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.

A-infinity algebras, strand algebras, and contact categories

(83 pages) – on the arXiv.

Abstract: In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure. In this paper we investigate such A-infinity structures in detail. We give explicit constructions of such A-infinity structures, and establish some of their properties, including conditions for the nonvanishing of A-infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.

A-infinity_algebras_strand_algebras_contact_categories_v1

Is the traditional mathematics blackboard lecture dead?

The Australian Mathematical Society Annual Meeting this year included a public debate on the topic “Is the traditional mathematics blackboard lecture dead?” I was on the affirmative team, arguing that the traditional blackboard lecture is in fact dead. Below is some approximation to my remarks. Being a case for one side of the argument, and in a context of an event as much for entertainment as for serious discussion, the below is a small part of my views on the matter.

We lost the debate rather convincingly — and arguing against blackboard lectures to mathematicians is a rather unpopular cause! — but nonetheless it was an entertaining event raising some important issues for tertiary mathematics educators. I thank my colleagues Birgit Loch and Marty Ross for their valiant prosecution of this unpopular cause; congratulations to Adrianne Jenner, Heather Lonsdale and John Roberts on their victory; and to Adam Spencer for moderating.

* * *

We’re here to debate whether the traditional blackboard mathematics lecture is dead.

We are asking whether something despised, deserted, and largely replaced, is dead. Is a corpse dead? Yes, a corpse is dead. Nonetheless, we can still debate the question.

Now I would dearly love this corpse to be resurrected — at least, in its better forms. But resurrection is well beyond my pay grade, and I look forward to my colleagues on the other side bringing forth its second coming.

In our affirmative case, I‘ll be laying out the issues: what the traditional blackboard lecture is, and how it’s dead or dying.

Birgit will provide the proof around attendance, technology, and how traditional blackboard lectures have been replaced.

And Marty will be summing up and sledging the opposition.

* * *

So what does a traditional mathematical blackboard lecture look like?

The students shuffle in, notebooks open. Some of them have understood the material. Most have not — we all know this, because we mark their exams. The lecturer mumbles incomprehensibly into the blackboard. The students are bored; probably the lecturer too. The lecturer copies their notes onto the blackboard. The students copy the notes from the blackboard into their notebooks.

In this way, traditional mathematics blackboard lectures are a transmission channel for maths notes. This gives maths notes a very good reproductive advantage. The same cannot be said for the lecturer.

* * *

Now, not all traditional backboard lectures are that bad, although as we all know, often they really are. A knowledgeable lecturer joyfully expounding the subject they know and love, in all its depth and beauty, can really shine.

But this is not what most people experience.

And how could it be – when most peoples’ experience is of first year units, where students enrol with imagination beaten out of them by a thousand algebra exercises, ever-decreasing background knowledge, ever-increasing financial stresses, ever-decreasing attention spans, and we have to complete their secondary education, fix their miseducation, and teach the actual intended content, if they ever turn up to lectures, in a tightly constrained timeframe, with limited resources, cramming in those fundamentals we’d be embarrassed for students not to know in the no time left before the end of semester?

The tragedy is that for most students, this is our only chance to get them into mathematics, and we lose them. They don’t learn much – even what mathematics is. And those are the ones who pass.

These circumstances make a mockery of our goals as mathematics educators. How can we nurture that free creativity, that tightly constrained logic, that we recognise as the glory of mathematical thought – that joy at playing with new ideas and problems? No, in these circumstances, the traditional blackboard lecture reinforces all the worst of secondary schooling: regimented curriculum; passivity; boredom. But now at scale.

Tertiary education is a mass social institution. University maths departments teach thousands of students each semester. The bulk of these students have only ever taken lower level subjects where they’ve gotten a taste of the traditional blackboard lecture and then run a thousand miles away.

And it is this, the mass social phenomenon that we mean by the traditional blackboard lecture. It is this tradition which is dying. It is already dead. And there is no reason to mourn its demise.

* * *

Let me turn to traditional blackboard lectures as they currently exist, in my own experience, at Monash.

Well, they don’t — at the first year level. They are literally dead, as a matter of cruel hard fact. Large first year classes are taught in lecture theatres with no blackboards at all… or whiteboards, for that matter.

However… I don’t know if I should say this, but some blackboards actually remain at Monash.
There is a small holdout, a rebel building, camouflaged in 1970s mission brown. It is, so I’m told, slated for demolition. No doubt soon it will come into view of the death star. The building next door is already gone.

But in the meantime, traditional blackboard lectures are still taught on this holdout rebel planet. In these theatres — these arenas! — the chalk still flies, the dusters still dust, the blackboards sail up and down in banks of threes.

