Abstract algebra nursery rhyme

In the spirit of hilariously advanced baby books like Chris Ferrie’s Quantum Physics for Babies, I have taken to incorporating absurdly sophisticated concepts into nursery rhymes.

To the tune of the ABC song (or, equivalently, Twinkle Twinkle Little Star):

The axioms of a group go 1, 2, 3
Identity, inverse, associativity!
The identity times any element g is g,
Inverse of g times g is identity,
Associativity says ab times c
is equal to a times bc.

The last resort of scoundrels

Samuel Johnson said it was “the last resort of scoundrels“; Emma Goldman, a menace to liberty. Leo Tolstoy said it “as a feeling is bad and harmful, and as a doctrine is stupid“. Patriotism, at least in its usual sense of love of one’s country over others, veneration of the virtue of its people over others, and adoration of its flag, is awful, irrational nonsense.

How on earth one can deduce moral values, or even a positive emotional response, from a geographic entity — indeed, such powerful emotions as to move men to war (yes, usually men) — has always eluded me.

It may be that there may be various administrative reasons to divide a geographical area (like the earth, or a continent) into official or legal sub-regions (like countries, or states).

More importantly, it may be that, for one born in a land oppressed by a colonist, an occupier, or other oppressor, the natural solidarity among those oppressed peoples in their legitimate resistance may be expressed in the language of patriotism.

And it may be that there can be good, even uniquely good, things about a nation’s culture, and that it is worth recalling them occasionally — though there will equally be bad, even uniquely bad aspects also. One must never forget that people everywhere are roughly equally good and equally bad.

It may also be that countries may have sporting teams, or the like, and it can be fun to barrack for them.

Beyond that, there is nothing positive to say about patriotism.

Even if a country is physically beautiful, others are too. Even if a country’s culture or people are wonderful, others are too. There are wonderful people and wonderful ideas everywhere, just as there are horrible people everywhere. Venerating only those nearby, to the exclusion of others, is insular, narcissistic, and leads naturally to racism, chauvinism, and xenophobia.

Even if the highly dubious conceit of orthodox patriotism is true for a country — that this nation is great and to be preferred over others, despite all the other ones believing the same — it does not follow that that one ought to venerate this nation: if one wants to venerate something, one should venerate good things and good people, whether here, there or anywhere.

(Incredibly, orthodox patriotism means that vast numbers people in every land can believe precisely this, despite those elsewhere thinking the same. They cannot all be right, but they can all be wrong — living “in a gross and hamful delusion“. It is the same with all religions claiming to be the one true religion, of course. It discloses something deep, and deeply worrisome, about the human condition, that vast numbers of people are capable of this conceit.)

What matters are universal moral values, equity, justice, freedom, and so on; not the country in which they are expressed. One’s specific birthplace or homeland or nation is irrelevant.

This is kindergarten level morals; except that the corresponding kindergarten situation, of a group of children each boasting they are the best, will be resolved by a game or by a distraction, rather than by oppression, detention archipelagos, or war.

Perhaps the worst aspect of patriotism is in the cultural realm. It creates mythologies, with deep and powerful emotions latent within its manufactured communities. These emotions, fueled also by resentment of outsiders, can be manipulated by regressive political forces to reinforce inequalities, persecute outsiders, and stoke wars.

These mythologies are created when a nation’s history is recounted as virtuous, dramatic and heroic. But it is the same with other nations; and if retelling the story of one nation excludes other peoples and nations (or worse, disparages or invokes hatred of them), then it leads in the direction of, at best, insularity and stagnation, and at worst, militarism, oppression and war.

Then there is Australia.

Here, the magnitude of the artifice required to tell the nation’s history as a virtuous story is itself heroic. The result is an increasingly viciously enforced cultural orthodoxy, together with a crushing cultural cringe.

An island continent, home to hundreds of Indigenous nations, until colonised by an imperial power to create an antipodean jail; the original inhabitants and rightful owners dispossessed by the accumulation of property and capital and microbes, by genocidal policy, and by over a century of smouldering frontier war; no galvanizing wars fought for independence, only complicity in the motherland’s imperial ambitions, and a standard role in humanity’s propensity for worldwide violence; with all the bravery, heroism, obedience, murder and atrocity that entails. The overall arc of post-settlement history must be twisted beyond recognition to confect an orthodox patriotic mythology.

There are plenty of heroic Australians, to be sure; just as there are plenty of villains, and everything in between. And there are plenty of legitimate sources of pride in that nation’s achievements, just as there are plenty of horrific sources of shame.

Nothing more and nothing less; special in some ways and not in others; which is precisely the negation of every orthodox patriotic myth.

Limitless as that space too narrow for its inspirations

On 22 February, 1877, James Joseph Sylvester gave an “Address on Commemoration day at Johns Hopkins University”.

Sylvester, the very excellent English mathematician, worked in areas of what we would today call algebra, number theory, and combinatorics. He is known for his algebraic work in invariant theory; he is known for his work in combinatorics, such as Sylvester’s Problem in discrete geometry; and for much else. He invented several terms which are commonplace in mathematics today — “matrix”, “graph” (in the sense of graph theory) and “discriminant”. He was also well known for his love of poetry, and indeed his poetic style. (He in fact published a book, The Laws of Verse, attempting to reduce “versification” to a set of axioms.)

