Talk at Knots in Washington 49.75

On 22 April 2022 I gave a (virtual) talk at the 49.75’th (!) Knots in Washington conference, an international conference on knot theory held regularly since 1995.

Title: A symplectic basis for 3-manifold triangulations

Abstract: Neumann and Zagier in the 1980s introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We will discuss an interpretation for this symplectic structure in terms of the topology of the 3-manifold. This joint work with Jessica Purcell involves train tracks, Heegaard splittings, and is related to Ptolemy varieties, geometric quantisation, and the A-polynomial.

Slides from the talk are below.


A Symplectic Basis for 3-manifold Triangulations, AustMS 2021

On 8 December 2021 I gave a (virtual) talk in the Topology session of the 2021 meeting of the Australian Mathematical Society.

Title: A symplectic basis for 3-manifold triangulations

Abstract: Walter Neumann and Don Zagier in the 1980s introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold. We will discuss an interpretation for this symplectic structure in terms of the topology of the 3-manifold. This work, joint with Jessica Purcell, involves train tracks, Heegaard splittings, and is related to Ptolemy varieties, geometric quantisation, and the

Slides from the talk are below.


Five-minute surrealist antiwar exposition of topological data analysis

On Remembrance Day 2021 (11 November) I have a talk at a session of Lightning Talks at a session on “Mathematics for Data Analysis, AI & Machine Learning” organised by the Monash Data Futures Institute.

This was a “Lightning Talk” — 5 minutes only. In which I attempted to explain what topological data analysis is and how it works.

It had to be impressionistic, but it turns out surrealism is better for this kind of thing. For what is topological data analysis — or more explicitly, one of its main tools, persistent homology — if not the Persistence of Memory of Topological Contortion?

Being Remembrance Day, a day for ending war, and Topological Data Analysis having been funded for military applications, no better time to mention the campaign against lethal autonomous weapons systems.

Five minutes.

The slides from my talk are available below (2mb pptx).

An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes

In this article, by my Monash colleague Jerome Droniou, and Daniel A Di Pietro, I am acknowledged for providing an explicit basis for the space \( \mathcal{G}^{c,l} (T) \), which is used in a software implementation.

Di Pietro, Daniele A, Droniou, Jérôme, An arbitrary-order discrete de Rham complex on polyhedral meshes: exactness, Poincaré inequalities, and consistency. Found. Comput. Math. 23 (2023), no.1, 85–164.

The article has open access online and is available here at the journal website.

General tips for studying mathematics

I don’t know that I would have anything to say that’s not a platitude, but here are some thoughts.

It’s a hard topic. Recognise that it’s hard. Don’t expect to learn everything in one go. Don’t expect to solve every problem on the first attempt. Take the time to let ideas sink in.

Don’t feel intimidated by others who do things faster, nor superior to those who do things slower. Don’t let anyone put anyone else down for finding the subject difficult, comprehending things more slowly, etc. Try to include people who feel excluded. It’s difficult for everybody, and everybody has different difficulties. We don’t need to make them worse.

Ask questions of your teachers. Chances are if you’re thinking something in a lecture was unclear, many other people are too, but not all of them are brave enough to raise their hand. And ask outside of class too if you want. For most lecturers, most of the questions they have to answer are about homework extensions, repeating for the hundredth time some administrative detail, etc. They love answering questions about their actual subject. Ask them.

If you can, find other students to work with. People vary, but for many people, working together productively with others on problems makes the whole experience much more pleasant, and it’s better for solving problems and bouncing around ideas too. Learn from others, and they can learn from you.

Be curious. Ask how and why things work. Think about the big picture and how everything fits together. Ask why we study the things we study.

Enjoy the subject. Mathematics is full of patterns, curiosities, profound truths, breathtaking theorems, hard-fought triumphs, ancient mysteries. Its roots in culture stretch back as far as humans have walked the earth. Play with the patterns, play with as much of it as you can get your hands on, it’s much better if you an understand it that way. Understand it your way, and tell stories about it.

Wonder at the profound beauty and elegance of deep theories. Take the time to properly understand and appreciate what theorems say. They surprise, they flabbergast, they connect oceans of thought. They reduce the incomprehensible to the world of the living. Their proofs are often even more instructive than their statements. These things are among the greatest achievements of humanity. They are the eternal harmonies of this tragic universe.

Also, savour the enjoyment when you solve a difficult problem. It’s a hard subject and we need to take all the wins we can get.

Summarise your maths research in one slide, Dan

I’ve never much been one for concision. To say that the contemporary attention economy, not to mention professional communication, not to mention mainstream politics, trades on absurdly short, necessarily incomplete, if not wrong, attention-grabbing quips and barbs and slogans and pitches, effectively giving most social discourse a lobotomy, is a cliche. That doesn’t make it less true or less important. Equally, to insinuate that you have much more to say, and have such nuanced and complex ideas, that you’re above that sort of thing, that you’re so sophisticated as to be impossible to summarise or condense to that form, is not only irritating and self-important, it’s probably false most of the time too.

