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Thick Calculus textbooks finally come in useful!

June 17, 2015

Who needs a standing desk, after the publishers learn you teach Calculus to 1st-years!


Pursue Impossible

May 13, 2015

Dad, what does that sign mean?

Pursue Impossible

Pursue Impossible

This is what my 9-year old daughter asked when she saw these signs adorning a wall on campus when walking to her music class on Saturday. Without waiting for an answer from me, she immediately gave her opinion:

It doesn’t make any sense – and it’s not even proper English!

She’s right of course – what on earth does it mean?  Pursue the impossible? Pursue impossible dreams? And why are the campus walls decorated with this ungrammatical imperative? And what are we meant to do if we acquiesce – start attempting to square the circle or find a 5-chromatic planar graph? In mathematics at least, pursuing the impossible is not such a great idea.

Then I remembered seeing some people installing various objects in different places about campus last week. At the time I had assumed that this was the output of an art project, but now I went for a closer look. The first thing I saw was a collection of parallel metal plates on the lawn with a sign saying “Take Photo Here”.  IMG_0085So I did, and when I took the photo from the recommended place, the plates lined up and I got a picture of John Winthrop Hackett, who was UWA’s inaugural chancellor.

IMG_0086Because my first view of the structure (sculpture?) was at almost the same angle as the “correct” viewing angle, there wasn’t all that much difference between the views. The next one was more interesting though – again a set of parallel plates, and obviously a person, but who?

IMG_7429 Even poking my camera through the hole in the “take photo here” spot, I couldn’t line it up perfectly, but managed to get a recognisable face.

IMG_7431The photo is of Barry Marshall, UWA’s one and only Nobel Laureate (shared prize with Robin Warren) and he really does have one of the great stories of scientific persistence triumphing against the odds. While working as a young internist, he noticed high concentrations of the bacterium Helicobacter Pylori in many of the biopsies he performed on tissue from people with stomach ulcers and other stomach problems, and he developed the theory that stomach ulcers might actually be caused by the bacterium. At that time, stomach ulcers were universally regarded as arising from excessive stomach acid caused by stress and diet. The medical establishment greeted his theory with ignorance, indifference or condemnation.  After all, who would pay any attention to this maverick whose nonsensical theories about bacteria living in the stomach causing ulcers were obviously so wildly misguided that they should be summarily rejected.

One of the important and most dramatic steps that he took to prove that he was right after all was to create a Helicobacter Pylori broth, and drink it himself, while documenting the survival of the bacteria in the stomach (deemed impossible) and the almost immediate onset of a number of stomach complaints. The ultimate consequence of his persistence, indeed intransigence, is that stomach ulcers have been transformed from a painful, often-chronic condition making life miserable for tens of millions of people to an easily-treatable condition just requiring a short course of antibiotics. Occasionally I see a car driving round Nedlands with the vanity plates H PYLORI on it.

By now,  it’s dawning on me that this is not some final year project in Visual Arts, but is actually the unveiling of the University’s new branding – yes, our new “slogan” or “motto” or whatever you call it is actually “Pursue Impossible”. I didn’t attend the meeting at which this branding was unveiled and explained, so I had to fill in the gaps myself. The installations are all about how by looking at things in just the right way, the impossible becomes possible. To my mind, the sentiment is fine, but not immediately apparent from “Pursue Impossible”.

Also some of the installations don’t seem to really be conveying the right idea. In this one, the word POSSIBLE on our staff club wall becomes IMPOSSIBLE if you manoeuvre the letters IM (sitting on a plinth some distance away) into place. So by looking at it in the recommended way, the possible becomes impossible! Hurray!


So what do we all think of our new motto? Everyone that I’ve spoken to has reacted with either disbelief, bewilderment or derision, but I’m not sure which is winning at the moment. I don’t know if anybody (other than the marketing firm that pocketed the cash) likes it, but if they do, they haven’t told me.

