic a Bach fugue.

Okay, so we have four books. But now imagine that you were reading all of these books at the same time. Don’t picture having four separate volumes down on the table, and reading a chapter of each in turn, but really think about what it would mean to read them simultaneously, sentence by sentence, word by word even, with ideas from one book cropping up in another, constantly, for over 700 pages. For a start, those Dialogues are spread evenly throughout the GEB text – there are 21 of them in total – and they all contain analogies, analogies within analogies, which make ready the topics of the forthcoming chapters. It’s a bit like reciting a well-chosen Aesop fable before embarking on a heavy discussion of moral philosophy, only much more intricate, as the academia often finds its way into the dialogue, and vice versa, and all those other themes – Bach, DNA, quantum physics, Zen, logic, computer programming – get thrown along too, into the blender. Ah yes, I forgot to mention that Hofstadter’s PhD was actually in Quantum Physics…

Perhaps you are beginning to get a flavour of just how wildly rich GEB is.

As a brief example, let me describe the plot for the ‘Prelude’ ‘Ant Fugue’ – pun certainly intended – which are the two Dialogues beginning Part II of GEB. To enjoy these delights as a reader you would have to survive 274 pages, and make it through the definition of propositional calculus; although, to give some perspective, you would still be a long way off the half-way mark. The ‘Prelude’ begins with Achilles and the Tortoise having tea with the Crab, and being introduced to the Crab’s friend, the Anteater. They have brought the Crab a present, which turns out to be two records of Bach himself playing the harpsichord, created by means of ‘acoustic retrieval’, a wonderful newfangled technique, where from studying the movements of the molecules in the atmosphere one can retroactively model vibrations, simulating exactly the quivering of the air from many centuries ago. They start listening to the Prelude and, following the music on the Crab’s score – which happens to have some old Escher prints between the pages – Achilles talks about whether, in the fugue that is to come, one can ever really appreciate the intricacy of each individual line whilst also considering the effect of their combination, in the piece as a whole. There is also a strange little discussion about Fermat’s Last Theorem, though that is by the by.

In the Dialogue, the prelude ends and the fugue begins. Popping out one level, the ‘Prelude’ ends, and – after an excellent intervening chapter about computer systems – the ‘Ant Fugue’ begins. Pushing back in, the characters have discovered a peculiar illustration in between the leaves of the score, on which the word MU in written. [MU is a weird recurring element in the book, not unlike ‘Bad Wolf’ in the first series of the revamped Doctor Who – the one with Christopher Eccelston, remember? Thus I write my own submission to the canon of obscure references.]  On closer observation it seems that the word MU is actually comprised of small letters, which spell Reductionism and Holism, and under still closer observation these turn out to be written from smaller letters again, spelling Holism and Reductionism, which under much much closer observation are seen in turn to consist of the word MU once more. The characters discuss their different perceptions of the image.

Meanwhile the fugue is going on, and the characters are also commenting on particular effects that have occurred – the entrance of each voice, stretto, an organ point, etcetera. The Anteater begins discussing his recent trip to see the ant hill ‘Aunt Hillary’, and describes how he can communicate with the colony, and the colony communicate with him, even though the individual ants have no intelligence of their own. Mechanisms for such a process are posited, how different complexities of thought can coexist and intermingle, and Reductionism and Holism are once more discussed. Fermat appears again, with his famous marginal note being translated into musical parlance and placed in the mouth of a fictional Buxtehude, who claims to have composed a 24 part fugue which modulates through all 24 keys. But all the while, the text mimics the fugue which is being played in the background. Let me quote from the start:

“Achilles: I know the rest of you won’t believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word – but what an important one: “MU”!

Crab: I know the rest of you won’t believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word – but what an important one: “HOLISM”!

Achilles: Now hold on a minute. You must be seeing things. It’s plain as day that the message of this picture is “MU”, not “HOLISM”!

Crab: I beg your pardon, but my eyesight is extremely good. Please look again, and then tell me if the picture doesn’t say what I said it says!

