To mathematics champions

(I was the guest of honour at the Victorian prize ceremony for the Australian Mathematics Competition in November 2014. These were my remarks to the students, families, teachers and others at the ceremony.)

Thank you for all coming along tonight to recognise the achievements of these prizewinners.

The Australian Mathematics Competition is one of the largest school maths competitions in the world –– since 1978 it’s grown to have around 400,000 competitors in Australia alone, not to mention all the other countries.

Prizes go to the top 1 / 300’th of students. That’s the top 0.3%. To one decimal place, anyway.

So the students receiving prizes tonight are really extremely clever and talented. It’s really something to be quite proud of.

* * *

Well, I’d like to talk to you about maths for a bit. That’s what we’re here for after all.

I am a mathematician. Maths is what I do for a living. I do it every day. I like this fact.

If you’re here to receive a prize, you’re clearly quite good at doing maths problems. And I might even surmise that you enjoyed it.

It’s fun to solve problems. Extending your abilities in something in which you’ve got talent, is good for you, it feels good. You move closer to achieving your potential — just as a talented tennis player improves their abilities by playing against challenging opponents.

All of you who have topped this competition know that solving challenging maths problems is not simple, not routine, not formulaic. It doesn’t mean you can add really big numbers or know really big formulas. There are no formulas to solve hard maths problems you’ve never seen before.

You know that to solve difficult problems you have to think laterally, think creatively, find different approaches, ask what’s going on, ask why –– until you break through the barrier, and find an insight that illuminates the problem. And then you have the idea, and you get the answer.

Well, unless it’s one of those dastardly trick questions — but otherwise, you’ll get it!

Many people don’t really understand what this sort of mathematics is about. But when you do understand this, you also understand what wonderful achievements we are recognising tonight.

Because what does it look like from the outside, as you blaze through the Australian Maths Competition? From the outside, all someone sees is you sitting at a desk working with pen and paper. Or maybe even a pencil.

But from the inside? From the inside, it’s an epic struggle against a series of increasingly powerful adversaries, a struggle so great that most of the time you can’t even see what you’re struggling against.

There’s a good pop culture reference that for what’s going on here. I think some of our prizewinners might be a bit young to know it, but most of the parents here should know about it.

What they’re doing looks passive from the outside; but it conceals a great adventure going on in an inner dimension, a virtual dimension, on the inside. While they’re plugged into these problems their minds exist solely in this inner, virtual dimension.

Just like… the Matrix. That’s right parents, your children, in taking on these maths problems, are essentially heroes in the Matrix.

Well — except without the violence, without being trapped in a computer simulation where you’re enslaved by killer robots — but apart from that! And in fact, on the contrary, in solving these problems, you learn critical thinking skills that will help you be a better citizen.

So while you thought your children were just solving maths problems, actually they were taking on the Matrix using nothing but their own minds. They are all The One.

Neo’s got nothing on that. But of course Neo wasn’t doing mathematics.

* * *

I think that the Matrix is a pretty good analogy for how we ought to think about solving maths problems. But, to be frank, our society tends to think of solving maths problems rather… differently.

We live in a society where it’s socially acceptable, even normal, to declare “Oh I hate maths!”

How often do you hear, on the other hand, someone say: “Oh I hate geography! I was never any good at it!” If someone’s going on a holiday to a different country, people don’t blurt out how they never heard of that place and don’t really care about it but they just hate the whole subject and then they rant about a teacher they didn’t like. No, that usually doesn’t happen. But if anyone mentions maths, you watch how quickly, and how often, this happens.

If someone does admit their ignorance of geography, they’re generally ashamed of it — ignorance is not socially acceptable. But sadly, sometimes it is with mathematics.

Mathematics is the most feared subject around. There’s even a psychological condition known as “mathematical anxiety”, where people have such fear and loathing of mathematics that even the mention of it can fill them with anxiety of such a level that it requires counselling.

Well, what mathematical problem-solving means to me, and what I learned through mathematics activities like the AMC, is that one can take a mathematical problem, and play with it.