I lectured there as nearby buildings crumbled around me — literally. Lemmas were interrupted by jackhammers, propositions by demolitions, and proofs were built up as roofs came crashing down. Now there’s a postmodern metallic learning space next door.

All that remains, for us, of the traditional blackboard lecture, is this forlorn mission brown outpost. And it is really a metaphor for tertiary mathematics education today.

Support for traditional lectures has crumbled. Their time is past. And whatever we think of them, we need to think about what will take their place. Because if we don’t decide for ourselves, there are plenty of others willing to decide for us.

* * *

Now, just because something is dead does not mean it is bad. Evariste Galois is dead, but he was awesome. And so it is for traditional blackboard lectures: some were good; but most, in practice, were bad; and all are dead or dying.

What is to be done, and the Paradox of Choice

“What should I do?” This question is commonplace around, at least, the richer and more privileged parts of the world: there is so much wrong with the world, but I don’t know what to do. How to respond?

To a first approximation, this seems like a bad question. There is too much to do, not too little. The immediate answer seems to be: if you are asking this question, you are probably thinking too much. Stop asking, and get doing! There’s plenty to do, so do something.

But, to a second approximation, I think this question is quite an important one, and reveals some desperate failures of political philosophy and thinking across the world. It’s always worth taking some time to think about the broader picture, and how our actions fit into it.

* * *

My understanding is that this is a much more common question in the global North, in the richer western nations, than elsewhere. (I use the phrase “richer world” as a shorthand for people in the global north, the richer nations, the “developed” nations, though this is only a rough approximation and applies to anyone with a sufficiently privileged socio-economic position to have the freedom of some choice over what they do.) Chomsky often notes that, unlike elsewhere, he is never asked this question in the poorer parts of the planet, in the “third world”, the global south. And indeed, if oppression is clear and present, if you can, you go and do what needs to be done.

Broadly speaking, people in the richer world have more time, more resources, more wealth available to them. There is, on the whole, much more free time and excess cognitive capacity to figure out what to do. But there is no shortage of things gone wrong, of iniquities inflicted on other parts of the world. From climate change, to economic inequality, to civil liberties, to Orwellian surveillance, to creeping militarisation, to indigenous rights, to corporate power, to the erosion of democracy, to unjust wars and terrorism waged around the world — the litany is familiar.

Indeed, there is arguably *more* to fight against. To ask “What should I do?”, when there is so much to be done, rings shallow.

But there is also greater *distance* to the problems. The problems are further away, if only in a psychological sense, and lives are lived in a more comfortable consumer cocoon. The circumstances are no doubt familiar to most people reading this.

Wars are waged far away. Asylum seekers and refugees are locked up far away. Economic institutions by design, and by law, keep information about production, distribution and income far from public view. News media presents a skewed view of the world in keeping with their owners and largely consists of irrelevant distractions. Governments govern from far away, keeping their distance from the people. Never is a citizen given any impression that they can have any effect on any of it — except perhaps between a particular marketing effort called an “election” every few years. These problems, at least, then seem far away and it seems hard to have any effect on them.

Meanwhile, media, advertising, and custom combine to create a consuming culture of fear, guilt, infantilisation, anxiety, economic burden, and social apathy. Simultaneously marketing promotes instant gratification — and shame as to one’s own weight and self-image; unhealthy habits — and then guilt about it and products to relieve it; an array of consumer choices — but all equivalent brands of the same thing; beautiful houses to live in — and then non-stop wage slavery to pay off the debt; alienation from meaningful work — and then stress relief through shopping; “empowerment” by studying and taking approved positions of authority — while intimidating, overpowering and ridiculing those who take collective political action; a thin veneer of consumer goods — as a means to a thin veneer of happiness; turning fiction into “reality” television shows — and then turning reality into fiction at the news hour. There is fear of bad health, fear of not bringing up chidren properly, fear of boredom, fear of not having the latest consumer item — and then fear of terrorism, fear of foreigners, fear of Islam, fear of serial killers, fear of the latest enemy. It goes on. This dazzling, confusing, pervasive, terrifying, disorienting politico-economic-cultural system normalises apathy, renders political action barely imaginable, and alternative analyses of society barely thinkable. It is the bubble in which the richer world lives.

It may explain the lack of activity, but it is no excuse.

For those who have got to the point of asking the question, to some extent they have successfully resisted the consumer bubble. They are prepared to do something. They have found — provided they can navigate it critically — there is a huge amount of information about all the problems of the world, whether through learning, books, or (more usually) online. They have the sense that everything is wrong, and at least to some extent, know some details, though they are possibly overwhelming. But the barriers are still high, and the tendency to apathy all the more so.

