Maths for the people!

Including introductory talks and articles for undergraduate students, outreach talks at schools, popular articles, and the occasional mathematical comic or poem.

Maths articles for the people
• 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.
• Summarise your maths research in one slide, Dan
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.
• One line Euler line
A fun fact from Euclidean geometry that I thought was a wonderful enough gem to share. It’s standard, but it’s nowhere near any curriculum. I’ll try not to get too snarky about the curriculum.
• From Here to Hensel
Here’s a nice maths problem, which I thought it would be fun to discuss. The question doesn’t involve any advanced concepts, but it leads on to a very nice result called Hensel’s lemma.
• Return of the Euler-Fermat theorem
A long long time ago, in a galaxy far away, I wrote up an account of the Euler-Fermat theorem for school students.
• Sitting out the math wars
Very few professional mathematicians have been involved in the “math wars”, and when they have, they have not always inspired confidence. I wondered why.
• 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. But there should also be others.
• Breakthroughs in primary school arithmetic
Humans have known how to multiply natural numbers for a long time. In primary school you learn how to multiply numbers using an algorithm which is often called long multiplication, but it’s called “long” for a reason! Recently, a new paper purports to give an algorithm to multiply faster.
• From Liouville geometry to contact geometry
(Technical) We’re going to take Liouville structures and move them into 3 dimensions, to obtain contact structures.
• Lovely Liouville geometry
(Technical) I’d like to show you some very nice geometry, involving some vector fields and differential forms.
• Emmy had a theorem (mathematical nursery rhyme #2)
In the spirit of previous work in abstract algebra, I have, erm, adapted another nursery rhyme. To the tune of “Mary had a little lamb”, a discussion of Noether’s theorem.
• Golay Golay Golay (Top of the autocorrelation world)
In 1949, Marcel Golay was thinking about spectrometry. Here’s what happened next…
• Topological entropy: information in the limit of perfect eyesight
Entropy means many different things in different contexts, but there is a wonderful notion of entropy which is purely topological. It only requires a space, and a map on it. It is independent of geometry, or any other arbitrary features — it is a purely intrinsic concept. This notion is known as topological entropy.
• Abstract algebra nursery rhyme
In the spirit of hilariously advanced baby books like Chris Ferrie’s Quantum Physics for Babies, I have taken to incorporating absurdly sophisticated concepts into nursery rhymes.
• Limitless as that space too narrow for its inspirations
In which I recall, via neurologist Oliver Sacks, some musings of Sylvester from 1877 on the limitlessness of mathematics.
• The Brain makes Contact with Contact Geometry
It’s always nice, intellectually, when two apparently unrelated areas collide. I had an experience of this sort recently with an area of mathematics — one very familiar to me — and an ostensibly completely distinct area of science.
• “The beauty of mathematics shows itself to patient followers” — The work of Maryam Mirzakhani
The recent passing of Maryam Mirzakhani came as a shock to many of us in the world of mathematics. Together with Norman Do, we attempt to share something about Mirzakhani’s work.
• Tutte meets Homfly
I’ll tell you about some extremely clever methods to tell graphs and knots apart, involving polynomials: the Tutte and HOMFLY polynomials. And they’re closely related.
• Holy h-principle, Batman!
In which I attempt to explain some of the ideas behind the h-principle.
• Every world in a grain of sand: John Nash’s astonishing geometry
After the recent tragic death of John Forbes Nash Jr, many tributes have been paid to this great mathematician, who was made famous by the movie “A Beautiful Mind”, and much has been said about his work on game theory. But less has been said about Nash’s other mathematical achievements.
• Paranoid defence controls could criminalise teaching encryption
You might not think that an academic computer science course could be classified as an export of military technology. But under the Defence Trade Controls Act – which passed into law in April, and will come into force next year – there is a real possibility that even seemingly innocuous educational and research activities could fall foul of Australian defence export control laws.
• Why your calculator (and computer, and phone…) is a weapon
The Australian government may have classified your calculator — and phone, and computer, and every electronic device you own — as military weapons.
• The Lost Art of Integration Impossibility
Integration is less a science and more an art form. It high time we shed some light on this lost art.
• Hyperbolic links
A set of links I collated about hyperbolic geometry.
• Quadratic geography, algebraic extreme sports and magical Farey trees
Published in the Australian Mathematical Society Gazette “Mathellaneous” column.
• Games with Galois
In 2004 for the Mathellaneous column of the AustMS Gazette, I wrote an article about games and Galois theory.