I came across this address of Sylvester, not through mathematical investigations or in the references of a mathematical book, but rather in the footnotes of the book “Awakenings”, in which the late neurologist Oliver Sacks discusses, in affectionate and literary detail, the case histories of a number of survivors of the 1920s encephalitis lethargica (“sleeping sickness”) epidemic — an interesting and mysterious event in itself — as those patients are treated in the 1960s with the then-new drug L-DOPA and experience wondrous “awakenings”, often after decades of catatonia, although often followed by severe tribulations. (These awakenings were the subject of the 1990 Oscar-nominated movie of the same name.) These tribulations, in each patient, form an odyssey through the depths of human ontology, in which the effects of personality, character, physiology, environment, and social context are all present and deeply intertwined.

Sacks comes to the conclusion that a reductionist approach to medicine, focusing on the cellular and the chemical, is wholly deficient:

What we do see, first and last, is the utter inadequacy of mechanical medicine, the utter inadequacy of a mechanical world-view. These patients are living disproofs of mechanical thinking, as they are living exemplars of biological thinking. Expressed in their sickenss, their health, their reactions, is the living imagination of Nature itself, the imagination we must match in our picturing of Nature. They show us that Nature is everywhere real and alive and that our thinking about Nature must be real and alive. They remind us that we are over-developed in mechanical awareness; and that it is this, above all, that we need to regain, not only in medicine, but in all science.

Indeed, Sacks quotes from W H Auden’s “The Art of Healing”:

‘Healing,’
Papa would tell me,
‘is not a science,
but the intuitive art
of wooing Nature.

In an accompanying footnote, Sacks notes that mathematical thinking is real and alive, in just the same way. He quotes the aforementioned address of Sylvester.

Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as as the worlds which are forever crowding in and multiplying upon the astronomer’s gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness, the life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud and cell, and is forever ready to burst forth into new forms of vegetable and animal existence.

Sylvester is right, and if anything his argument is not forceful enough. Mathematics has always been limitless — and even more limitless than the seemingly (to Sylvester, at least) infinite possibilities of astronomy and biology — for, unlike the experimental or observational sciences, it requires no substrate in reality beyond the imagination of those who think it. Liberated from the necessity to study only this world, mathematics studies all the worlds it can imagine, which include our own but go far beyond our own one. (It is perhaps surprising, and even “unreasonable”, as Wigner argued, that we can count our own world as among those which are mathematical; but it is not surprising that its worlds transcend ours.)

The progress of science has displayed, in an absolute sense, how mathematics outstrips the limitlessness of other sciences.

However many may be the worlds of the astronomer — now teeming also with exoplanets and gravitational waves — they are still finite; the observable universe has a finite radius.

Sylvester’s panpsychism (everything has consciousness) is now out of fashion, but seems focused on biology — and we now know that biological life is constrained by genetics, and at the molecular level by DNA and related biochemistry. Mathematics knows no such constraint.

Taking panpsychism more generally, there is an argument — and a strong one, in my view — that understanding consciousness will eventually require a radical revision of our understanding of physics. But even then, I very much doubt any such radical revision would completely transcend mathematics — and I very much doubt that mathematics would not encompass infinitely more.

It is worth noting, though, that mathematics is, in a certain sense, reductionism par excellence. Even accepting what we know about incompleteness theorems and the like, mathematics, theoretically at least, can be reduced to sets of axioms and logical arguments, in the end consisting only of formal logic, modus ponens and the like. That is not how mathematicians do mathematics in practice, but that is the orthodox view on what mathematics formally is. Even the standard theorems that mathematics “knows no bounds” — the Godel incompleteness theorems, the Cantor diagonalisation argument, the set-theoretic paradoxes like Russell’s, for instance — can themselves be expressed, reductionistically, in this formal way.

All the infinite possibilities, the unboundedness, of mathematics, then, can be expressed in a very finite, very discrete, very reductionistic way. This is not surprising — even with finitely many letters one can construct an infinity of sentences, one can burst all brazen clasps, one can empty all veins and lodes, one can exhaust all soils, there is no end to the harvest, however dizzying and rarefied the altitude at which it is sown.

And as for definitions of permanent validity? At least in terms of the experience of learning, doing and discovering mathematics, I cannot go past Ada Lovelace’s definition of the “poetical science” as “the language of the unseen relations between things”.

There is much else of interest — and not just historical interest — in Sylvester’s address. Mathematics impedes public speaking; university study and research ought to avoid monetary reward and public recognition; students should avoid “disorder or levity”; all researchers should simultaneously engage in teaching; anecdotes of arithmetic in the French revolution; every science improves as it becomes more mathematical; and the taste for mathematics is much broader than one might think. So argues Sylvester, poet, mathematician; perhaps I will return to these arguments one day.

The Doors of Crime Perception

Crime is uniquely susceptible to the manipulation of perceptions.

It is common, it is bad, it is fascinating.

