A-infinity algebras, strand algebras, and contact categories

(83 pages) – on the arXiv — published in Algebraic & Geometric Topology.

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. (more…)

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.

Knot Invariants and Cluster Algebras, AustMS Dec 2017

On 12 December 2017 I gave a talk in the Topology session of the 2017 meeting of the Australian Mathematical Society.

Title:

Knot invariants and cluster algebras

Abstract:

Cluster algebras are a type of commutative ring appearing across mysteriously many different areas of mathematics, with numerous intriguing properties.

Recently, cluster algebras have recently been noticed arising in certain knot invariants. We’ll discuss some of these developments.

Slides from the talk are below.

17-12-12_AustMS_knot_invariants_cluster_algebras

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

Tight contact structures on Seifert surface complements

(47 pages) – on the arXiv – published in the Journal of Topology.

Abstract: We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda’s method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured Floer homology. Finally, we relate our methods and results to Kauffman’s formal knot theory.

tight_contact_structures_on_seifert_surface_complements_published_version

Plane graphs and Grassmannian positivity, September 2017

On 20 September, 2017 I gave a talk in the Monash topology seminar

Title:

Plane graphs and Grassmannian positivity

Abstract:

In recent years there has been excitement about some developments in physics relating to an object called the “amplituhedron”. This talk is not primarily about the amplituhedron or physics, but about some of the mathematics behind it. This mathematics, much of it due to Alexander Postnikov, involves wonderful combinatorics and topology, connecting plane graphs and networks with the theory of Grassmannians and their non-negative cells. I’ll try to explain some of this mathematics; no knowledge of Grassmannians or physics will be assumed.