Anteater: I know the rest of you won’t believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word – but what an important one: “REDUCTIONISM”!

Crab: Now hold on a minute. You must be seeing things. It’s plain as day that the message of this picture is “HOLISM”, not “REDUCTIONISM”!

Achilles: Another deluded one! Not “HOLISM”, not “REDUCTIONISM”, but “MU” is the message of this picture, and that much is certain.

Anteater: I beg your pardon, but my eyesight is extremely clear. Please look again, and then see if the picture doesn’t say what I said it says.”

Thus the Dialogue continues with three interweaving fugal voices, meandering off into the discussion of ant hills. A few pages later we get the following, with a killer stage direction:

“Achilles:  Oh, dear! We’re getting nowhere fast. Why have you stayed so strangely silent, Mr. Tortoise? It makes me very uneasy. Surely you must somehow be capable of helping straighten out this mess?

Tortoise: I know the rest of you won’t believe this, but the answer to the question is staring us all in the face, hidden in the picture. It is simply one word – but what an important one: “MU”!

(Just as he says this, the fourth voice in the fugue being played enters, exactly one octave below the first entry.)

This delight lasts for 25 pages.

One cannot overlook the fact that every single character mentioned in the book is male. The Dialogues in particular seem to propagate the establishment notion that intellectual coffee-time talk is the exclusive preserve of men.  To his great credit, Hofstadter dedicates a large passage of the 20th anniversary-edition preface to this terrible state of affairs, and berates his younger self – he was in his early thirties when he wrote GEB – for being so blind to this inadequacy. Fortunately the issue has a happy ending, to a degree, which stems from the translation of GEB into French. The translators suggested that ‘Mr Tortoise’ needed to become ‘Madame Tortue’, as  the word ‘tortue’ is feminine. Thus the French reading public will open GEB to find the intellectually brilliant ‘Madame Tortue’, running rings around her dimmer-witted friend Achilles.

GEB is overlong – much like this review. There is easily enough content to fill 550 pages, 600 at a push – I am not suggesting that GEB should be a short book – but there are several sections which do drag awfully, particularly some of the bio-philosophy later on, and there are themes which are introduced and never quite brought under the wing of the whole. The diversion early on into fractal graph plots from the author’s PhD thesis – by ‘early on’, I mean after 140 pages or so – is a case in point. More drastically, it is worth noting that all the references to Bach could be eschewed completely and the book would still function, be entertaining and be informative. I mention this only in passing, though; such a heinous act would rob the text of much of its unique charm.

With all its different themes, GEB does struggle with issues of unity. However, one should not overly criticise this aspect, but rather gawp in amazement at how it manages to hang together at all.  Though the writing style is very different, I am reminded a little of Bertrand Russels’ ‘History of Western Philosophy’, in which the language is so engaging, limpid and witty that the reader will gladly follow the author anywhere, even through the very deepest of intellectual thickets. Though he does waffle in the neuroscience sections, consider Hofstadter mid-flow, explaining his ‘Typographical Number Theory’:

“This completes the vocabulary with which we will express all number theoretical statements! It takes considerable practice to get the hang of expressing complicated statements of N in this notation, and conversely of figuring out the meaning of well-formed formulas. For this reason we return to the six sample sentences given at the beginning, and work out their translations into TNT.”

If that doesn’t get you manipulating formulae, I don’t know what will.

GEB has fundamentally altered the way I think about so many ideas – not least the art of writing popular science – and I can well understand why my boyhood friends were so enraptured by it. Though it is a flawed work, I can think of few other long reads that have changed me so much (‘Earthly Powers’ maybe, but not ‘Augie March’). Every day I find myself drawn to using Hofstadter’s analogies, for thought, for music, for self-reference, and I have to constantly catch myself, as I realise with a shock that not everyone will get the reference.  GEB has even made me re-evaluate my belief in free will. Consider the book’s ultimate claim: any system that is capable of sufficiently powerful computation must necessarily be self-aware. Whether you agree with this or not, it is a truly startling pronouncement.  Read ‘Gödel, Escher, Bach’, and see how Hofstadter, in his own utterly inimitable way, develops his theme.  I’m glad that I finally did.