A wide spectrum of this common, bad, fascinating activity exists, and the fixation of fascinated attention on certain narrow portions of this spectrum serves numerous powerful political interests. Those numerous, already-aligned, authoritarian political interests — tabloid media, conservative politicians — are only too happy to indulge the public’s fascination. No similar political interest is usually served by attempting to understand other portions of the spectrum. Attempting to understand the spectrum as a whole, or the overall picture and causes of crime, might serve the purpose of building a better society, but that purpose is one which, all entrenched political powers agree, must remain unthinkable.

Which types of crime are they, on which power so fixates attention? Preferably those which are sensational, preferably involving violence and fear, preferably with perpetrators who are suitably villainous and “not like us”, where “we” means the “good folk” who are normalised within society. Powerless, marginalised groups form perfect villains: immigrants, ethnic minorities, racial minorities, Indigenous people, and in general, “others”.

In Australia at present, that means asylum seekers and refugees, it means African Australians, it means Aboriginal Australians.

Accordingly, the fixated attention of society on this narrow portion of crime — and its villains — blows it out of all proportion. Perceptions of crime in society can warp radically, tending towards fear and paranoia of the fixated type of crime, and the fixated vilains — and generalises to a fear of society at large.

The propaganda powers of media campaigns, their political protagonists, and their guerrilla online counterparts, are substantial. The far right delights in it.

A fearful populace is one that is easier to control. It is one which will more easily submit to existing oppression as justified or necessary, and accept further devolution towards a surveillance or police state. Fearful people will tend to look out only for themselves, diminishing the bonds of social solidarity, and furthering capitalist atomisation. And as the public holds a paranoid, distorted idea of reality, the desire to understand society, and in particular the root causes of crime, diminishes, or becomes unthinkable. Hysterical overreaction to the villains is the urgent goal, anything else is wasting time against this menace. The already marginalised will be oppressed further.

* * *

What is the situation in Victoria?

Crime statistics are freely available in Victoria.

What do they say?

(Let us put aside broader questions, such as whether existing laws are good laws, whether the criminal justice system is a good one, what better systems might exist, and so on.)

We can, for the moment, put aside subtle questions of methodology. (Do people report more crimes now, especially domestic violence? Should we refer to the number of criminal incidents, recorded offences, or offenders?) Because in any case it the statistics tell a fairly clear story.

To a first approximation, in Victoria, crime rates have decreased since 2016. They were roughly level from 2009 to 2015, at a rate of just under 6,000 incidents per 100,000 population, with a jump in 2016 to over 6,600. The rate has since decreased, and the current crime rate is similar to the rate of 2009-15. This crime rate is roughly similar to other states in Australia.

Some categories of crime, however, have not decreased from 2016-18. Assaults have remained steady at around 610 incidents per 100,000 population, and sexual offences have increased from about 110 to 132 incidents per 100,000 population. On the other hand, theft and burglary have decreased dramatically (from about 2,500 to 2,100, and from about 840 to 620 incidents per 100,000 population, respectively).
(More detail can be found from the Age here or in the statistics themselves.)

These numbers are too high. They mean thousands of sexual offences, tens of thousands of assaults and burglaries, and hundreds of thousands of thefts, happen each year, in Victoria. Each such crime is potentially a source of outrage.

Society ought to work so that these numbers, in the long run, tend to zero. It is not at all clear that more draconian laws or policing will help that goal. It requires addressing the root causes, which include, among others, poverty, misogyny, racism, authoritarianism, capitalism, and a culture which glorifies greed and violence.

But nonetheless, the point about perception remains. If one felt that, despite the continual rate of ongoing crime, that Victoria was a generally safe place to live in 2015, and one is consistent, then (putting aside local variations) one must feel the same at the beginning of 2019.

Indeed, Melbourne ranked in the top 10 safest cities in the world, in a 2017 Economist study.

If one feels that “African gangs” are a menace to society, as right-wing politicians and tabloid media continue to claim, despite the protestations even of the police to the contrary, then one is living in an alternate reality — a reality that at least has provided some social media entertainment, but whose racism is profoundly damaging to African communities in Melbourne.

The algebra and geometry of contact categories, Melbourne July 2018

On Monday July 23 2018 I gave a talk in the Geometry and Topology seminar at the University of Melbourne.

The slides from the talk are available here.

Title:

The algebra and geometry of contact categories

Abstract:

Contact categories, introduced by Ko Honda, are a type of cobordism category related to 3-dimensional contact geometry. Geometrically, they encode contact structures in an elementary combinatorial way. Algebraically, they are related to triangulated categories, A-infinity algebras, Floer homology, and other wholesome fun. In this talk I’ll tell you something about them and report on some recent developments. No knowledge of contact geometry or topology will be assumed.

melbourne_talk
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The Brain makes Contact with Contact Geometry

It’s always nice, intellectually, when two apparently unrelated areas collide.

I had an experience of this sort recently with an area of mathematics — one very familiar to me — and an ostensibly completely distinct area of science.

On the one hand, contact geometry — a field of pure mathematics, pure geometry.

And on the other hand, the brain and its functioning. More particularly, the visual cortex, and how it processes incoming signals from the eyes.