Anyway, as part of an upcoming workshop participants were asked to introduce themselves with a one-page slide. I took it as an extreme form of concision: summarise your maths research in one slide, Dan. In practice it turned into a word cloud with a bunch of pictures taken from my papers.

Anyway, I thought the pictures were, if not pretty to look at, a bit fun. Well, of course I would say that, I do this stuff for a living. (Or, at least, I’m supposed to. What it seems I actually do for a living is drown in email and admin and committees and so on.)

So why not post the one slide here too?

slide summarising my maths research

A-Polynomials of fillings of the Whitehead sister

Joint with Joshua A. Howie, Jessica S. Purcell, and Em K. Thompson – (20 pages) – on the arXiv – published in the International Journal of Mathematics

Abstract: Knots obtained by Dehn filling the Whitehead sister include some of the smallest volume twisted torus knots. Here, using results on A-polynomials of Dehn fillings, we give formulas to compute the A-polynomials of these knots. Our methods also apply to more general Dehn fillings of the Whitehead sister.


One line Euler line

Well, I probably saw it as a student years ago, and apparently it’s standard, but I came across this today through Olympiad wanderings and thought it was wonderful enough to share.

It’s a fun fact from Euclidean geometry — the sort of gem which in a decent world would be taught in an advanced secondary school maths class. But of course it is not, at least nowhere near me — and indeed even the basic ideas on which it relies (including almost all geometry and indeed the very notion of proof) have more or less been ruthlessly exterminated from the high school curriculum. As is happening now, with the content of the Australian school mathematics curriculum being shifted further back and, so it seems, further watered down, in the name of teaching abstract problem solving skills… skills so abstract that there is less and less actual concrete knowledge to apply them to.

And while not taught in schools, this sort of thing is also too “elementary” to appear in most university courses — unless the university has a specialised “elementary” mathematics course, perhaps aimed at future mathematics teachers, that might cover it. Even then the subject matter can’t be taught by those future teachers, as the content lies in a different universe from that where the curriculum lives, a superlunary realm inaccessible from such a small base of material.

It is just triangle geometry, in some sense no more than a clever elaboration of similar triangles and similar ideas. So it falls through the cracks, only to be learned by those precocious enough to be taught it in an Olympiad programme, or already interested enough to read Euclidean geometry for their own interest.

Anyway, the fact here is a fact about triangles. I’ll try not to get too snarky about the curriculum.

Triangles in the plane. Just the nice, flat, Euclidean, plane.

It’s a fact about centres of triangles. Triangles have centres.

But not one, oh no, no one. They have many centres. There is a whole world of triangle centres out there, none of which you are likely ever to see in any curriculum any time soon, being too far from mundane everyday applications. (Imagine if just one of these was included in a curriculum! Ah, we can dream. Imagine the problems you could solve, the proofs and abstractions and explorations you could do, if you even knew what these things were. Which you don’t.)

So yes, as it turns out there are many ways in which you might define the “centre” of a triangle. Here are three of the most standard ones. Let the triangle’s vertices be \( A,B,C \), so its sides are \( AB, BC \) and \( CA \).

  • The centroid. It’s often called G. This is what you get when you join each vertex to the midpoint of the opposite side. Those three lines, called medians, meet at this point G, called the centroid. It’s also the centre of mass of the triangle. (Oh what, a practical application? Who would have thought.)

  • The circumcentre. It’s often called O. As it turns out, there is a unique circle which passes through the three vertices A,B,C. A basic fact, again, which is not in any curriculum anywhere near me. This circle is called the circumcircle of ABC. (Well, it circumnavigates the vertices, doesn’t it?) Its centre O is called the circumcentre.

Circumcircle and circumcentre of a triangle

  • The orthocentre. It’s often called H. This is what you get when you drop a perpendicular from each vertex to the opposite side. Those three lines, called altitudes (because they measure the height of the vertex above each side), also meet at a point H. That point H is called the orthocentre.

(All diagrams public domain from wikipedia.)

Now, it’s not even clear that any of these points exist. (No, the pictures do not constitute proofs.) If you take three lines in the plane, they may or may not meet at a point. Usually they won’t. But in special cases they will. As it turns out, the three medians of a triangle are sufficiently special that they do meet at a point. As are the three altitudes.