# Reflections on X+Y

Hollywood is governed by fashion, but I was as surprised as anyone by the glut of intellectual biopics with which we were presented this winter: the Oscar-winning The Imitation Game and The Theory of Everything, of course, but also X + Y, and an adaptation of the excellent Ramanujan biography The Man who knew Infinity, currently in post-production. Stranger still that I should have personal connections with several major characters in these films. Considering The Theory of Everything, say, I sang with Jonathan Hellyer Jones for three years whilst an undergraduate at Cambridge, and conversed with Jane Hawking many times over excellent paté in their delightful conservatory. Considering X+Y, the subject of this current musing, as a schoolboy I ventured through the dreamy world of Olympiad mathematics myself, meeting several figures whose fictional doppelgangers have now been immortalised in film. You wait 20 years for a movie about your life, and then two come along at once; a peculiar delicacy, to have your past served up for a repeat viewing.

X+Y – directed by Morgan Matthews and starring Asa Butterfiled, Sally Hawkins and Rafe Spall – charts the journey of an autistic boy Nathan, his family and his teacher, as he grows up and vies for a place on the UK team at the International Mathematical Olympiad. It is a fictionalised version of the documentary Beautiful Young Minds, also directed by Matthews, which followed the selection and success of the real UK IMO team in 2007. I am not a fan of this documentary: whether by accident or design it has a rather warped focus, never managing to capture the essence of what it is to participate in UK mathematical enrichment. I approached the fictionalised version, in which all inadequacies would surely be amplified, with no small amount of trepidation.

However, I can safely report that the film is excellent, and I would thoroughly commend it to everyone. It has its failings, which I shall come to divulge, but the world – and the world of mathematics – is much improved by its existence. I saw it with my friends David, Oscar and (his fiancé) Rachel; I met David at a weeklong maths camp at the University of Bath in 2008, when we were both 16.  He shares my misgivings about Beautiful Young Minds, but remarked most perceptively that, if anything, the film is more accurate than the documentary on which it was based, at least in its rendering of the Olympiad experience. I agree: it is as if the extra freedom and intimacy afforded by filming actors rather than people – as Stoppard might put it— has allowed Matthews to honestly represent what he observed while following the team in 2007, rather than to be tied to what he managed to get documented on film. More speculatively, I know that several UK Maths Trust big-shots were involved in the production of the film, possibly influencing the quality of the portrayal of the mathematics, whereas during the documentary the mathematicians were reduced to being subjects of Matthews’ editorial reign.

The most scintillating aspect of X+Y is the quality of the acting, particularly from the child-stars. In the course of my school maths enrichment – six camps and two international competitions – I must have interacted with at least 100 other young mathematicians of my age, several of whom have become my dearest friends. Whether by intensive research or blind luck, the actors manage to capture so many of the indescribable physical and vocal idiosyncrasies of this real-life cast-of-characters that, when watching the scenes at the maths camp in Taipei, it was as if I was 16 again, and meeting David for the first time. Mathematicians suffer from a peculiarly stilted speech pattern: we like to conceive of what we are going to say, and then to say it, causing our conversation to consist of long pauses followed by a cascade of all the words we were so painstakingly arranging in the interim. In Butterfield’s intonation you can sense just this very feature — silence indicating the presence, rather than the absence, of thought.

From the adults there is quality too: Rafe Spall brings a refreshing humanity to the role of the crippled former-genius maths teacher, which could so easily have fallen into tired cliché, and Sally Hawkins is thoroughly convincing as the much-suffering single-mother seeking happiness both for her son and herself. It is slightly unfortunate that Hawkins played an, albeit very different, mother figure in the winter’s biggest blockbuster; at any moment I was half-expecting Nathan to sprout hairs and start wittering about darkest Peru. Eddie Marsan is believable too, although his portrayal of the team-leader is rather an evil turn, which could have the unfortunate outcome of discouraging parents from allowing their children to participate in UK Maths Trust events. A risk one runs when putting ones publicity out-to-tender.