Now, contact geometry has lots of applications: arguably it goes back to Huygens’ work on optics. It is closely related to thermodynamics. It is the odd-dimensional sibling of symplectic geometry, which is related to classical mechanics and almost every part of physics.

But applications to neurophysiology? Now that’s new.

Well, it’s only new to me. It’s been in the scientific literature for some time. It goes back at least to a paper from 1989:

And the discussion below is largely based on this article:

What’s the connection?

Contact geometry is the study of contact structures. And a contact structure on a 3-dimensional space \(M\) consists of a plane at each point satisfying some conditions. That is, at each point in the space, we have a plane sitting there. But not just any plane at each point. The planes have to vary smoothly from point to point — having such smoothly varying planes forms a (smooth) plane field. But moreover, the plane field, which we can call \(\xi\), is required to be non-integrable.

There are various ways to explain non-integrability. To “integrate” a 2-plane field is to find a smooth surface \(S\) in space so that, at every point of \(S\), the tangent plane to \(S\) is given by the plane of \(\xi\) there. At every point \(p\) of the 2-dimensional surface \(S\), the tangent plane is a 2-dimensional plane, which we write as \(T_p S\). If we write \(\xi_p\) for the plane of \(\xi\) at the point \(p\), then the integrability condition can be written as \(\xi_p = T_p S\).

Well that’s what integrability means (roughly) — \(\xi\) is integrable if you can always find a surface tangent to \(\xi\) in this way.

But a contact structure is just the opposite: you can never find a surface tangent to it in this way! The planes of the plane field \(\xi\) somehow twist and turn so much that you can’t every find a surface tangent to it. You can always find a surface tangent to \(\xi\) at a single point, and you might even be able to find a surface which is tangent to \(\xi\) at some of its points, (perhaps even along a curve on \(S\)), but you’ll never be able to find a surface which is tangent to \(\xi\) at all its points.

(If you’re familiar with differential forms, then the plane field \(\xi\) can be described (locally, at least) as the kernel of a 1-form, \(\xi = \ker (\alpha)\), and then the non-integrability condition is that \(\alpha \wedge d\alpha\) is a volume form. If you’re not familiar with differential forms, don’t worry.)

Contact structures can be hard to visualise. Here is a picture of one contact structure on 3-dimensional space:

A contact structure on 3-dimensional space. Public Domain, wikipedia

You’ll note that, if you consider going from left to right in this picture in a straight line, you can actually stay tangent to the contact planes. A curve like this is called a Legendrian curve. Let’s call the curve/line \(C\). But the planes twist around \(C\) as you travel along \(C\). This is a characteristic property of contact structures (and in fact, with a few extra technicalities, can be made into an equivalent characterisation).

Another example of a contact structure is a projectivised tangent bundle. Let’s say what this means. (Actually we’ll only consider one such contact structure: on the projectivised tangent bundle of a plane.)

Consider a 2-dimensional plane; let’s call it \(P\). Let’s even be concrete and call it the \(xy\)-plane, complete with coordinates. So all the points on \(P\) can be written as \((x,y)\).

Now, lay \(P\) flat on the ground, in 3-dimensional space. (More precisely, embed it into \(\mathbb{R}^3\).) We would usually denote points in 3-dimensional space by \((x,y,z)\), but I want to suggestively call the third coordinate \(\theta\), because it will denote an angle. In any case, the points of \(P\) now lie horizontally along \(\theta = 0\); so they lie at the points \((x,y,0)\) in 3-dimensional space.

Now in 3-dimensional space, through every point of \(P\) there is a vertical line. For instance, through the point \((1,2,0)\) of \(P\) is a line, and the points on this line are all the points of the form \((1,2,\theta)\).

And now the “projective” part of the situation comes in. Pick a point on the plane \(P\): let’s say \((1,2,0)\) again. Now consider lines on \(P\) through this point. There are many such lines; in fact, infinitely many. But we can specify a line by specifying its direction. And that direction can be specified by an angle \(\theta\). We could have various conventions to measure the angle \(\theta\), but let’s do it in the standard way: \(\theta\) is the angle (measured anticlockwise) from the positive \(x\)-direction, round to the line.

Now at each point \(p = (x,y, \theta)\) in 3-dimensional space, we’ll define a plane \(\xi_p\) as follows. The plane \(\xi_p\) contains the vertical line (i.e. in the \(\theta\) direction) through \(p\); and it also contains a horizontal line through \(p\) in the direction given by the angle \(\theta\). The result is as shown below.

Image by Patrick Massot.

Starting from \(p\) (and the plane there), if you move vertically upward you get to other points of the form \(p’ = (x,y,\theta’)\), with the same \(x,y\) coordinates but different \(\theta\) coordinates. The plane at \(p’\) still contains a vertical line, but the horizontal line has rotated from angle \(\theta\) to angle \(\theta’\). Thus, as you move upwards along a vertical curve, the planes spin around the vertical curve — just as shown in the animation.