This situation, very broadly speaking, entails that (a) citizens have relatively good access to information about what is going on, so that (b) they may more easily be are aware of the number and scale of things gone wrong and in need of fixing, while (c) living all too easily in a bubble of consumerism, marketing, media and superficiality, and (d) restrained by various social forces, from capitalist employment and debt to family ties and culture, so that knowledge of the scale and depth of the problem may combine with apathy and relative comfort and other social constraints to result in an outcome of little or no action. If the social outcome is no action, then the accumulation of knowledge and understanding, the asking of the question, was for nothing — it would, in fact, have been better to know nothing at all, since knowledge combined with inaction creates only stress and guilt, which are of negative value.

We are considering here the citizen’s point of view — the restraints and inhibitions upon them, to whatever extent voluntary, social or involuntary. In this view, the means and opportunity for doing something useful is just as important as the motive — though no doubt they all run together to some extent. This analysis is prior to any important particular issues such as climate change, civil liberties, workers’ rights, economic democracy, and so on — or any choice between them. (Of course there are many other valid analyses of political action, and how issues relate. Most notable in recent times is Naomi Klein’s argument that confronting climate change “Changes Everything”, so that in a certain sense it includes many other important social questions.) This is just another way of looking at it.

In any case, this means that to the extent that people are restrained from doing something useful by their job, their debts, and their culture, (i.e. as in (c) and (d) above), one objective of activism should be to minimise these restraints, by whatever struggle or action is appropriate. This includes maximising workplace rights, with the associated economic freedom it brings, and minimising debts, acting against exploitative financial institutions and mechanisms, and against the regressive influence of (at least) conservative religion. This much is straightforward.

And I don’t think many would argue that having more information, or being aware of what’s going on in the world, (i.e. (a) and (b) above) is a bad thing. In fact, on the contrary — if anything, people need more and better information. There is much to be said of how terrible the corporate media remains in reporting on important issues. There is much to be said of what is said, and what remains unsaid, in the corporate media. There is much to be said about the problems and issues facing “alternative” and online media. But I don’t think anyone could argue that the volume, or scale, of information is something that needs to be reduced. Perhaps perhaps less disinformation, better focused information, perhaps better presented, perhaps made less disorienting and more coherent, but not less actual information.

But the problem goes deeper. Even assuming that (a) citizens get good information about what’s going on, and (b) are highly socially aware, and (c) willing to break out of their superficial, consumer, media bubble, even against (d) any social restraints on their action, it does not seem to me to follow that many people would become effective change agents.

Why? There is still so much uncertainty about what to do. The problem of “too much information” becomes a “paradox of choice” paralysis.

So, to a first approximation, yes, the answer to “What should I do” is “stop thinking too hard and do something useful!” But to a second approximation, it probably is useful to have a way to make some sense of all the information thrown at us, the hundred slings and arrows of outrageous misfortune facing the world, the hundred different dimensions of outrageous misfortune and crime and tragedy and dysfunction.

To put it mathematically, the problem thrown at us has too many dimensions — it has the *curse of dimensionality*. We need to cut down the dimensions with a better analysis, order the data somehow, introduce an objective function or equivalent.

Yes, we need an objective.

* * *

The “paradox of choice” is a psychological phenomenon. When confronted with too many choices, the brain is overloaded. There is too much data to consider. Capacity is overwhelmed. Cognition is choked. There is too much to think through, too much uncertainty, and the response is paralysis. Nothing is done.

And indeed, one immediately observes this phenomenon in the situation facing the concerned citizen today: should I go to a climate rally? Volunteer for an environmental organisation? Participate in direct actions? Study to become a facilitator, human rights lawyer, scientist, renewable energy expert? Become a journalist, academic, social entrepreneur, hacker, engineer? Put my time into opposing the surveillance state, or international trade policy, or Aboriginal deaths in custody, or a hundred other things?

And, if we are to be honest, in many cases these are imponderable questions, on a par with “what shall I do with my life?” Except in rare cases, they are answered by some combination of one’s talents, pre-existing interests, people one meets, ideas one is exposed to, accidents of history, by fumbling through, choosing what seems best at the time, each time.

Even if we restrict to the question of what to do today, or this week, or in the next few months, there are too many problems, there is too much to do, there are too many problems, too much to learn about each of them, too much thinking to decide what to do. The result is too much time spent thinking and doing nothing, too much uncertainty about the best thing to do. It is exhausting to learn about one problem, let alone many. After learning, perhaps having a conversation or two, an argument or two, perhaps even a burst of social media, exhaustion sets in. The outcome again being nothing except more stress and guilt.