Most likely any reader here will either know about all three of these centres, or about none of them. If you know all three, congratulations my friend. If not, let me simply say this about them:

  • The fact a centre of mass of a triangle exists shouldn’t be too surprising. And a median splits a triangle into two halves of equal area (same height, half of the base each), so it shouldn’t be too surprising that the centre of mass should lie on a median, hence on all the medians, so that it seems they at least ought to meet at a point. In fact, if you know vectors, then if we take an origin \( O \) and position vectors \( \vec{OA}, \vec{OB}, \vec{OC} \) for A,B,C, then the position vector of the centroid G is their average, \[ \vec{OG} = \frac{ \vec{OA} + \vec{OB} + \vec{OC} }{3} \].
  • The fact that you can put a circle through three points shouldn’t be too surprising. Given two points, you can draw many many circles through them, in fact so many that they can pass through any other point as well. Or, for a more misanthropic take on the matter, have you ever tried to keep equally away from 2 people? Well, I suppose it depends on how obnoxious or irritating are the people near you or wherever you happen to go, but in any case it’s not that hard, you just have to stay on a line bisecting those two people (it’s called a perpendicular bisector). Well, now take it a step further and suppose there are 3 people. It can be done. When you do that you intersect another perpendicular bisector and at their intersection, in fact all three perpendiculars have to meet, and that point is the same distance from all the people. You are now at the circumcentre and you can put a circle through all those who repel you. (Well, you’d likely feel trapped between all 3, and you’d probably end up running away, but then you won’t be equally far away from the repellent ones. Anyway…)
  • The fact that the orthocentre is much less obvious, at least to me, without using some other mathematical theorems. It is a bit surprising, without knowing mathematics. But no matter, because I’ll prove it as part of the gem I’m about to explain.

It’s not just the orthocentre. Mathematics is full of surprising, counterintuitive, even apparently ridiculous ideas and statements that turn out to be true. And the process of turning them from apparently ridiculous to understood — then even intuitive — ideas, is the idea of proof. Which you will never learn in a mathematics curriculum any time soon, except in the very uppermost advanced stream in the last year or two, if you’re lucky. Without the language of proof, there will be no words in which to express this process.

So, the centroid \( G \), the circumcentre \(O \), the orthocentre \( H \). Three very different but equally reasonable claimants to the title of *centre* of a circle. They are not the same. But as it turns out, amazingly, they always lie on a line. This was proved by Leonhard Euler in 1767, and the line is called the Euler line of the triangle. Drawing everything on one diagram results in a suitably impressive picture of this line, and hopefully gives some impression of how surprising it is. (Source: wikipedia.)

Euler line of triangle

There are many proofs of this fact. Many of them are very nice, using some combination of similar triangles, or transformation geometry. And as with all proofs of facts that are initially surprising, it converts a possibly surprising, possibly ridiculous statement into an incontrovertible, natural, and true one.

The proof I’m going to share uses vectors. So to understand it, you need to know something about vectors: including position vectors of points; addition, subtraction and scalar multiplication of vectors; and the dot product, and the fact that the dot product of two nonzero vectors is zero if and only if they’re perpendicular. If you don’t, oh well, go and learn it. You likely won’t be able to derive it yourself, even with your mad 21st century skillz.

Remember, \(G\) is the centroid, \(O\) is the circumcentre and \(H\) is the orthocentre. But I’m not even going to assume we even know the orthocentre exists yet.

Let’s set up some position vectors. We’ll take the circumcentre \(O\) as the origin. (After all, we often use \(O\) for the origin and \(O\) for the circumcentre. Why not both? Quite a convenient elision.) As \( O \) is the centre of a circle passing through \( A,B,C \) we have \( | \vec{OA} | = | \vec{OB} | = | \vec{OC} | \). And, as I mentioned above, as \(G\) is the centre of mass of the triangle, \( \vec{OG} = ( \vec{OA}+\vec{OB}+\vec{OC})/3 \).

We’re now ready to state a theorem.

Theorem: Let \(H\) be the point with position vector \( \vec{OH} = \vec{OA}+\vec{OB}+\vec{OC} \). Then the orthocentre of \( ABC \) exists and is equal to \(H\), and \( \vec{OH} = 3 \vec{OG} \).

This theorem tells us a lot. Firstly, that the orthocentre exists, i.e. that the three altitudes of ABC intersect. Secondly, that they intersect at this point \(H\) defined by \( \vec{OH} = \vec{OA}+\vec{OB}+\vec{OC} \). Thirdly, from \( \vec{OH} = 3 \vec{OG} \), recalling how scalar multiplication works, it tells us that \( O,H,G \) all lie on a line, with \( O, G, H \) in order along it, and with \( H \) 3 times as far away from \( O \) as \( G \).

