University forever

I don’t have anything against people who want to stay at university as long as possible: this is, after all, my life. Well, if you are failing everything, than perhaps there is a reasonable argument you shouldn’t be there.

I think long term economic trends point in this direction though.

With improving technology and automation and so on, it may well be that there aren’t enough jobs to go around for everybody. I don’t think we’re anywhere near a science-fiction type situation where robots can do all the work and humans can live in paradise (would it be paradise?), but it’s certainly the case that we don’t need all adults working 40 hour weeks to produce our current level of wealth. The pandemic has conclusively demonstrated that we can essentially shut down vast portions of the economy, without shortages of food, without shortages of essentials, without losses of basic amenities or utilities, and in fact without shortage of most luxury consumer products either.

There are of course many types of hardship going on, great sadness and loneliness and alienation, and many of the failings of our society have been exacerbated.

But the economic point remains: our society really can produce more than everybody needs, with only a relatively small proportion of the working population.

In that case, pandemics aside, we don’t have to “work”, in the usual sense of the word, very hard to produce all our society’s requirements. Rather than sitting around unemployed, a civilized society would spread the work around, shorten the working week, and provide more and more opportunities for self-development, through art, literature, science, and education — and whatever else, for that matter.

There are plenty of platitudes spoken by politicians about “lifelong learning” — why not make it formally possible to continue learning throughout your entire life?

Capitalism cannot do this. A civilized society could.

Sitting out the math wars

I recently had the misfortune to revisit some episodes of the “math wars” — the ongoing struggle over mathematics education and curriculum, mostly at the school level. It’s an important struggle, with great ramifications for our education system, not to mention one in which I have a personal stake, being a mathematician and all. But very few professional mathematicians have been involved in them, and when they have, they have not always inspired confidence. The very worst behaviour in these “wars” has come from professional mathematicians.

Putting aside the worst behaviour (but only temporarily), I wondered why. Indeed, I’ve never gotten myself involved in it. I think the reasons are worth examining.

Immediate disclaimers are necessary regarding (1) “math” and (2) “wars”.

I say “math” rather than “maths”, because, as with many of struggles over human culture in the West, the focal point of the struggles have been in the US. Of course, those struggles have their resonances and reverberations in Australia — the turbulence splitting off an Oceanic vortex across the Pacific.

And I detest the “war” analogy. It’s not a war in any meaningful sense. It’s a policy debate with some important political-economic and ideological undercurrents. And war analogies are overused and our society is over-militarised.

So, this is a bad usage, but it’s a standard usage. Hence the scare quotes. It’s “math wars” all the way through.

So why would mathematicians sit out the “math wars”?

For myself, the “math wars” are not a debate in which I’ve often felt I have much to contribute — despite having been involved in mathematics teaching, in one way or another, for over two decades. There are several reasons.

Firstly, my experience of learning mathematics is far from average, in any sense, and so I am naturally skeptical of any attempt to generalise to others those learning or teaching techniques that have worked for me. Arguably, I look like the best but am in fact the worst qualified person to opine on the topic.

Secondly, at the school level, my teaching experience is almost entirely limited to the teaching talented or advanced students. This type of teaching presents its own challenges, but it does not reflect the average teacher’s experience in the slightest. The challenges of extending precocious geniuses are essentially completely irrelevant to the challenges of getting the average class interested in and proficient at mathematics.

Thirdly — of which I am reminded every time I actually teach — I’ve never felt like a natural teacher: a natural explainer, perhaps, but teaching is much more than that. At most, I’ve become an experienced lecturer, I know what works for me and think I do a good job of it, but I recoil from the idea that this gives me any authority to speak on what others should do.

Because, finally, all my scientific and political commitments militate against pronouncements on the topic. The mathematician’s allergy to over-generalisation, the scientist’s skepticism to the applicability of findings beyond their domain, the anarchist’s aversion to telling others what to do, the socialist’s solidarity with a class of underpaid and underappreciated workers, and the general democratic injunction against thinking you know what is best for others in their own profession — all these, when I have some idea about classroom teaching, scream in unison that in fact I don’t know the first thing about teaching the average maths class in Australian schools today, and that greatly limits what I could say about the curriculum that should be taught.

Essentially, I have enormous respect for primary and secondary teachers — they do a thankless but crucially important job, are underpaid and overworked — and for all that, suffer the disrespect of parents and students every day. Telling them what to do, what curriculum is best for their class seems not only distasteful and uninformed, but also, in a certain sense, wrong.

On a deeper level, I’ve always felt that battles over curriculum and teaching are not the real issue, and that it is really simply the policy end of a much deeper cultural problem.