In support of the excellent performances, the screenplay is an amusing menagerie, commendably well-written and poised for the most part but, on occasion, catastrophically dire. David and I agreed to play a drinking game – with popcorn replacing spirits, law-abiding citizens that we are – in which one would eat a handful at any grating cliché. Our gargantuan supplies lasted until the closing credits, but only because we more-than-once refrained from upturning the container and pouring its entire contents over our heads. The film espouses the maths/music identification, of course – “Music is just maths, really,” says one Bach-loving contestant whilst murdering the Prelude in C – and the burgeoning romance between Nathan and his Chinese counterpart, initially handled with a deft touch, passes into the ridiculous once Nathan starts googling, “A mathematical formula for love.”  The final speech is also disappointing, exploring the differences between the arithmetical and the anthropological notions of ‘value’; a bold and interesting theme, which regrettably is never really lifted above a base pun. There is also an infuriatingly simple-minded appeal to Keats’ famous equivalence – although, as maths has been grappling unsuccessfully with the relationship between Beauty and Truth for two-and-a-half millennia, perhaps we can forgive Matthews this one.

However, in and amongst these occasional patches of drivel are sensational touches of insight. The joy of possessing an exquisite pair of compasses; the hunched shoulders and tight neck arising from hours spent poring over Olympiad problems; the manner of speech that can find no middle-ground between timidity and over-assertiveness; the UK Maths Trust logo, so evocative for my generation of English mathematicians; the detail of the Australian IMO team uniform;  the unique atmosphere of the opening ceremony; the lines of vacant desks topped only with a bottle of water, an obscure foreign cereal-bar, and an ominous piece of A5 paper, turned face down; the colour-coded help cards. Sights and sounds I had long forgotten, conjoured before me once again.

To head off tangentially for a brief moment, we always wondered what manner of international crisis one could engender by, in a fit of hysteria, waving all of the help cards simultaneously – “Excuse me, I need to ask a question about going to the toilet, whilst getting more paper and more water, and I’m waving this other red card just to show how distressed I am about this whole question-toilet-paper-water situation!”. In fact this reminds me of a story told to me by my maths teacher, from his university days. In the maths cafeteria at Oxford University there was a rudimentary electronic tea-coffee-soup machine, which dispensed one of a few simple concoctions if one imputed the appropriate number from the attached list. In a flash of mathematical insight, Andrew and his friends realised that the machine was merely converting the imputed number into binary and then mixing together the ingredients which corresponded to 1s. Naturally, they typed in 255. Perhaps such anarchic inclinations are endemic in mathematicians.

To return to my theme, the resonances with another past life came from the film’s third Act, shot in Cambridge, evoking so many happy undergraduate memories. Most wonderfully, the denouement– a conversation between Nathan and his mother about the nature of love – takes place in ‘Hong-Kong Fushion’, the Chinese takeaway on St. John’s Street in which I ate many a glorious meal. Trinity College, my alma mater and the mathematical hub of the country, always refuse filming permission to any movie makers – the remarkable exception being The Man who knew Infinity, perhaps because it features a former Fellow – so the X+Y team make clever use of fixed camera stills of Trinity’s courts beefed up by internal scenes in the neighbouring St. John’s and live action sequences in its numerous courts and bridges. Both Trinity and St. John’s are extremely beautiful, but Trinity is the only college mentioned by name, and so the casual observer may well believe that all the shots are from different corners of the mighty Trinity. What a shame to have to deconstruct this pleasant illusion.