Our previous but now-outdated motto was “Achieve International Excellence” which is pretty clunky but at least the intent is clear. Even earlier we had a much more succinct motto with which surely no-one can disagree  – “Seek Wisdom” – and to which I think we should return, if we really think a motto is important. But actually, what is the point of a university motto/tagline at all? Do students choose universities based on the motto? Is the motto intended to convey to the public some deeply held core value? If so, should it really be chosen by some marketing consultant?

It seems endemic though, because almost everywhere has a motto – for example, both of my daughters’ schools have mottos, one of them is Semper Altius and the other Savoir C’est Pouvoir.  (Obviously, a motto in another language automatically has more gravitas than one in English.) The good news though is that we’re not the worst – when I went to UNSW a few years ago, I was surprised to be continually urged to Never Stand Still, which is their motto. To me, it always brings to mind the image of someone standing giving a lecture while hopping frantically from foot to foot, as though desperate for a pee.

I do have to feel a bit sorry for the VC though. Shortly after unveiling the “Pursue Impossible” brand, he was forced by UWA staff and students to reluctantly return $4m (to the government) that had been given to UWA to set up a think-tank analysing problems in the developing world and making recommendations on which development projects give the best bang for the buck. The catch was that the proposed centre was to be associated with Bjorn Lomborg, a Danish political scientist who performs “Freakonomics” style analysis of environmental and developmental issues, often  with counter-intuitive and/or controversial results.

But UWA staff spoke loud and clear – they absolutely do not want to be associated in any way with this maverick whose nonsensical theories are obviously so wildly misguided that they should be summarily rejected.

Aha, at last I’ve got it – the motto that we should have if they were subject to truth-in-advertising rules.

UWA – you just can’t make this stuff up!

An interesting formula in combinatorial representation theory

January 6, 2015

Some research projects give you more than others in terms of reward and enjoyment. I’ve been very lucky to have been part of an enthusiastic team in Stephen Glasby, Luke Morgan, and Alice Niemeyer on a problem concerning automorphisms of p-groups; but more of that in a later post. In our investigations, we needed to know some basic data on polynomial representations of the general linear group GL(d,p). Consider the natural action of GL(d,p) on the tensor power T^n V where V is the vector space \mathbb{F}_p^d. It is a well-known result of Schur (and I won’t elaborate on it here) that if p>n, then we can parameterise the irreducible GL(d,p)-modules by the partitions of n. (Well actually, the characteristic 0 analogue is due to Schur, but it was folklore for a long time until a paper of Benson and Doty). For example, let us take the tensor square (n=2). If p is odd, then it is a classical fact that T^2 V breaks up into two smaller GL(d,p)-modules, namely

T^2 V \cong S^2 V \oplus A^2 V

where S^2 V is the symmetric square of V and A^2 V is the alternating square of V. The partitions here are the trivial partitions of the number 2. The partition (1,1) corresponds to A^2V and the partition (2,0) corresponds to S^2 V.

We are interested in something slightly more difficult. We actually want to know the irreducible constituents of the free Lie algebra L(V) generated by V. The connection between the two settings is cute. Define a bracket operation on the tensor algebra T(V) by [u,v] = u\otimes v-v\otimes u. Then we obtain a graded Lie algebra L(V)=\bigoplus_{n=1}^\infty L_n where each L_n is just L(V)\cap T^n(V). Each L_n then breaks up into irreducibles indexed by partitions of n, except we don’t know the multiplicities of each submodule. Write V^\lambda for the GL(V)-module corresponding to a partition \lambda of n. Note that the multiplicities here could be 0, that is, the V^\lambda  may not even appear. So think of the partitions now as being a superset of parameterising objects. For n=2, it is well known that L_2(V)\cong A^2V, that is, the symmetric square vanishes (just think of what the Lie bracket does here!).

