Category: Mathematics

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:

http://www.ukmt.org.uk/pdfs/Newsletter%2047.pdf

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.

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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 ‘2m-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 2m-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.

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