It’s a contact structure. Indeed, you can even, if you want, identify the point \((x,y,\theta)\) with the line through \((x,y)\) in the plane \(P\) with direction given by \(\theta\). In this way, the points in 3-dimensional space correspond to the lines in the plane through various points, and this is the thing referred to as the “projectivised tangent bundle”. (Strictly speaking though, a line at angle \(\theta\) and a line at angle \(\theta + \pi\) point in the same direction, so we should identify points \((x,y,\theta) \sim (x,y,\theta+\pi)\).)

What does this have to do with the brain?

Well I’m no neurophysiologist, but the claim is that the neurons in the visual cortex can be regarded functionally as exactly this kind of contact structure. This is not to say that the neurons are planes, or spin around quite like the picture above. But it is to say that neurons in some ways, functionally, behave like this contact structure.

When you look at an image, the photoreceptors in your eye send signals into your brain. These signals are processed, at a low level, in your visual cortex. They are then processed at a higher level, extracting features, objects and eventually reaching the level of consciousness as the unified visual field which is part of ordinary human experience. However, here we are only interested in the lower-level processing, which extracts basic information from the image projected on the retina. This low-level processing extracts features like which areas of the visual field are light and dark, the shapes of light and dark areas, and importantly for us here, the orientation of any lines or curves that we see.

The particular area of interest in the visual cortex seems to be an area called “V1”. This area of the brain contains many structures. It contains several “horizontal” layers 1-6, each divided into sublayers; the most important is apparently the sublayer 4C. We’ll call this the “cortical layer”, as it’s the one important for our purposes.

Now it turns out that different points on this cortical layer relate to different points on the retina. Each point in your visual field corresponds gets projected to a different point on your retina, which (roughly speaking) connects to a different point in the cortical layer. The map from the retina to the cortical layer is called a retinotopy. In fact, beautifully, this map from the retina (which is a surface at the back of your eye) to the cortical layer (Which is a surface in your brain) is a map which appears to preserve angles (but not lengths). In other words, the retinotopy is a conformal map.

Even better, the cells of the cortex are organised into structures called columns and hypercolumns. Along each hypercolumn, the cells detect curves which point in the same orientation. So there are not only cells which are specialised to detect images arriving at particular points on your retina; there are also cells which are specialised to detect a curve at a particular in a particular orientation.

Functionally, then, the visual cortex behaves like a contact structure. The neurons aren’t arranged in a contact structure, but they behave like one. And this means that various processes in low-level visual processing can be understood in terms of contact geometry.

In particular, the “association field” can be understood in terms of contact geometry, as perhaps also can certain hallucinations — including those seen under the influence of psychedelics like LSD.

Well, it’s definitely the most psychedelic application of contact geometry I’ve seen.

Some further references:

“The beauty of mathematics shows itself to patient followers” — The work of Maryam Mirzakhani

(this article is jointly written with Norman Do)

I don’t think that everyone should become a mathematician, but I do believe that many students don’t give mathematics a real chance. I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it. I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers.

— Maryam Mirzakhani, 2008

The recent passing of Maryam Mirzakhani came as a shock to many of us in the world of mathematics. Not only was she in the prime of her life, she had also been intensely active in her work, posting research articles online on the arXiv right up until November 2016.

Although we did not know Maryam Mirzakhani personally, we were both fortunate to have met her. Like many others, we were impressed by her friendliness and enthusiasm. One of us (Dan) met her as a prospective PhD student in 2004, just as she was finishing hers. Her PhD advisor Curtis McMullen was very keen to explain that his student had been doing great work. Both Mirzakhani and McMullen’s discussions of mathematics remained incomprehensible to this young student, but he was impressed by their excitement about her work. And for the other author (Norm), Mirzakhani’s work formed the basis of his PhD thesis.

We have both been inspired by her mathematics and her example, and heartened to see her work gain recognition. Mirzakhani was a deserving recipient of the Fields Medal in 2014. As the first woman to do so, she is a true trailblazer.

Both the authors were influenced by Mirzakhani’s work, or rather, that small portion of it which we understand. Even that small portion, sitting as it does on the cutting edge of research mathematics, towers in its abstraction — but it is not so far removed from everyday experience that we think it impossible to explain some small part of it for a general mathematical audience.

So bear with us as we attempt to share something about Mirzakhani’s work. As Einstein said, “Nature hides her secret because of her essential loftiness, but not by means of ruse.”

From the art of Escher to conformal geometry

A great deal of Mirzakhani’s work has illuminated our understanding of so-called moduli spaces, which are “spaces of shapes” important in mathematics and physics.

But first things first. Consider the picture below. It’s a 1956 lithograph by the Dutch artist M. C. Escher entitled Print Gallery.

All M.C. Escher works © 2017 The M.C. Escher Company – the Netherlands. All rights reserved. Used by permission. www.mcescher.com

This strangely distorted image contains several surprising features. It depicts a man looking at a picture, and in that picture is a city, and in that city is a gallery, and in that gallery is… a man, the very same man we started with!

The picture is delightfully self-referential. But the way the picture is distorted is particularly interesting. (For an extended discussion, see this article of B. de Smit and H. W. Lenstra Jr.) If we compare to a “straight” version, we see that all of the angles in Escher’s drawing are accurate! Although lengths have been distorted, angles have not.