Of course there are various other inaccuracies in the film, created for expedience of both plot and narrative. Regrettably the UK team will never be able to compete on equal terms with the Chinese, not just because of the difference between our countries’ respective populations but also owing to the insane Chinese training regime (which seems to be damaging their mathematicians in the long-run – but that’s another story). The UK team selection process is also considerably lengthier than the film has room to represent. In reality there are a plethora of different camps for different ages and abilities, both at home and – for interesting historical reasons involving a chance meeting in a Turkish bath— in Hungary; a long chain of different rounds of internal exams, whittling down from the entire national school age population, to 2000, to 200, to 20, to 9 and then to the final IMO team of 6; and several other international competitions. The richness of this culture, built from the ground up by the efforts of the UK Maths Trust over the past 20 years, affects positively a large number of students, many of whom never reach the final stages of selection for the IMO team but nonetheless benefit in their future intellectual lives. Nathan’s choice of reading material is also rather too eclectic: he is seen devouring the fearsome IMO Compendium whilst barely out of short pants, whilst the night before the contest he settles down to ’10 Years of Mathematical Challenges’, described accurately by the incorrigible Dr. Geoff Smith as, “an excellent but elementary text.” Geoff is the friendlier real-life version of the team-leader character Richard, and records his brief-but-beautifully formed comments on X+Y here:

Finally, and for a Trinity-man like myself most unfortunately, Cambridge would never host the IMO – the competition is just too large, far larger than is represented in the film. Think Gandhi’s funeral procession. Combining contestants, team leaders and observers, not to mention organisers, invigilators and markers, hosting the IMO means being able to accommodate cheaply in excess of 1000 people, most of them highly volatile adolescents, within a small geographical area, not to mention providing an examination hall in which 600 contestants can sit two four-and-a-half hour exams in the utmost security. To do this in Cambridge would mean block-booking several colleges – a feat in itself – but even if this were accomplished the participants would be spread all over a town in which the only reliable way of transporting large quantities of people from one side to the other is by foot. The UK will next host the IMO in 2019, with the competition being housed somewhere more capacious.

Before briefly relating my own IMO experience, here are a few parting morsels about X+Y. For connoisseurs of the original documentary, there are a variety of choice lines lifted verbatim. There are also several cameos by real-life UKMT heroes. One of the exam invigilators may be spotted as none other than Dr. Joseph Myers, double gold-medal winner in his youth and trainer of the UK team. More bizarrely, one of the contestants featured in the documentary, Lee Zhao, featured as one of the contestants in the film, despite being 9 years older – still fresh faced even after completing his PhD, most impressive. There is a two-second cameo given to Andrew Carlotti, the most successful UK contestant in the history of the IMO. Currently a second-year undergraduate at Trinity, he fills the screen a few moments into the first IMO exam, masquerading as one of the usual mortals struggling over the questions. Most confusingly, the actor playing the Olympiad hopeful Isaac – Alex Lawther – also played the young Turing in The Imitation Game; I think this gave Isaac an unfair mathematical advantage in the team selection.

And my own IMO story? Well, like the fictional Nathan, I managed to scrape onto the team. I achieved the basic aim of any UK participant – to answer the two ‘easy’ questions, a feat that eludes half of the competitors overall each year. Unfortunately, to my eternal embarrassment, I failed to achieve a single other point on the other questions, thus falling one point short of the bronze-medal boundary. The academic part of the experience was horrible: I dispatched two of the six questions in little more than an hour, and then spent the remaining eight long hours floundering helplessly against my mathematical naivety. However, I did get to spend a crazy fortnight in Kazakhstan in the company of 600 of the most dynamic and interesting young people in the world. Such opportunities come only once in any lifetime. I wrote a report on my experiences there, which you can read at your leisure at

http://www.imo-register.org.uk/2010-report-aled.html

There is also another truth: the IMO doesn’t really matter. This is perhaps a message that, understandably, the film masks. The IMO is the crowning jewel in pre-university mathematics, but it is only a step on the road to becoming a fully-fledged mathematician and, in the grand scheme of things, a rather minor one. Many Fields medallists had outstanding IMO records in their younger days – a fact that makes me more-than-a-little jealous –but such success in the Olympiads is neither necessary nor sufficient for success in research. There are many other skills, much advanced learning, and many diverse styles of thinking required to become a ‘grown-up’ mathematician which  the Olympiad does not, could not, test. Such was the view espoused by the current IMO Patron Terry Tao when we discussed the topic over dinner, in one of the more surreal nights of my life.