The mantra of direct action in this context, which cuts through the bullshit and gets something done, seems excellently wise.

* * *

The paradox of choice, while representing part of the problem facing the informed citizen, does not quite seem to entirely capture the issues. It is a good description of the consumer facing thirty brands of breakfast cereal — but there are extra considerations when it comes to the citizen’s social thought.

Which cereal do I feel like, says the consumer, and how much? How healthy are they? How good do they taste? How much do they cost? What is their fat, sugar, gluten, etc content? What about the brand? What about the important social information that I am never given, says the socially-aware consumer? What were the working conditions under which they were produced? Where were they made? How were they transported? What is their carbon footprint? Were animals harmed? Was there environmental damage?

This is an overload. It is an unnecessary overload. It’s too much data; usually any of the choices will be similar, or similarly bad. Sometimes there are a few excessively unhealthy or expensive or unethical choices, which can be quickly eliminated. Any of the others will be not much different from each other. It is ridiculous to devote more than a minute to deciding which breakfast cereal to choose, standing in the aisle. It may be less ridiculous to spend an hour researching which breakfast cereal is most socially beneficial — this information should be found in the supermarket, but is not. Nonetheless, the point remains that it would seem something quite a waste for an average citizen to be spending more than a miniscule fraction of their time deciding what to eat for breakfast.

With politics, it is different. Of course, the first thing to note is that comparing political action to consumer choices of breakfast cereal is odious and demeans the idea of citizenship. This of course has not stopped some social choice theorists from doing turning precisely that reduction into mathematics and calling it science.

But there are other differences between politics and breakfast cereal, other than their intrinsically different moral register. Even in the specifics, there are major differences. What do I think of these ideologies? What is my position on the State? On law? On political structures? Democracy? Private property? Organisation of work? Distribution of wealth? Economic mechanisms? This is a much, much greater overload — but it is a necessary one. Again, there are a few particularly unethical choices, which can be quickly eliminated — fascism, religious fundamentalism, colonialism, white supremacy, and so on.

Unlike other “paradoxes of choice”, it is not ridiculous to devote a serious amount of time to deciding what one thinks on these questions — and to use these answers to decide what to do politically. In fact it is necessary. The questions are complicated. The questions are important. The questions carry deep significance for our own individual and collective lives. And it’s not just a “choice” between existing alternatives. If it turns out that the “choice” (to the extent it exists) of what political ideas to follow, what actions to take, is a choice of the least bad alternative, then it is a bad set of choices. In this case, if every option is bad, one can alternatively try to do something *good* — one also has the option of imagining new ideas, building new alternatives, doing new things.

With the twenty brands of breakfast cereal, we can cut through the decision-making process by realising that, having thrown away obviously unacceptable answers, and in the absence of an obviously least bad choice, any of the remaining answers will probably be roughly equally acceptable. And in fact in practice one will tend to choose what was okay in the past.

With the hundred and more political issues to work on, one approach is similar, and perhaps one can sometimes cut through the question of what to do in a similar way. Sometimes there are obviously unacceptable answers — regressive, fundamentalist, sectarian, etc — and one can discard them. Sometimes there is an obvious choice given one’s own circumstances, talents and interests — for a computer hacker with a background in journalism, for an indigenous person living under Australian military “intervention”, for a member of an ethnic minority facing discrimination, there may be some choices that are clearly better than others. And if no such answer applies, after the clearly bad have been discarded, then probably any of the remaining choices will probably be positive, and beneficial to society. And in fact if we’ve been involved in one particular issue we may tend to continue with that, since we know more about it.

But this still seems like an inadequate way to think about it. Woefully inadequate.

* * *

Talk of “choice” is always problematic. Of course, for people in marginalised positions, suffering from various particular forms of oppression, if there is nothing to do but fight your oppressor, then you do it, there’s not much “choice” about it. To speak of “choice” in politics is to intrinsically speak about a privileged point of view. Nonetheless, in the richer world, with its wealth, cognitive surplus, and leisure time, there is significant choice of this type. And the choices are significant — they can help to determine what sort of world we live in, whether it is good or bad, whether it is worth living in, even whether it is alive or dead.

The inadequacy of the “paradox of choice” view of citizens thinking about their own social action, or activism, is that it conceptualises issues as separate “choices”. In fact what happens on one social issues affects several others, and they affect others, to that really all these “choices” are aspects of a single organic social whole. They are all connected.