Source: Escher Droste

Mathematicians have long been interested in this kind of geometry, in which angles are important, but lengths are not. It’s known as conformal geometry and has all sorts of applications throughout mathematics and physics.

Source: Escher Droste

Now let’s consider something simpler than Escher’s artwork. Let’s take a circular disc \(D = \{(x,y) \in \mathbb{R}^2 \colon x^2 + y^2 \leq 1 \}\) and ask about its conformal symmetries: how can the points of the disc move around, so that all angles between curves remain the same? More precisely, we ask for bijections \(f \colon D \rightarrow D\) that preserve angles.

Perhaps surprisingly, there are quite a few ways to do this, one of which is demonstrated below.

There is a conformal symmetry that takes the curves of constant radius and angle shown in the disk on the left to the curves shown in the disk on the right.

However, while there are many such transformations of the disc, there are not really that many. In fact, there are interestingly many.

If we consider the bijections \(D \rightarrow D\) which preserve distances as well as angles — also known as isometries — then the only such maps are the rotations. (We ignore reflections, since they reverse orientation and so negate angles). There is a 1-parameter family of rotations of \(D\).

On the other hand, if we simply consider smooth bijections \(D \rightarrow D\), there are many, many such maps. This set of maps is, in a suitable sense, infinite-dimensional.

As it turns out, the set of conformal symmetries \(f \colon D \rightarrow D\) is three-dimensional. Conformal geometry sits somewhere between the isometries of Euclidean geometry, and the smooth maps of topology: isometries are too rigid, smooth bijections are too flexible, but conformal symmetries are just right. Conformal geometry strikes an interesting balance between rigidity and flexibility.

One way to understand the degree of flexibility in conformal maps is as follows. Take any three distinct points \(p_1, p_2, p_3\) on the boundary of the disc \(\partial D = \{(x,y) \colon x^2 + y^2 = 1\}\). Take another three distinct points \(q_1, q_2, q_3 \in \partial D\). Then there is a unique conformal transformation \(f \colon D \rightarrow D\) such that \(f(p_i) = q_i\) for \(i=1,2,3\). This property is known as triple transitivity.

Triple transitivity means that conformal transformations are specified by three points. Being specified by three parameters, the set of conformal symmetries of \(D\) is indeed a three-dimensional space.

Tessellations of the hyperbolic plane by triangles

Hyperbolic geometry and the shape of space

It turns out that conformal symmetries of \(D\) preserve circles and lines: the image of any circle or line on \(D\) under a conformal symmetry is again a circle or line.

In fact, the situation is even better than that. One can put a metric on the disc — defining a new notion of distance, different from the standard Euclidean metric — so that all conformal transformations are isometries. This metric is known as a hyperbolic metric; it is a scalar multiple of the Euclidean metric, but the scalar depends on \(r\), the (Euclidean!) distance from the origin.

\(\displaystyle \text{Hyperbolic distance} = \frac{2}{1-r^2} \; \text{Euclidean distance}.\)

With this metric, the disc is known as the Poincaré disc model of the hyperbolic plane, and in both pictures above, all the triangles are congruent. The “tiny” triangles are the same size as the “big” ones; as \(r \rightarrow 1\), indeed \(\frac{2}{1-r^2} \rightarrow \infty\), and in fact \(\partial D\) is infinitely far away from the points inside \(D\)!

Between any two points on \(\partial D\), there is a unique circle or line perpendicular to \(\partial D\). These are in fact the “straight lines” or geodesics of hyperbolic geometry: with the hyperbolic metric, they are shortest distance curves between points.

By drawing such geodesics joining three points \(p_1, p_2, p_3 \in \partial D\), we have an ideal triangle. (Ideal just means that the vertices lie on \(\partial D\).) The triple transitivity of conformal symmetries means that if you take any ideal triangle, and specify where those vertices are to go on \(\partial D\), then the rest of the triangle, and in fact the whole disc, comes along for the ride — and in fact this transformation preserves hyperbolic distance.

Thus, from the point of view of conformal geometry, all triangles of this type have the same shape: all triangles with vertices on \(\partial D\), and sides given by lines or circles perpendicular to \(\partial D\), are conformally equivalent. From the point of view of hyperbolic geometry, all ideal triangles are congruent. But from either point of view, we are looking at the same thing: the conformal symmetries of the disc are the same thing as the isometries of the hyperbolic plane.

While ideal hyperbolic triangles are all congruent, the same could not be said of quadrilaterals. There are many different shapes of quadrilaterals that cannot be related to each other by conformal transformations. If we consider all possible shapes of quadrilaterals, then they form a space called the moduli space of quadrilaterals. Roughly and in short, a moduli space is a space of shapes.

Mathematicians are interested in moduli spaces because they describe all the possible shapes something can have. The moduli space viewpoint is not only crucial in understanding the mathematical nature of particular shapes, but also features in physics, such as when string theorists want to understand all the possible ways that a string can evolve in time.

In any case, the notion of moduli spaces extends far beyond triangles and quadrilaterals. We can instead consider much more complicated surfaces, such as the surface of a donut or a pretzel. The moduli space of pretzels is then the space of shapes of all pretzels, in the worlds of conformal or hyperbolic geometry.