To play is to not be afraid, to not worry about grades, but simply to explore.

Fear breeds anxiety, suspicion, and hostility; but play breeds joy, fun, and imagination.

One traps the mind; the other — frees it.

And our prizewinners tonight have such free, dexterous, resourceful minds as to succeed against some devilishly difficult problems.

* * *

For myself, it was through problem-solving that I saw how interesting, surprising, and creative mathematics could be.

Along the way I learned astounding facts. I learned that the square root of 2 is not a fraction. I learned that I could add as many numbers as I wanted by clever tricks. I learned not only what Pythagoras’ theorem is, but why it is true. These were not only some of the most surprising things I’d learned, but also the most certain facts I had ever known.

In high school I did the Australian Maths Competition, and the enrichment programs run by the Australian Maths Trust and Olympiad Committees. The problems got harder, the challenges became tougher, the mathematics became more interesting. I understood why the philosopher John Locke described mathematical proofs as “like diamonds:” they were very hard, and very clear — brilliant.

OK, well, my proofs were not always very clear. As diamonds went they were sometimes pretty opaque. Or maybe they were just rocks.

Anyway…

I eventually made it into the Australian team for the International Mathematical Olympiad. I went overseas twice representing Australia. I got a bronze and then a silver medal.

Studying other subjects, like chemistry and physics, I learned that mathematics underlies them. I learned, in fact, that at the deepest levels the laws of our universe appear to be mathematical in nature. In learning mathematics I was not only learning something that was wonderful, creative and surprising: I was also learning the ideas which run the universe.

I learned about why things are the way they are, the forces that shape our world.

For me, this applied whether those forces were mathematical, or not –– they apply as much to history, to society, as to mathematics.

I went on to university. I wanted to learn everything. I studied… maths, of course. But I also wanted to learn the laws of the universe. So I studied physics. And I wanted to learn the laws of our society. So I studied… law.

Now, there may well be a few parents here who are lawyers, but I’m sure that they would agree that the law is not exactly a very… mathematical kind of thing.

But in fact, in addition to being a mathematician, I’m “almost” a lawyer. For me, they are both about understanding how the world works — just at different levels. Well, you can’t change physical laws, but you can change law laws –– just as well, because the law is sometimes not very logical! Or very good!

But it turns out that you can use the critical and creative thinking skills that you learn in mathematics everywhere you look. Even in the most unlikely places… like law.

After I finished my degrees I went overseas to do my PhD in mathematics. I went to Stanford University, which is in California. There are a lot of good mathematicians at Stanford University and it was good to learn from them. Also, one good thing about studying mathematics is that you can do it anywhere. Also, one good thing about California is that it has a lot of beaches. Lying on the beach can be “working on my PhD”, sure it can.

From there I went to do research in mathematics in France, and then in the US again, in Boston — and then I came home. I now work at Monash University.

So now, when you finish school, you can study at Monash from their excellent maths department.

* * *

So, being a mathematician is pretty good. You get to do maths problems every day. You get to decide what you want to work on, each day. Not many jobs are like that. When you like to play with maths problems, then the job of a mathematician is basically always playtime.

Mathematics can be put to many uses, for better or worse. It’s used to land a spacecraft on a comet hundreds of millions of kilometres away; to contain the spread of disease; to protect your communications; to model the climate; and much, much more. At the same time, it’s also used to design lethal weapons; to eavesdrop on your communications; and bad mathematics played a role in causing the global financial crisis.

All of you winning awards tonight have got some wonderful gifts. It’s worth asking: what is the best use you can make of your talents, for yourself and for others?

Now –– and this is especially for parents –– while I’ve said that not everyone treats mathematics with the respect it deserves, there is one particular group that does — employers! So, parents –– If you’re worried that your children won’t get a job because of studying a subject which isn’t always directly applicable to the real world, I refer you to the Wall Street Journal list of “Best Jobs of 2014”:

Number 1, mathematician. (It’s nice to be on top.)

Number 2, university academic. That include university mathematicians, you know.

Number 3, statistician. A different kind of mathematician.