So for me the IMO itself was a rather disappointing climax to the first stage of my mathematical journey. Yet, watching the film, I nonetheless felt a deep sense of wellbeing, and I realised that the main purpose of my mad sojourn through the Olympiad world hadn’t been to win medals, but to make friends. At dinner, after our popcorn had gone down, I raised a glass, “to mathematical friendship!” and marvelled at my luck at having been introduced to such wonderful people at an early age. And that is the message of the film, if it has a message at all: people are more important than maths. It is a safe message, to be sure, but it recognises that mathematics is a profoundly human activity with human challenges and human joys, an insight rarely understood by those living on the outside.

The Olympiad does leave the best mementos:  my friends, my memories, and my 823 page-long IMO Compendium, which looks down from my bookshelf as I write.

An alternative perspective from a former IMO contestant — with a bit more maths — can be found here: http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/mar/11/x-plus-y-insider-view-maths-tournaments

# Heegner’s solution to the ‘Class Number 1 problem’

I have just completed a short monograph on the so-called ‘Class Number 1 problem‘. It was written to fulfill the EPSRC ‘broadening requirement’, having attended the Part C Modular Forms course this term in Oxford,  and was therefore, by design, a little outside my comfort zone — I hope this disclaimer will temper the disdain of any serious algebraic number theorists who happen across this article. No expert, I set out to try to write the kind of exposition of this topic that I , as an interested mathematician with a slightly different specialism, would have liked to have read myself. Although there already exist some very thorough accounts of this topic in the literature — we make reference to a book by Cox, and essays by Booher, Green and Kezuka — I know of no shorter survey which nonetheless gives a detailed description of the entire argument and sketch proofs of most of the important results.

Introduction

Gauss found nine imaginary quadratic fields with class number 1, and conjectured that he had found them all. In 1952 Heegner published a purported proof, based heavily on the work of Weber from the third volume of his landmark, but fearsome, treatise Lehrbuch der Algebra. Heegner was unknown to the mathematical community at the time, and it was felt that his proof contained a serious gap. Stark and Baker independently published the first accepted proofs in 1966, but then Stark examined the argument of Heegner and discovered it to be very similar to his own. Indeed, he went on to show that the ‘gap’ in Heegner’s proof was virtually non-existent. Furthermore, he noticed that enough technical machinery could be avoided to have enabled Weber to prove this result some 60 years earlier.

A detailed historical overview of progress on the problem has been written by Goldfeld. It is worth noting that Heilbronn and Linfoot knew in the 1930s that there were at most 10 imaginary quadratic fields with class number 1.

The aim of this short essay is to outline Heegner’s argument, prove a few of the important constituent lemmas, and to relate some of the theory to that covered in the Part C Modular Forms Course. With this latter aim in mind, we shall focus mostly on the modular functions involved in the proof, rather than the input from algebraic number theory — indeed, two particularly technical propositions will be left entirely unproved. However, we will assume familiarity with basic results concerning the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field, and concerning non-maximal orders $\mathcal{O}\subset \mathcal{O}_K$. This theory is well covered in Chapter 7 of Cox’s ‘Primes of the form $x^2 + ny^2$‘. Regarding the input from modular forms, we will make heavy reference to Eisenstien series and the Ramanujan $\Delta$-function, and introduce other modular functions which are invariant under other congruence subgroups $\Gamma\leqslant SL_2(\mathbb{Z})$. There will also be an analogy to a lemma from the theory of Hecke operators.

Serre has an approach to the class number 1 problem which is much more geometric, constructing a particular modular curve and then counting special points on it, which has more of a flavour of the first half of the Part C Modular Forms course. However, the approach is extremely involved and this is only a short project — Booher  gives details, and also discusses the relationship to Heegner’s argument.