Well I came across a beautiful result of Kraśkiewicz and Weyman that yields the answer. Consider standard tableaux of type \lambda. As an example, consider the partition (2,2,1,1) of the number 6. Then there are 9 possible standard tableaux:

tableauxA descent is an entry i of a standard tableaux such that i+1 appears in a row lower than it. So for example, 2, 4, and 5 are descents of the first tableau above. The major index of a standard tableaux is the sum of its descents. So for our nine tableaux we have major indices 11, 10, 9, 13, 9, 8, 12, 7, 11 accordingly. Then the magic number for the multiplicity of a V^\lambda is the number of standard tableaux of type \lambda that have major index congruent to 1 modulo n. In our example, there are two such young tableaux, the 4th and 8th one. So V^\lambda has multiplicity 2.

Purple patch for Michael (and Gabriel)

November 18, 2014

Congratulations are due to Michael who is currently in a purple patch riding a wave of recognition and achievement over the last two or three weeks. (I had look up why it’s called a purple patch and not a green patch or a chartreuse patch etc.)

First, we heard that his latest ARC Discovery Grant application (along with Li and Gabriel) has been successful for a project involving symmetries of directed graphs, about which much less is known than for undirected graphs.

Secondly, he won a UWA Research Award in the “mid-career” category – these are internal awards across the whole university designed to recognise excellence in research; Gabriel also won the same award, but in the “early-career” category.

Then to top things off, we heard yesterday that his promotion to “something” was granted. Unfortunately it is not entirely clear what “something” should be. UWA used to use the British system of “lecturer/senior lecturer/associate professor/professor”, with only a handful of people in latter category. In other words, “Professor” is quite prestigious and only a few people will reach that level.

Then a few years ago, we decided to change to the US system of “assistant professor / associate professor / professor”. I’m not entirely sure why this change was made, but I think that it was partially to allow UWA to attract strong American academics (looking for jobs due to the hiring freeze in the US during the GFC) for whom dropping down to an “associate professor” would be seen as a backward step, and generally to align ourselves more with the US nomenclature than with the UK.

But then the local medical research funding body, the NHMRC, said that anyone who called themselves a professor would be evaluated as though they were an old-style prestigious professor, suddenly making some perfectly good research records look relatively mediocre.

So the edict has come down to, well, we don’t actually know what we’re meant to do. New jobs are to be advertised under the old titles, but holders of old jobs who chose to go to the new titles can continue to use the new titles or revert to the old titles depending on their preference or the day of the week or something. At least, I think I’ve got that right.

Anyway, there’s one thing that is clear. Congratulations to Michael on his promotion to Level D, his ARC Discovery Grant and his UWA Research Award. And to Gabriel for the latter two.

Is there a 5-chromatic cubelike graph?

November 9, 2014

I’m in Singapore at the moment, spending the weekend watching my 11-year old daughter Amy in a gymnastics competition and the weekdays visiting Dave Roberson (currently at NUS) and Fengming Dong (currently at NTU).

Meeting up with Dave and thinking once again about the annoyingly resistant conjecture that the core of a cubelike graph is cubelike, reminded me of another “cubelike” question that is still unresolved.

First some terminology: a cubelike graph is a Cayley graph for the elementary abelian 2-group Z_2^n; the term “cubelike” arises for two reasons:

  1. The n-dimensional cube is cubelike
  2. If you talk, or write, anything about these graphs, you rapidly need something snappier for the phrase “a Cayley graph for an elementary abelian 2-group”

It has been known for decades that a cubelike graph cannot have chromatic number 3, and this was proved in a beautifully elegant fashion by Payan. A very ambitious conjecture that all cubelike graphs have chromatic number equal to a power of 2 can be disproved by exhibiting a cubelike graph on 16 vertices with chromatic number 7, and cubelike graphs on 64 vertices with chromatic number 6.

But what about 5?