Number 4, actuary. A specialist mathematician, important in insurance companies.

So — Want to explore interesting problems, challenge yourself, learn amazing things — and, as it turns out, have the best economic prospects?

Then have I got the subject for you!

And lucky for all of you, you’re all winning prizes in it.

Thank you.

The G20 and the Sanity Deficit

There is an important summit being held in Brisbane this week.
At this summit, some of the most important issues facing humanity will be discussed: economic issues of growth and sustainability; the environment and climate; the rights of indigenous peoples and first nations; human rights; and war and peace.
Leaders from all around the world will come to this summit to discuss these issues. Events will be held on austerity and growth; transnational corporations; international trade and investment policy; democratic rights; political vision; food and agriculture; Latin American integration; land ownership and tenure; labour rights; gender equality; and of course climate change.
There is also a G20 summit.
***
It’s trite to say that the world faces an enormous range of increasingly urgent challenges. There is good reason, then, for a summit of world leaders — or indeed, for many summits of many groupings of world leaders. Even if the G20, as a grouping of nations, is woefully unrepresentative of the world’s poorest nations — with Africa in particular represented only by one nation with a rather inglorious history — one could still argue that there is a place for it, as one grouping among many.
Such legitimacy comes on the assumption, of course, that their meeting discusses the enormous range of urgent challenges the world is facing.
Climate change; renewable energy; food and agriculture; and global inequality. These, among many others, must be top of the list. How many of them are on the agenda of the G20 summit in Brisbane?
Zero.
Nothing so potently reveals the political insanity of our planet as much as international meetings of powerful leaders.
***
What then, is on the agenda a the G20 summit in Brisbane?
The G20 meeting in Brisbane describes its mission as a “Growth Challenge”. By its own advertising, its aim is to promote economic growth and “break the cycle of low growth and diminished consumer confidence”.
There may be a serious debate about whether economic growth can be “decoupled” from dependence on non-renewable resources; but such subtleties are off the G20 agenda. As an exponentially growing consumer economy’s use of non-renewable resources comes up against hard limits, the enlightened rulers of the world set as their top priority the simple encouragement to produce more stuff, making sure people are confident to continue spending money shopping.
The key themes, according to the G20 itself, will be promoting growth and employment, and making the global economy more resilient against future shocks. Making the global economy resilient against future climate shocks would, of course, be a useful theme; but climate shocks are not what they have in mind. Rather, the G20’s version of “economic resilience” is focused purely on finance capital. Important as it may be to avoid future financial crises, the most well-oiled fiscal, monetary and financial engines will be the ones most likely to drive the economy off the carbon cliff.
(The US and China may have signed a bilateral climate deal; but that is just 2 nations, not the G20; and the deal has its own problems.)
We might suspect that the G20 is an unaccountable and elite forum, what with the world’s poor having no voice there. But thankfully, one of the top priorities on the G20 agenda is “strengthening global institutions”.
And indeed, it would be a useful policy goal to strengthen the voice and influence of the world’s people in their global institutions, from the UN, to the International Court of Justice, to UNESCO, aid agencies, global charities — or even, god forbid, to make the world’s governance more democratic.
Lest we be under any illusions about what “strengthening global institutions” means, charities and aid agencies are locked out of the summit. Strengthening the international rule of law — even though the international financial institutions are forever requiring respect for the rule of law as a precondition to providing funds — is certainly not on the agenda. There is no proposal to strengthen the rule of law in international relations by increasing the enforceability and jurisdiction of the International Court of Justice; no suggestion of granting the International Criminal Court powers to investigate the war crimes of all nations, large and small.
Minimally sane ideas to “strengthen global institutions”, such as removing the anti-democratic veto power of Great Powers in the UN Security Council, or dismantling the private “investor-state dispute settlement” mechanisms in many international agreements, by which corporations can subvert democratically-passed laws impeding their profits, are far off the agenda.
As it turns out, the G20’s agenda to “reform global institutions” refers to precisely one institution: the International Monetary Fund.
Nonetheless, it would still be a good idea to bring the IMF under more democratic control. For the IMF has never had a principle of “one person one vote”, or even “one nation one vote”. Rather, the IMF is built upon the sterling principle (literally) of “one dollar one vote”.