A donut has one hole and a pretzel has three: we say the surface of the donut, or torus, has genus 1, and the surface of the pretzel has genus 3. There is a moduli space of tori and a moduli space of pretzels.

Clearly discs are one type of surface, donuts are another type, and pretzels are another type again. They are different in a topological sense — strictly speaking, they are not homeomorphic. (That is, there is no bijection \(f \colon \text{Donut} \rightarrow \text{Pretzel}\) with \(f\) and \(f^{-1}\) both continuous.) A classical theorem of topology says that a surface is specified topologically by its genus \(g\) and its number of boundary components \(n\). The case \((g,n) = (0,1)\) is the disc; \((1,0)\) is the torus; and \((3,0)\) is the pretzel.

A moduli space consists of surfaces of a given topology, up to conformal symmetry. Slightly more precisely, consider the set of all surfaces of genus \(g\) with \(n\) boundary components. Consider two such surfaces \(S\) and \(S’\) to be equivalent if there is a conformal bijection \(f \colon S \rightarrow S’\). The set of all the equivalence classes of such surfaces is the moduli space \(\mathcal{M}_{g,n}\).

Thus, the moduli space of pretzels \(\mathcal{M}_{3,0}\) is the set of all pretzels, but we consider two pretzels equivalent if they are related by a conformal bijection.

As it turns out, the moduli space of pretzels is 12-dimensional. In other words, there is a 12-dimensional space of shapes of pretzels. In general, the moduli space \(\mathcal{M}_{g,n}\) has dimension \(6g-6+2n\). This was essentially discovered by Bernhard Riemann in the 19th century.

As mathematicians continued to explore these moduli spaces, they discovered that they in turn have their own natural geometry, given by the so-called Weil–Petersson metric.
This geometry — the shape of moduli space — is the shape of a space of shapes!

Mirzakhani investigated the geometry of moduli spaces, and made a raft of discoveries. The moduli spaces \(\mathcal{M}_{g,n}\) of different types of surfaces — i.e. for different \(g\) and \(n\) — are all related to each other in an intricate way. Introducing brilliant new methods to study moduli spaces, she was able to prove several results about not only the geometry of moduli spaces, but also elementary questions about curves and surfaces.

One such theorem concerns simple closed geodesics on a hyperbolic surface. A hyperbolic surface is a surface which at every point looks locally like (i.e. is locally isometric to) the Poincaré disc. A geodesic, or “hyperbolic straight line” on the surface, is closed if it goes around in a loop (i.e. begins and ends at the same point, without a corner), and simple if it has no self-intersections. So a simple closed geodesic is a straight non-intersecting loop.

Now, there are uncountably many simple closed curves on a surface, but if we “straighten them” into geodesics, the number of simple closed geodesics is countable. And if we count all the simple closed geodesics with length up to some value \(L\), the number of simple closed geodesics is finite and we can find it.

By the works of Delsarte, Huber and Selberg that started in the 1940s, we know that on a hyperbolic surface, the number of closed geodesics of length at most \(L\) is approximately \(e^L/L\).

\(\displaystyle \# \{ \text{closed geodesics of length at most L} \} \sim \frac{e^L}{L}\)

(Strictly speaking, we mean oriented, primitive closed geodesics.) This theorem has a fascinating analogy with number theory, where it is known that the number of primes that are at most \(e^L\) is also approximately \(e^L/L\). Indeed, the theorem above is often known as the prime number theorem for hyperbolic surfaces.

Note, however, that this “classical” theorem says nothing about simple closed geodesics. Does the number of simple closed geodesics of length at most \(L\) also grow like \(e^L/L\)? Mirzakhani answered this question with a definite no: in fact, she showed that on a closed hyperbolic surface of genus \(g\),

\(\displaystyle \# \{ \text{simple closed geodesics of length at most L} \} \sim L^{6g-6}.\)

(Actually, this was essentially known previously, depending on what \(\sim\) means here…) That is, while the closed geodesics grow exponentially with length, simple closed geodesics grow polynomially, and the degree of the polynomial depends on the genus.

This theorem was obtained by a deep understanding of moduli spaces and their geometry. Mirzakhani even went so far as to calculate the volume of moduli spaces. Since \(\mathcal{M}_{g,n}\) has dimension \(6g-6+2n\), this amounts to calculating a \((6g-6+2n)\)-dimensional volume. She showed that these volumes possess a rich structure. For instance, the volume of the moduli space of pretzels is precisely

\(\displaystyle \text{vol} (\mathcal{M}_{3,0}) = \frac{176557}{1209600} \, \pi^{12}.\)

Actually, this particular volume was known prior to Mirzakhani. But Mirzakhani found a way to calculate the volumes of all \(\mathcal{M}_{g,n}\) by an intricate recursive method. Moreover, she showed that if you fix the lengths of the boundary components to be \(L_1, L_2, \ldots, L_n\), then the volume of \(\mathcal{M}_{g,n}\) is a polynomial in \(L_1, L_2, \ldots, L_n\) of degree \(6g-6+2n\).