Note on references: We have relied heavily on the excellent essay of Booher, the paper by Stark, and the astonishing book ‘Primes of the form $x^2 + ny^2$ by David Cox, which despite its unassuming title provided a wealth of insight into all aspects of the argument. The master’s essay of Kezuka is comprehensive but is rarely more than a recitation of Cox. Green has an essay which covers much of the background regarding complex multiplication, although from a more high-brow viewpoint than we shall pursue here.

# A Lemma of Bateman-Katz

Here we present a formulation of a lemma from Bateman-Katz‘s work on the cap-set problem, which may be of independent interest in additive combinatorics at large. The proof is also novel — Ben Green knows of no other instances of this argument — and may yet find other applications. Indeed, Thomas Bloom recently used this lemma together with other bounds to give his improvement to Roth’s theorem.

[Our main result will be ‘Corollary 3’ below, which will follow from Lemma 1 and Theorem 2. ]

For any abelian group $G$ and finite $A\subseteq G$, we define the ‘2$m$-fold additive energy’  of $A$ by

$E_{2m}(A)=\left\lvert\{(a_{1},\cdots,a_{m},a_{1}^{\prime},\cdots,a_{m}^{\prime})\in A^{2m}:a_{1}+\cdots+a_{m}=a_{1}^{\prime}+\cdots a_{m}^{\prime}\}\right\rvert$

Any $(a_{1},\cdots,a_{m},a_{1}^{\prime},\cdots,a_{m}^{\prime})$ counted on the right-hand-side we call an ‘additive tuple’. The energy is one way of measuring the additive structure of $A$. The main theorems of this note link data concerning the energy to data concerning another notion of additive structure, namely (in the case where $G$ is a finite vector space) dimension.

Lemma 1

If $\lvert S\rvert = d$ and $\text{dim}(\text{span}(S))=d-k$, with $1\leqslant k\leqslant d$, then $E_{2m}(S)\leqslant 2^{2m+1}m^{4m+3}k^{2m}d^{m}$

Remarks on Lemma 1

1. There are about $m^{2m} d^{m}$ diagonal solutions counted in the additive energy, and — thinking of $m$ as much smaller than $d$ — the claim is exactly saying that (up to some tame factors) these dominate when $k$ is very small, i.e. when $S$ is almost completely linearly independent. This matches our intuition.
2. The claim is (worse than) trivial if $k$ is the same order as $d$.
3. We shall see that information about dimension meshes most naturally with equations in the elements of $S$, and not additive tuples. The count for the former is roughly equivalent to Bloom’s notion of ‘restricted energy’, and we will, like him, have to undergo a messy calculation to convert information about this restricted energy  into information about  $E_{2m} (S)$. The issue that $E_{2m} (S)$ cannot be calculated directly does genuinely seem to have been missed by B-K. Broadly speaking this means that their definition giving what it means for the large spectrum to be ‘additive smoothing’ is overly optimistic. I am currently trying to establish whether this has any major effect on their work; this will probably be the subject of a future blog post.
4. My $m$ dependence is extremely wasteful, and no doubt one can do much better. However, don’t be too scared of extra $m^m$ factors; in Bloom’s argument they roughly correspond to $\log\log N$ factors in the final bound. Also remember that there are $m^m$ factors in the trivial lower bound, so we cannot eschew them completely.
5. The reason we go to the trouble of proving these theorems for $E_{2m} (S)$ rather than the restricted energy is that  $E_{2m} (S)$ has a particularly pleasant expression in terms of the Fourier transform, which allows for many useful manipulations (in particular Holder’s inequality).

Viewing Lemma 1 in the contrapositive, it is saying that sets with larger-than-trivial energy have lower-than-trivial dimension. Theorem 2 is a more refined version, namely that sets with large energy have subsets with very low dimension.