So, is the G20’s proposal to bring democratic principles to the IMF, thereby strengthening it — and potentially giving the world’s people some say in its decisions?
Of course not: the G20’s policy is simply to have a relatively few more dollars from developing nations in its voting base. The goal, in the G20’s own words, is to make sure the “IMF has enough resources and is a credible and legitimate institution to fulfil this role”. More specifically, the G20 proposes to tweak the IMF voting share formula so as to marginally increase the voice of central bankers from Brasilia and New Delhi, as against those from Frankfurt and Washington DC. This is their version of “institutional reform”.
But the G20 does at least correctly identify that the IMF has a credibility problem. The IMF has almost has no role left to play in south America, since those nations threw off its yoke and sought monetary and fiscal solutions elsewhere.
It is difficult, even reading their own stated purposes, goals and agendas with the most charitable of inclinations, to find anything worth defending in the G20 agenda. The plan is to accelerate economic growth and pursue this goal ahead of all others. There may be a token mention of such growth paying attention to climate change, but most likely there will not be any at all.
Putting aside the question of whether an organisation dominated by central bankers is likely to pursue an aggressive pro-equality agenda providing a dignified livelihood for all, we can simply ask: at this crucial point in history, if the G20 promotes economic growth without any heed of environmental limits, how quickly will we burn through our remaining carbon budget — before we have locked in levels of global temperature rise which endanger the entire planet?
***
The event at which the important issues will be discussed, rather than the G20 summit, is the People’s Summit organised by the Brisbane Community Action Network.
Those seeking solutions to the world’s pressing problems in Brisbane are more likely to find them at the People’s Summit. After all, one cannot find any solutions at all, when the questions are off the agenda.
Such questions, in addition to being obvious, are important to Australians. Despite a decade of propaganda, 63% think the Australian government should be taking a leadership role in international policy on global warming and carbon emissions, and 81% think climate change is an important threat to Australia. But this is only what the general public thinks, and so is irrelevant.
Australians have many other thoughts on policies relevant to the G20 too: some progressive, and some conservative. Fully 82% who think US foreign policies are an important threat; 75% think that Australian foreign aid programs should be used to help reduce poverty in poor countries; 76% think the gap between high and low incomes is too large; all irrelevant again. Rather more xenophobically, the Australian population is strongly against foreign involvement in the Australian economy (86%), and worried about competition from low-wage countries (81%). The G20’s agenda is studiously against every single one of these policy preferences of the Australian population. Public opinion, whether left or right, progressive or conservative, simply does not matter.
Then again, Australians do appear also to have some insight into about the value of democracy in its present form. Nearly half the Australian population does not believe in democracy — presumably, understanding “democracy” to mean as it appears in the current “representative” institutions which make a mockery of it. More precisely, 40% of Australians believe that “democracy” is not preferable to other forms of government. These Australians believe overwhelmingly that there is no real political choice available to them, and that our “democracy” only serves the interests of a few.
Those who want to understand where this sentiment comes from need look no further than the parade of world leaders coming to Brisbane under military lockdown.
***
The G20 exemplifies a tragic and radical disconnect in the world today: between who is excluded and who is not; between whose voice is heard and whose is not; between human activities, and the environmental limits imposed upon them by physics; between what people want, and what their leaders do; between societies containing awesome levels of technology and scientific knowledge, and awesome levels of ignorance and inhumanity among those who govern them; between what needs to be done, and what is being done; between sanity, and the national and international political-economic elite.
There has long been a notion of a “democratic deficit” in international institutions — their distance from the people, the layers of representation in which lobbies, powerful institutions and elite interests exert their influence; their rarefied lifestyle far from those on the receiving end of their policies.
That is all well and good, but we are well past that now. It is by now a sanity deficit, and the only corrective is the other world superpower: the world’s people. Some of their work will be proceeding at the People’s Summit — the main show in town in Brisbane this weekend.