Australian culture, in general, despises mathematics. It is the only essential school subject where an Australian can declare its uselessness, and generally be assured sympathy. It is the only school subject on which, generally speaking, one can confess their ignorance or incompetence and be assured of understanding rather than embarrassment, without a hint of guilt. One can confess to not being much good at geography, or history, or other sciences — and even if it is more a celebration of ignorance than charming self-deprecation, it is still a confession, with a hint of remorse. Not so with simultaneous equations and the like, upon which opinion may vary from collective recollection of terrible classroom experiences, through to vengeful abuse of the educational system.

Given that cultural status, is it any surprise that students might be uninterested in mathematics? That there might not be many students studying mathematics to a high or advanced level? That there might not be many people studying to be mathematics teachers — and hence, a shortage of qualified mathematics teachers? And hence, the reproduction of a populace which despises mathematics?

On cultural matters I tend to throw up my hands, if not in despair, then because of my inability to do anything to change it. I can explain mathematics to people, why it is interesting and fascinating and beautiful and so on. But, in news which will come as a surprise to no one, I don’t really like my chances of making it cool.

Of course, all these difficulties — general derision for the subject, declining enrolments, teacher shortages — run together. An improvement in one can lift all the rest, and a failure in one can lower all the others. But being so total makes the problem a formidable one.

And that leads to a great sense of disconnection. I feel an enormous chasm between my experience and interests, and those of the general population. Why wouldn’t everyone want to try to figure out what on earth (on earth!) this universe is, and what it is, and the mechanisms that underlie it? How can it be that not everyone is captivated by mathematical truth? Antipathy, or even indifference, to mathematics is to me an alien concept.

Of course I understand that people have different interests, and diversity of human interests is a good thing. And this chasm does not of course prevent me from trying to convince others that these topics are fascinating, nor from trying to promote understanding of these topics. And of course I’m aware that the secondary mathematics curriculum can be boring and lifeless. (At school I much preferred history classes to my official school maths classes.)

But all this does give me pause as to how to address such a radically different audience. The reality is that I am the radically different one, not everyone else.

So that is my navel-gazing examination of my reticence on the subject. And much of it will clearly generalise to the mathematical community, though much will not. Not every mathematician shares my politics or my cultural background, but I think many would feel a long way removed from classroom teaching of their subject, a sense of being cultural outsiders in their embrace of mathematics, and despite their expertise in the subject, a general lack of knowledge about what would actually work in classrooms.

But this reticence, though perhaps justified in each individual case, may collectively be counterproductive. Arguably mathematics curriculum and assessment have descended into various domains of stupidity. Arguably the whole debate would benefit from more involvement by knowledgeable mathematicians.

And as per Bertrand Russell, “The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt.” Not to suggest I am at all intelligent on this matter, or that anyone else involved is not — but I am certainly full of doubt. (It’s about the only thing about which I have no doubt!) Perhaps Russell’s observation is a reason for some of us who are knowledgeable on the subject matter, at least, to put aside some doubts occasionally, as to what we can and should say.

And even though mathematicians may not know the answer to every question, one does not need to know the answer to every question. It is sufficient to be able to say something. Or even, simply to say something is bad.

This is nothing new. I grew up learning mathematics in a curriculum that was essentially universally understood by all those who taught me, and all those who I respect, to be heading in a ridiculous direction. Everyone decried the increasing reliance on calculators as a substitute for mathematical understanding. That has continued, with whole exams now written as a test of calculator usage. (Who even uses calculators any more? Phones are more powerful devices.)

Trends continue. My recent years of teaching first year university students have been marked by the steadily decreasing knowledge of incoming students. There is always some need to remove topics from the secondary curriculum, and never any room to strengthen it or include anything new. It can always be hollowed out and weakened, but never filled out or strengthened. I then have the task, in teaching university classes, of filling in the gaps and bringing students up o speed.

But these are not the main reasons I think mathematicians should say more about mathematics curriculum and teaching. Rather astonishingly though it sounds to say it, I think the most important reason why more mathematicians should say something on these matters is to make for greater civility.

Now, civility is one of the worst topics in political discourse.

One terribly overused, and bad, contemporary political argument is that extreme voices dominate political debate, and we need more civility. In one sense this is true: more and more right wing and even fascist voices have become prominent and even dominant in politics across the world in recent years. And debate increasingly happens on social media platforms which algorithmically promote extreme content because it generates “engagement”. But more usually the argument is made by centrists who decry this alongisde some supposed left-wing equivalent, such as the political movements behind Jeremy Corbyn or Bernie Sanders. There is just no comparison between their mild social democratic politics, and the resurgent reactionaries on the right. To the extent politeness or civility towards such reactionary forces normalises or strengthens them, we need less politeness or civility. Civility is not the problem here.