Mirzakhani showed us how to understand all this, and a whole lot more.

Magic wands and billiards

One theorem in particular, proved by Mirzakhani along with collaborators Alex Eskin and Amir Mohammadi (see here and here) has been described as a “magic wand” which can be used to attack a vast range of problems.

The magic wand theorem describes a surprisingly simple and rigid structure underlying certain group orbits in certain moduli spaces. A rigorous statement would take us too far afield, but it does have some more down-to-earth applications. For instance, this work has been used to significantly advance our understanding of mathematical billiards.

In mathematical billiards, just as in real billiards, we have a table, and we consider a ball rolling on the table, bouncing off the walls as it goes. (No jumps or spin allowed!) However, unlike standard billiards, we don’t just consider rectangular tables — we consider tables of whatever shape we like, limited only by our imagination.

The goal of mathematical billiards isn’t so much to get the ball into the pocket, but to understand the dynamics of where a ball can go, and how. For instance, if the ball starts in a particular position, can you hit it to any other position?

Now, for most billiard tables you can think of, you can probably find a way to get the ball from one point to any other point. (At least in theory, if you have better skills than us!) But in general it is a much more difficult problem than it seems.

In 1958, New Scientist published a Christmas puzzle column written by two people. One was an esteemed psychiatrist and geneticist; the other was to become an esteemed mathematician and mathematical physicist. They were the father–son duo of Lionel and Roger Penrose. They posed the following puzzle: can you design a billiard table on which you cannot hit the ball from every point to every other point?

Since a billiard ball bounces off a wall in the same way that a light beam reflects off a mirror, one can equivalently consider the following “illumination problem”: can you design a room with mirrored walls, in which a candle can be placed without illuminating the entire room?

The diagram above depicts a solution that involves a combination of straight sides and arcs of ellipses. But let’s now put ellipses aside, and only consider billiard tables that have straight sides. Can you design a room with straight mirrored walls, in which a candle can be placed without illuminating the entire room?

This problem was only resolved in 1995, when George Tokarsky successfully designed one with 26 sides.

Still, in Tokarsky’s construction you can illuminate almost the entire room. As it turns out, a candle placed at the red point on the left will illuminate every point apart from the red point on the right.

So we might ask: can you design a room with straight mirrored walls, in which a candle can be placed that leaves a whole region in the dark?

Mirzakhani’s work on moduli spaces has shed light, so to speak, on this problem. As it turns out, moduli spaces are deeply connected to mathematical billiards — because moduli spaces are spaces of shapes, and they can encompass the possible shapes of billiard tables.

The mathematicians Samuel Lelièvre, Thierry Monteil and Barak Weiss were able to apply the magic wand theorem of Eskin–Mirzakhani–Mohammadi to the illumination problem. They showed that, as long as the walls of the room meet at fractional numbers of degrees, then Tokarsky’s construction is as bad as it gets: a candle in the room will illuminate all but a finite number of points in the room. In other words, “almost everything is illuminated”.

And thanks to Mirzakhani’s work, a great deal more mathematics is now illuminated.

The disempowerment of positive thinking

In general I’m quite skeptical of the “positive psychology” movement, as it encourages the individualisation of some problems that are really social. For instance, one might be out of a job, improverished, be depressed, anxious, etc — essentially as a consequence of living in a capitalist/atomised/otherwise awful society, and this stuff often gives the message that the problem is not society, but yourself. You just need to be happier and practice positive thinking. That idea squashes all possible motivation to actually change the system, makes people feel terrible about themselves, but better if they do think in some positive ways that are predominantly individualistic (or based on interpersonal relationships).

In other hands, positive psychology is more immediately harmful — cancer sufferers, for instance, being told that if you don’t “think positive” you’ll get worse. Well of course no medical professional should actually say this (although the placebo effect etc is real), but I’ve certainly heard of it among those who are sick, survivors, etc. Barbara Ehrenreich has written some interesting stuff about this (see e.g. here).

Having said that, “positive psychology” often focuses on having better interpersonal relationships, building better connections with others. That is all to the good — although, to the extent it does so, it is a fairly minor palliative to the atomisation produced by capitalism, and the atomising technologies it has spawned (from television to social media, etc).

It is true of course that we all need to figure out the best way to live a happy life — just for a properly understood notion of “happy”. I tend to agree with Bertrand Russell on this — I recall him saying somewhere that the first step to happiness is, first of all, to realise that the world is terrible, terrible… terrible. Only having fully internalised that (everything from human mortality, to cosmic insignificance, to the ways capitalism and sexism and elitism/hierarchy destroy most of the things in life worth living for), can one really appreciate the tenderness of human kindness, the precarity of human life, the preciousness of human love and solidarity — what human life is and what it could be, and live a fulfilling, deeply satisfying, happy life.

I don’t think many people can fully internalise all that and come out unscathed, but I do think that any claim to happiness not grounded in these facts of the world is a superficial one. A wise and happy person is also a battered one. It’s a hardline view, but the world is still a hard place — though it should not be.

Only by the struggles of those unhappy with the world, can we make it a world which systematically produces happiness.