Strings, fermions and the topology of curves on annuli

(55 pages) – on the arXiv – published in the Journal of Symplectic Geometry.

Abstract:In previous work with Schoenfeld, we considered a string-type chain complex of curves on surfaces, with differential given by resolving crossings, and computed the homology of this complex for discs.

In this paper we consider the corresponding “string homology” of annuli. We find this homology has a rich algebraic structure which can be described, in various senses, as fermionic. While for discs we found an isomorphism between string homology and the sutured Floer homology of a related 3-manifold, in the case of annuli we find the relationship is more complex, with string homology containing further higher-order structure.

Topological recursion and a quantum curve for monotone Hurwitz numbers

With Norman Do and Alastair Dyer
(18 pages) – on the arXiv – published in the Journal of Geometry and Physics.

Abstract: Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In this paper, we prove that monotone Hurwitz numbers arise from the topological recursion of Eynard and Orantin applied to a particular spectral curve. We furthermore derive a quantum curve for monotone Hurwitz numbers. These results extend the collection of enumerative problems known to be governed by the paradigm of topological recursion and quantum curves, as well as the list of analogues between monotone Hurwitz numbers and their classical counterparts.

Climate Tragedy and the Three World Budget

Last week, two teams of scientists reported their findings on the West Antarctic ice sheet. Climate change has now melted this sector of Antarctica — comprising a tenth of the Antarctic ice mass — so far that it appears the melting has become irreversible. When fully melted — whether in the next century, or a few more — global sea level will rise over 3 metres, redrawing the coastline of the entire earth.
Over the previous few months, the Intergovernmental Panel on Climate Change has released the various volumes of its Fifth Assessment Report. Its findings were equally alarming, covering every aspect of the climate, its expected trajectory, impacts, and what can be done to avert the crisis.
We cannot now avert the crisis; permanent environmental and ecological crisis is our new status quo. This is not even to mention economic, psychological or moral crises.
Low-lying islands have been abandoned. Heat records fall in cascades like the meltwaters of ice sheets.
Rational presentation of the alarming facts has so far failed; political solutions through regular channels have failed; diplomatic negotiation of an international agreement has failed. None of these approaches show any signs of ceasing to fail in the near future.
Our collective failure so far is world-historic and monumental.
The mad rush of industrialisation, driven by the engine of capitalism, has dug up the fossilised and liquefied remains of animals and plants long past. We have feasted upon the fires of three thousand million years of our own evolutionary heritage, burning their remains to power our own development — and ending in a treadmill of acquisition, consumption, inequality, and futility. We have burned so much of this thousands-millions-years heritage that these paleontological remains now float in the atmosphere, tipping its composition so far as to affect the whole planet’s energy balance.
It is a sacrifice of such epic tragedy as the ancients could never have dreamed; we offer these charred remains on an intercontinental, world-destroying scale to placate the monstrous appetite of capitalism. This brutal monster knows only how to run faster and bigger and it is gobbling up the entire world. If it so much as slows, we call it recession, and it sends us wholesale to poverty. Our ancestors offered irrational sacrifices to placate forces they did not understand. We fully understand the irrationality, but offer the sacrifice of all our ancestors anyway, vaporised and liquefied — and fully aware that their vaporised remains carry an infrared absorption spectrum which will choke us.
The earth has shifted its average temperature by a few degrees, and will shift for a few more. How many more is in the balance. But the climate system itself has a great deal of inertia: what has already been done will affect us for generations to come.
The sins of the father shall be visited upon the son is not just brimstone fear-theology: it is also a statement about the inertia of the climate, a corollary of theorems on differential equations. The sins of the fathers, the climate forcings of all previous generations, will be visited upon the sons, and on down to the third and fourth generation; so says the book of functional analysis.
Some things cannot be changed by human will: the laws of chemistry, blackbody radiation, and photon scattering, to name a few. But some things can be changed by human will: foremost among these, our own human activities. There is no doubt that humanity, in its present state of knowledge and technology, has the technical capacity to stop burning carbon, and switch to renewable energy sources, in a sufficient time frame to avert disaster.