• A Beautiful Sequence
In 2004, I wrote a recreational article for the Australian Mathematical Society Gazette about one of my favourite sequences.
• The Exotic Realm of p-adic Numbers
A perhaps not-so-polished article I wrote for Paradox, the magazine of MUMS, the Melbourne University Mathematics and Statistics Society, as an undergraduate student. A very simple very brief introduction to p-adic numbers.
• Some Mathematicians Like it Hot: Fourier and Descartes
A perhaps not-so-polished article I wrote for Paradox, the magazine of MUMS, the Melbourne University Mathematics and Statistics Society, as an undergraduate student.
• Knot Man!
Of course the most important part of this webpage is the part devoted to mathematical superhero Knot Man. Known to others as Theodore J. Knott, in times of mathematical emergency, with his topological utility belt and supply of high-energy genus-1 donuts, he becomes Knot Man, Defender of the Mathematical Universe, saving the world from all manner of crazed physicists and economists! Illustrated by Priscilla Brown.
• Adventures with Pascal’s triangle and binary numbers
An article I wrote for Paradox, the magazine of MUMS, the Melbourne University Mathematics and Statistics Society, as an undergraduate student.
• Mathematical haiku
As an undergraduate student, I wrote some mathematical haiku.
Maths talks for the people
• “There is always room for a new theory in mathematics”: Maryna Viazovska and her mathematics
In August 2022 I gave a talk about recent Fields medallist Prof Maryna Viazovska and some of her mathematical work. This was a Monash LunchMaths seminar.
• May 12 talk on Maryam Mirzakhani
On May 12, 2022, I gave a talk at Monash for undergraduate students on some of the mathematics of Maryam Mirzakhani.
• 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 ...
• I got problems – congruence problems
On 7 December 2020 I gave a (virtual) lecture at the Australian Mathematical Olympiad Committee’s School of Excellence on congruences.
• Topology: The shape of space
Monash Open Day in 2020 was a purely online affair, thanks to COVID. I recorded a video talking about Topology: The Shape of Space.
• “I liked doing what I wasn’t supposed to do”: the life and mathematics of Karen Uhlenbeck
In September 2019 I gave a talk about the life and some of the mathematics of Karen Uhlenbeck, the great mathematician and first woman to win an Abel Prize. This was a Monash LunchMaths seminar.
• The beauty of mathematics shows itself to patient followers
In September 2018 I gave a talk on the life and mathematics of Maryam Mirzakhani in the School of Physical and Mathematical Sciences colloquium at NTU in Singapore.
• 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.
• Some pure mathematics and consciousness
In November 2017 I gave a talk to the Monash Consciousness Research Laboratory (Tsuchiya Lab). I talked about some pure mathematical ideas that have appeared in the literature on the frontiers of neuroscience and the study of consciousness — gauge theory, and category theory.
• Sciencey: Why do earphones get tangled?
An appearance in Sciencey, a new series from ABC that delivers quick, illuminating answers to some of the strangest questions in the universe. Why do earphones always tangle, and what does this tells us about the universe?
• The work of Maryam Mirzakhani
In August 2017 I gave a talk on some of the mathematics of Maryam Mirzakhani, the great Iranian mathematician, first female Fields Medallist. This was a Monash LunchMaths seminar.
• Maths week talk at PLC
I gave an outreach talk to secondary students at PLC about mathematics and mathematicians, talking about, among other things, topology, Maryam Mirzakhani, and billiards, in July 2017. Slides are available.
• Mathematics, mathematicians, philosophy
In December 2016 I gave a talk to secondary school students about mathematics and mathematical philosophy.
• The story of a paradox
My story of Bertrand Russell, given at The Laborastory, a monthly science storytelling event in Melbourne.
• Riddle. Mystery. Enigma.
I appear in a rather excellent and fun episode of the ABC Radio National program Radiotonic.
• Why your calculator is a weapon
I gave a talk about the Defence Trade Cooperation Act, encryption, and number theory, as part of the Monash University LunchMaths seminar series, in August 2015. Slides are available.
• To mathematics champions
In November 2014, I spoke at the Victorian prize ceremony for the Australian Mathematics Competition.
• LunchMaths topology, Aug 2014
I gave an introductory talk on Topology, as part of the Monash University LunchMaths seminar series, in August 2014. Slides are available.
• LunchMaths hyperbolic geometry, Sep 2013
I gave an introductory talk on Hyperbolic Geometry, as part of the Monash University LunchMaths seminar series, in September 2013. Slides are available.