In fact, the conclusion that \( \vec{OH} = 3 \vec{OG} \) isn’t really much of a conclusion. It is immediate from the way we defined \( H \). We defined \( H \) by \( \vec{OH} = \vec{OA}+\vec{OB}+\vec{OC} \). But since the centroid G satisfies \( \vec{OG}=( \vec{OA}+\vec{OB}+\vec{OC})/3 \), we observe that \( \vec{OH} \) is just 3 times \( \vec{OG} \). Of course 1 is 3 times 1/3…

So the thing to prove is that the orthocentre exists and is equal to \(H\), defined in this way. The proof is very short. And that’s what I mean when I say “one line Euler line”.

Proof: We show \(H\) lies on the altitude from \(A\) to \(BC\), i.e. that \(AH\) is perpendicular to \(BC\): \[\vec{AH}.\vec{BC} = (\vec{OH}-\vec{OA}).(\vec{OB}-\vec{OC}) = (\vec{OB}+\vec{OC}).(\vec{OB}-\vec{OC}) = |\vec{OB}|^2 – |\vec{OC}|^2 = 0. \] A similar argument shows that \(H\) lies on the other altitudes of \(ABC\), hence the altitudes intersect at \(H\).

By “one line” I mean the one line calculation above. (If indeed it fit on one line on your screen…) Let’s explain each step. First, \( \vec{AH} = \vec{OH}-\vec{OA} \); that’s how vector subtraction works. Second, we substituted \( \vec{OH} = \vec{OA}+\vec{OB}+\vec{OC} \), so that \( \vec{OH} – \vec{OA} = \vec{OB} + \vec{OC} \). Third, we expanded a dot product “difference of perfect squares”, and used the fact that the dot product of a vector with itself is the length squared of that vector. Finally, we used the fact that \(O\) is the circumcentre, so that \( |\vec{OB}|=|\vec{OC}| \).

If you’d like to make sure you understand this, try to do to the “similar” arguments that \(H\) lies on the other altitudes of \(ABC\). In other words, as an exercise, try to show that \( \vec{BH}.\vec{AC} = 0 \) and \( \vec{CH}.\vec{AB} = 0 \).

Well, as I say, this is all in some sense “standard knowledge”, just stuff which is inaccessible to pretty much every normal person since nothing remotely in the same universe is ever taught at school (and nor will it be any time soon anywhere near me). As I say I just came across and (re-)discovered it, and thought of it was worth sharing.

The fact that \( \vec{OH} =\vec{OA}+\vec{OB}+\vec{OC} \) is apparently known as Sylvester’s triangle problem, and what I’ve shown here is very similar to it; in essence it’s the same thing. This vector proof isn’t new, it appears in many places online and off. And you can find many other more standard type proofs, using similar triangles and such things, online and off. As for proving the existence of the orthocentre, cut the knot has a nice page of different proofs, including one along the lines above.

And there you have it. A one line Euler line. More or less. So to speak.

I would dearly love it if students could solve problems related to these ideas — or even much simpler versions thereof —  without knowing any geometry or the idea of proof beyond the pale shadow of which exists in our curriculum, simply by the application of “21st century problem solving skills”. Futuristic indeed they must be to derive everything from a few flawed lessons on proof and some drilling of ways to prove triangles congruent. But as it turns out, to do this sort of problem solving, the average person needs to learn some geometry first. You cannot expect the average student derive hundreds of years of mathematical progress and geometric ingenuity by themselves. With the words to say and the musical notes to play, you can sing your song and create your mathematical world. Once you have the ideas sorted out, you open up a whole universe of creativity, problem solving, a route to the stars, and you have a basis for learning 21st century problem solving skills of creativity, abstraction, precision, rationality, logic, powered by passion, insight, and intuition. But without them, what can you do but flail about. “Every year fewer and fewer words, and the range of consciousness always a little smaller.

Wouldn’t it be nice if ideas like these, or at an attenuated version, or at least something approximating them — or just something, something? — could at least make an appearance in our schools?

Ptolemy vs Thurston in Hyperbolic Geometry and Topology, AustMS 2020

On 9 December 2020 I gave a (virtual) talk in the Topology session of the 2020 meeting of the Australian Mathematical Society.

Title: Ptolemy vs Thurston in Hyperbolic Geometry and Topology.


Bill Thurston (1946-2012 CE) developed a system of great simplicity and power for understanding hyperbolic 3-manifolds. In particular, he introduced equations whose variables encode the shapes of ideal hyperbolic tetrahedra and whose solutions describe hyperbolic structures on 3-manifolds.

Claudius Ptolemy (c.100-170 CE), better known for developing a rather different system, proved in his Almagest an equation about the lengths of sides and diagonals in a cyclic quadrilateral. Such Ptolemy equations arise in numerous places across mathematics, including in 3-dimensional hyperbolic geometry and representation theory.

In this talk I’ll discuss how Ptolemy and Thurston equations provide complementary perspectives on hyperbolic geometry and topology.