Australia is an island continent occupying a small corner of this troubled planet. It is geographically large, demographically small, highly educated, relatively socially progressive, bathed in sunlight, hot, dry, and one of the wealthiest places ever to have existed. If ever there were a place with the means, motive and opportunity to take the actions necessary to stop burning carbon, it is Australia in 2014.
This week, the government of this island continent laid down its plans and priorities for the future. Its announcements were not mere rhetoric. A budget is not aspirational: it concretely apportions money and resources, materially allocates resources, appropriates public funds for its projects. There is a mythology that it expresses the collective will of the population.
Amidst irreversibly melting ice sheets, tumbling heat records, and ever-more-frequent extreme weather, the government of this wide brown sunburnt land, rich beyond the imagination of any previous society, chose to —
— repeal a very modest tax on carbon emissions, and disestablish the agency devoted to renewable energy.
Nothing more need be said.
But more should be said, and the litany is extensive.
One could — and should — go on at length about the cuts to universities, fee deregulation, and the end to the progressive principle of the HECS/HELP student loan scheme.
One could — and should — go on at length about the end to universal health care. About how a visit to the doctor, previously technically free, is now no longer, violating a fundamental social moral principle and deterring the poor from obtaining healthcare.
One could — and should — go on at length about the cuts to pensions, meaning further deprivation for some of the most vulnerable inhabitants of this land.
One could — and should — go on at length about the cuts to unemployment benefits, meaning further deprivation for another group of the most vulnerable inhabitants of this land.
One could — and should — go on at length about the cuts to legal aid, a nasty measure that makes a miniscule difference to the budget yet ensures that the most vulnerable suffer maximally.
One could — and should — go on about the removal of funding for indigenous affairs, an equally petty yet even nastier measure, an exquisite further humiliation after centuries of dispossession and violence aimed at the rightful owners of this land.
One could go on at length about these or many other outrages in this budget.
But in order to see the complete category error of this budget — the radical mismatch between physical reality and action taken — it is sufficient to consider the impending climate catastrophe, and the response to it: not a reluctance to act, not even total inaction, but a deliberate and systematic attack upon the tepid actions previously taken. It is full throttle in reverse; a categorical nihilism as to the future.
Moreover, unlike various other budget measures, this did not entail any broken promise or lie. The budget’s radical climate nihilism was one that was entirely premeditated and honest.
Having seen the watered-down carbon tax enacted by the Labor party, and its rapid destruction by the Liberal party, the possibilities for the necessary climate action to occur through mainstream electoral politics are exhausted — and there are many for whom it never seemed a viable option in the first place.
There remains one social variable providing us with a chance: resistance. The variable of political resistance is the crucial independent varaible — and it is the one over which people can have the greatest effect. Without the means, the capacity, and the will to resist the radical disconnect between our permanent climate crisis, and ongoing economic and governmental activity, we accelerate towards collapse. And so we must resist. We must restore sanity.
There was a hashtag on twitter lampooning the budget with Abbott-style three word slogans, #ThreeWordBudget. But this budget commits us to non-renewable energy, fossil fuels, and an acceleration of climate disasters.
At this rate we will need at least two more planets. This is a Three World Budget.
But we only have one.
So resist and fight for it!

An explicit formula for the A-polynomial of twist knots

(5 pages) – on the arXiv – published in the Journal of Knot Theory and its Ramifications.

Abstract: We extend Hoste-Shanahan’s calculations for the A-polynomial of twist knots, to give an explicit formula.

See also the erratum below – there was a mistake in one of the formulas in the original version.

Twisty itsy bitsy topological field theory

(52 pages) – on the arXiv – published in the International Journal of Mathematics.

Abstract: We extend the topological field theory (“itsy bitsy topological field theory”‘) of our previous work from mod-2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has information-theoretic (“itsy”) and quantum-field-theoretic (“bitsy”) aspects. In the process we extend some results of sutured Floer homology, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.