Even worse, the charge of incivility is often brought up against those who are genuinely and justifiably angry at the system. It can combine with racist and sexist attitudes to depict people of colour as irrationally angry, women as shrill, and so on. This sort of incivility, an expression of justifiable rage, is necessary for social change, and criticising it for its incivility is beside the point.

But the “math wars” are another matter. They arouse great passion and controversy in some mathematicians, educators and teachers. And this is a legitimate controversy, with deep political and ideological divides. But there is nothing to justify incivility, in the way that there is to justify rage against police brutality, misogyny, or fascism.

In my experience, the incivility is largely between mathematicians and mathematics educators. And the incivility is completely in one direction: by the mathematicians, towards the maths educators.

It’s not uncommon for mathematicians to express contempt for mathematics educators. It is not the only attitude, and in my experience in recent times not the most common one, but in my experience it is a standard attitude.

It’s disturbing how this contempt fits gender stereotypes. Professional mathematicians are overwhelmingly male, and mathematics education researchers are overwhelmingly female — and even if the contempt of the former for the latter is purely based on intellectual or political differences, it can easily look like misogyny.

Incredibly, the “math wars” have their own “cancel culture”. In the worst incident, though now over a decade ago, a group of prominent mathematicians tried to discredit a leading maths education researcher, and get her fired. Their accusations of misconduct which were found to be baseless, but not before succeeding in driving her out of a university position.

Legitimate criticism of the views or research of others is one thing. This was another.

This sort of incivility has to stop. If mathematicians on one side of the debate feel that they have been “losing” the “math wars”, this kind of incivility is part of the reason for it. More civil voices need to be heard. Mathematicians need to reject such incivility, and contribute in a democratic spirit.

Not human, but inhabited by humans: writing mathematics

Mathematics can be written in many ways. One approach, very popular with professional pure mathematicians, is to write as little as possible. Often the best proof of a mathematical theorem is the shortest and most elegant. This fact, combined with some of the history and culture of mathematics, leads to the classic terse mathematical style: theorem-proof, theorem-proof, lemma-theorem-proof, definition-prposition-theorem-proof, and so on.

(The fact that most mathematicians dislike writing may also have something to do with this!)

I think that, on every mathematical subject, there ought to be texts which are written in this way: short, crisp, elegant, minimalist. But there should also be others.

The standard terse style is, however, imperfect for learning mathematics — especially anyone below PhD level. Perhaps this style is tolerable for learning how the proofs go. It’s useful for understanding the exact steps in rigorous proofs of theorems. And it often works well with a highly motivated or sophisticated reader — one who understands that reading such books is not actually about reading, but about knowing when and how to ask oneself questions, filling in the details which have been omitted. The standard style is hard, both in the sense of “not easy” to read, and in the sense of “not soft”, with no surrounding story or context.

Such an approach is fine for insiders: those who already understand the culture and the conventions of mathematical literature. But for learners — particularly those with a weak background, as is increasingly the case — it is a different matter.

What tends to be left out in the standard terse style? Everything that makes mathematics human: history and context; motivation; commentary; connections within and beyond mathematics. And even a mathematician may not appreciate every book being so hard to read.

Other mathematicians may disagree, but given a choice between terse text, and a gentle version which is twice as long but twice as easy to read — and full of interesting details and tidbits — my preference is clear: genial is better than brutal and terse.

Why are we interested in the topic we are talking about? What are its implications and connections? Why do we cover the parts of the subject that we do? Why do we use the arguments we do, and why not others? Where did this proof come from? How could we use similar ideas to prove other results? These questions are often as important as the mathematical content itself.

More generally, contemporary curriculum and culture, at least in Australia, leads to the situation that students may know little about the background of their subject — even when they are studying at an advanced level.

Further, the classic terse approach can descend into a combination of intimidation and disrespect. Proofs and arguments are routinely omitted as “obvious” or “trivial”. Steps are skipped. Some readers may be fine with one or two gaps to fill in themselves, though no harm would have been done had the author included it; but every skipped step is a potential hazard, and a successful reader must navigate them all.

In extreme cases, authors leave a trail of breadcrumbs which the reader may be able to pick up and follow along, if they have enough knowledge or curiosity or insight or gumption or tenacity or luck. Mathematical writing then becomes a set of puzzles, where every sentence must be solved by the reader to progress to the next. Mathematicians in certain fields will know certain “classic” texts in the mathematical literature are precisely of this type. All this in the supposed pursuit of communicating mathematics as fast and efficiently as possible!

Such an approach makes reading mathematics, in its terse classic style, a completely different affair from reading almost any other subject.