Theorem 2

Let integers $d\geqslant 2m\geqslant\text{max}(2,2Ce)$ be arbitrary parameters and $C>0$ be independent of $d$.  Let V be finite vector space, and $A\subseteq V$ satisfy $E_{2m}(A)>\lvert A \rvert ^{2m}d^{-m}\left(2^{4m+3}m^{6m+4}C^{2m}+2m^{2m+1}\right)$. Then $\exists S\subseteq A$ with $\text{dim}(\text{span}(S))\leqslant d$ and $\lvert S\rvert \geqslant\dfrac{C}{d} \lvert A\rvert$

Remarks on Theorem 2

1. This theorem is good in the case where $d$ is a smallish power of $\lvert A\rvert$; B-K have $d\approx \lvert A\rvert ^{\frac{1}{3}}$. It gives some structural information even when the 2$m$-fold additive energy of $A$ is quite a long way below the maximum possible.
2. This is the first main ‘new’ result in Bateman-Katz’s paper, and is the main topic of this post.
3. $2m\geqslant 2Ce$ is just a minor technical condition that streamlines a particular estimation step. It could certainly be removed with a little more effort, and no effort at all if one didn’t care about how the constants in Theorem 2 depended on $m$.
4. One needs Lemma 1 as an ingredient in the proof of Theorem 2, applied to a suitably chosen random subset $S\subseteq A$.
5. For certain choices of the parameters we can use Theorem 2  to bootstrap an improved theorem regarding $E_{8}(A)$, say, which would be better than applying Theorem 2 directly with $m=4$.

Expanding on this last remark, we note that by Holder’s inequality we have

$E_{2m}(A)\geqslant \dfrac{E_{8}(A)^{\frac{m-1}{3}}}{\lvert A \rvert ^{\frac{m-4}{3}}}$

[This follows from the fact that $E_{2m}(A)=\sum\lvert \hat{A}(r)\rvert^{2m}$ and taking Holder in the form $\sum\lvert f\rvert ^{2k}\leqslant\left(\sum\lvert f \rvert^{2m}\right)^{\frac{k-1}{m-1}}\left(\sum\lvert f \rvert ^{2}\right)^{\frac{m-k}{m-1}}$ with $k=4$, followed by Parseval applied to the second bracket].

Therefore we can make the conclusion of Theorem 2 provided that, for some $m$,

$E_{8}(A)>\left(\lvert A\rvert ^{2m+\frac{m-4}{3}}\right)^{\frac{3}{m-1}}d^{\frac{-3m}{m-1}}\left(2^{4m+3}m^{6m+4}C^{2m}+2m^{2m+3}\right)^{\frac{3}{m-1}}$   (1)

Suppose we wanted to be able to take $d=\lvert A \rvert ^{\epsilon}$ for some fixed $\epsilon$. Applying Theorem 2 directly would only allow us to do so when $E_{8}(A)\gg\lvert A\rvert ^{8-4\epsilon}$ (so for example we would be forced to have $\epsilon>\frac{1}{4}$ to get a non-trivial result). However, equation (1) allows us to conclude whenever $E_{8}(A)\gg_{m}\lvert A\rvert ^{7+\frac{3}{m-1}-\frac{3\epsilon m}{m-1}}$. Taking $m$ large enough, depending on $\epsilon$, we win if $E_{8}(A)\gg_{\epsilon}\lvert A\rvert ^{7-2\epsilon}$, which is better for $\epsilon<\frac{1}{2}$. We state this as a corollary:

Corollary 3

Let $\epsilon\in (0,\frac{1}{2})$, and let V be finite vector space, and $A\subseteq V$. Then there exist a constant $K=K(\epsilon)$ such that if $E_{8}(A)\geqslant K(\epsilon)\lvert A\rvert ^{7-2\epsilon}$ then $\exists S\subseteq A$ with $\text{dim}(\text{span}(S))\leqslant \lvert A\rvert^{\epsilon}$ and $\lvert S\rvert\geqslant\lvert A\rvert^{1-\epsilon}$

Proof: Above discussion.

Proofs of Lemma 1 and Theorem 2 are below.