And to be sure, at present at least, most of these “political choices” are choices to stem a regressive tide that pushes on all fronts. The choices are usually to curb one or another social evil — curb the tide of corporate power, reduce carbon emissions, stop the latest proposed war, pass a mild reform to ameliorate one of many possible outrages. A choice between rearguard actions, a choice between only defensive actions, is a choice between urgent and more urgent emergencies, a choice between living under outrageous or mildly less outrageous circumstances.

To be fair, many, or even most, defensive actions are inspired by hope for a better future, and do have the effect of building movements that can both fight negative change and then push for positive change.

But nonetheless, would it not be wonderful to have an actual positive choice, one that concretely builds the type of world you want to live in?

would it not be better to have an idea about the type of world you want to live in, and then measure each possible choice against it?

Would it not be better to have an idea about the type of world you want to live in, and then say — if each choice available is not particularly good, then try to build some aspect of that world you want to live in?

At the very least, would it not be at least helpful, as one thinks about what to do in the world, to have some ideas — however tenuous, however contingent, however conditional — about how the world could be, ought to be?

Having a reasonably well-formed — but not rigid, not sectarian, not a blueprint — idea of a good society is, on the one hand, a useful guide to deciding what to do now: will this action help move us towards a good, or better, society?

Having an idea about the type of world you want to live in can provide inspiration. And that applies regardless of whether one has the privilege of “choosing” what to do, or not.

On some fronts there may not be that far to go. To the extent you can already act in accordance with the world as it should be, the world is that much closer to utopia.

* * *

As to what positive political vision might consist of, that is in itself an important question. It must consist of more than vague slogans like “equal rights” or “freedom for all” or even slightly sharper notions like a “social economy” or “economic democracy”. I think it must say something about the shape of institutions in a better world, of what better institutions would look like, how they might operate.

It then entails criticism of proposed institutions, proposed visions — and no doubt this can descend into something that looks like political science fiction. Some of us like political science fiction, but not all of us do. It will take continual thought, continual rethinking and analysis, and experiment, to keep apart the line between, on the one hand, fact, proposed fact, and what is potentially factually possible, — and, on the other hand, fiction, or the impossible; though fiction of course has its merits, not least of stimulating the imagination. And of course it requires open-mindedness and flexibility.

But all such discussion, of course, is haunted by the spectre of communism. Not the spectre that haunted Europe in the writing of Marx and Engels, but the one that haunts the entire world in the wake of the horrors of the Soviet Union and other nations that called themselves “socialist”.

But history moves. That collapse was over 25 years ago now. It is time, historically, to pick up the pieces. The wounds have stopped bleeding. And even especially in the formerly “socialist” sixth of the earth is there a clear understanding of the horrors — different in kind, and less intense in many ways — of the current global economic system.

It has never been a logically good argument to respond to any thought of a radically better world to point to the Soviet Union. Yes, the first time humanity successfully overthrew capitalism, it was a disaster; but so what? History moves on, and there is no impossibility theorem that any non-capitalist system must be a disaster. On the contrary, capitalism is increasingly evidently a disaster, setting human society on a collision course with the biosphere’s physical limits. To many this has always been a disaster, and many have been making this argument, persuasively, for a long time.

At the very least, there is the converse: without any well-formed idea of what a good society would look like, there is a profound uncertainty about any political action whatsoever, a lack of inspiration, and uncertainty about what we stand for or why we should do anything. How do we know we are doing anything useful? What are we doing? Where are we going? Why should we do anything at all?

This seems to me to capture another portion of the apathy, the indifference, and the ennui in many societies today. And I think it goes to the heart of the problem.

While it may look like a paradox of choice, at its heart it is not. The real problem is not that we are overloaded with too many ideas about what to do. The real problem is that we do not have enough ideas about where we want to go.

Plane graphs, special alternating links, and contact geometry, Sydney Oct 2017

On Thursday October 5 2017 I gave a talk in the Geometry and Topology seminar at the University of Sydney.

The slides from the talk are available here.

Title:

Plane graphs, special alternating links, and contact geometry

Abstract:

There is a beautiful theory of polytopes associated to bipartite plane graphs, due to Alexander Postnikov, Tamas Kalman, and others. Via a construction known as the median construction, this theory extends to knots and links — more specifically, minimal genus Seifert surfaces for special alternating links. The complements of these Seifert surfaces also have interesting geometry. The relationships between these objects provide many interesting connections between graphs, spanning trees, polytopes, knot and link polynomials, and even Floer homology. In recent work with Kalman we showed how these connections extend to contact geometry. I’ll try to explain something of these ideas.

sydney_talk_slides