Why erect walls of unexplained argumentation, and browbeat those who cannot scale them with cries of “obvious” and “trivial”?

For students first arriving upon the abstract world of pure mathematics, it can seem a harsh, even brutal subject. That is because it is a harsh, brutal subject. Mathematics does not forgive your one mistaken observation: your proof will come crashing down despite your pleadings. Most of your thoughts on mathematics will be wrong — to the extent they are even precise enough to be wrong. To do mathematics is to work through all the wrong thoughts to make them right.

Mathematical arguments are true independent of what humans think of them: in this sense, the truths of mathematics live in their own world, a world that has no feelings and is not human. The independence of mathematics from the human world is the source of an austere beauty, but it can also make the subject seem cold and desolate.

It is a cold world, it is a harsh world, but it is a beautiful world, and its statements are pure, honest, and beautiful. And while it is not human, it is a world inhabited by humans. It also provides the language of science and the universe.

Some can brave entry to this world themselves. But why should we not provide some guidance as to the nature of this world, as we enter it?

A-polynomials, Ptolemy varieties, and Dehn filling, Melbourne June 2020

On 15 June 2020 I gave a talk in the topology seminar at the University of Melbourne.

Title: A-polynomials, Ptolemy varieties, and Dehn filling, Melbourne June 2020

Abstract: The A-polynomial is a 2-variable knot polynomial which encodes topological and hyperbolic geometric information about a knot complement. In recent times it has been shown that the A-polynomial can be calculated from Ptolemy equations. Historically reaching back to antiquity, Ptolemy equations arise all across mathematics, often alongside cluster algebras.

In recent work with Howie and Purcell, we showed how to compute A-polynomials by starting with a triangulation of a manifold, then using symplectic properties of the Neumann-Zagier matrix to change basis, eventually arriving at a set of Ptolemy equations. This work refines methods of Dimofte, and the result is similar to certain varieties studied by Zickert and others. Applying this method to families of manifolds obtained by Dehn filling, we find relations between their A-polynomials and the cluster algebra of the cusp torus.


Monash topology talk on Circle packings, Lagrangian Grassmannians, and Scattering Diagrams, April 2020

On 1 April 2020 I gave a talk in the Monash topology seminar.

Title: Circle packings, Lagrangian Grassmannians, and scattering diagrams

Abstract: I’ll discuss some recent work, in progress, relating the theory or circle packing to various ideas in geometry and physics. In paticular, we’ll show how ideas of Penrose and Rindler can shed light on circle packings, describing them by spinors or by Lagrangian planes satisfying various conditions. We’ll also touch on how the resulting spinor equations are related to on-shell diagrams in scattering theory.


A-polynomials, Ptolemy varieties and Dehn filling

(45 pages) – on the arXiv

Abstract: The A-polynomial encodes hyperbolic geometric information on knots and related manifolds. Historically, it has been difficult to compute, and particularly difficult to determine A-polynomials of infinite families of knots. Here, we show how to compute A-polynomials by starting with a triangulation of a manifold, similar to Champanerkar, then using symplectic properties of the Neumann-Zagier matrix encoding the gluings to change the basis of the computation. The result is a simplicifation of the defining equations. Our methods are a refined version of Dimofte’s symplectic reduction, and we conjecture that the result is equivalent to equations arising from the enhanced Ptolemy variety of Zickert, which would connect these different approaches to the A-polynomial.

We apply this method to families of manifolds obtained by Dehn filling, and show that the defining equations of their A-polynomials are Ptolemy equations which, up to signs, are equations between cluster variables in the cluster algebra of the cusp torus. Thus the change in A-polynomial under Dehn filling is given by an explicit twisted cluster algebra. We compute the equations for Dehn fillings of the Whitehead link.


The sensitivity conjecture, induced subgraphs of cubes, and Clifford algebras

(4 pages) – on the arXiv

Abstract: We give another version of Huang’s proof that an induced subgraph of the n-dimensional cube graph containing over half the vertices has maximal degree at least , which implies the Sensitivity Conjecture. This argument uses Clifford algebras of positive definite signature in a natural way. We also prove a weighted version of the result.


Talk in Monash discrete mathematics seminar, September 2019

On 16 September 2019 I gave a talk in the Monash discrete mathematics seminar.


The sensitivity conjecture, induced subgraphs of cubes, and Clifford algebras


Recently, Hao Huang gave an ingenious short proof of a longstanding conjecture in computer science, the Sensitivity Conjecture, about the complexity of boolean functions. Huang proved this conjecture by establishing a result about the maximal degree of induced subgraphs of cube graphs. In recent work, we gave a new version of this result, and slightly generalise it, by connecting it to the theory of Clifford Algebras, algebraic structures which arise all across mathematics.