I simply cannot find any cubelike graph with chromatic number 5, and according to Brouwer’s website, this is unknown. Chris Godsil and I thought about this a few years ago, and someone said they had heard someone tell someone else that they had heard that perhaps some Russians had found such a graph on 128 vertices.

So, what’s the problem? Why not just construct all 128-vertex cubelike graphs and check their chromatic number?

The trouble with this is that the number of cubelike graphs grows very rapidly. There are only 1372 such graphs on 32 vertices, and we can find all the chromatic numbers. This jumps to 475499108 on 64 vertices (actually a handful less, this overcounts the number by about 10, due to some “accidental” isomorphisms), and although I don’t know the exact chromatic number of all of these, I can rule out enough of them as potential 5-chromatic graphs to complete the search.

But on 128 vertices, we have 1038397981840994509577948 graphs to work through (that’s just about 10^{30} and so unless we stumble on one somehow (perhaps someone knows who “the Russians” are) or make some theoretical advance, this problem is likely to remain unresolved for the time being.

Peace and quiet

November 4, 2014

In a colleague’s research grant proposal, under a heading about resources and equipment, he wrote something along the lines of:

All a mathematician needs is some paper, some pens, good access to online journals, and most importantly, a quiet place to work.

Of course, we also rely on good coffee, a buzzing environment of enthusiastic colleagues, natural light, and administrative support.

For other disciplines, lab equipment and technicians are extremely important, and not having the best equipment would severely cripple an experimental chemist, for example. So if you needed to find a way to stifle the progress of a group of mathematicians, what changes to their work environment would you make?

  • You could take their blackboards/whiteboards away, but they might be just as happy with their endless supply of foolscap paper.
  • You could take their paper away. This is difficult to do, as paper is easy to buy and very cheap. Mathematicians would bring their own paper, or just write on their desks and other flat surfaces.
  • You could take their pens away. Again, pens are cheap and easy to buy, so it would be difficult to outlaw pens in the workplace. And anyway, where would you draw the line? Pencils, chalk, and crayons would have to be outlawed too.

Perhaps the best way is to create tension within the group, and somehow put in place a situation where the ambient noise in the workplace was disruptive and unpredictable. For those who wear headphones, they would need to be distracted by visible movement in their periphery. How can this be done effectively? Even more insidiously, we could create an environment that would increase the rate of infection due to colds, influenza, or other airborne viruses. Have you tried to solve a difficult mathematics problem whilst your head is congested and your joints feel like jelly?

So I leave the question to you: how can you create a work environment that not even a resource-minimalist mathematician can bear? To summarise:

  1. it needs to organically create tension between colleagues,
  2. it needs to be cost efficient,
  3. it needs to be noisy,
  4. it needs to have visual distractions,
  5. it needs to foster airborne viruses.

I can only think of one solution to this problem. What is your solution?

Some Tikz pictures

October 9, 2014

Most LaTeX-ers know about Tikz, which allows the user to create images in LaTeX without having to embed images created from an external program. The main advantages are that

  1. The ambient LaTeX fonts are used in the image, so labels and such conform to the ambient style of the document.
  2. The size of the .tex file is kept small, since it is only text you are creating.
  3. It yields a picture that is smooth and that looks good upon zooming in (i.e., the resolution of the picture is good).
  4. It is functional code so that you can automate the drawing of many pictures by giving commands such as “draw a line between these two points”.

The main disadvantage, is that there is a steep learning curve. The best way to learn is through examples, and even though I’m still a hack, my tikz code has improved via my copying segments of other people’s code. For geometry, there isn’t much out there, so I thought that I would dump some images here. Below are some Tikz pictures of configurations in finite geometry that I’ve collected and think should be available for everyone else to use. A big thanks to Stephen Glasby who went to a lot of trouble to make the two generalised hexagons of order 2. If you have suggestions on how I can simplify my code, please let me know.

Desargues’ configuration, two ways


\tikzstyle{point1}=[ball color=cyan, circle, draw=black, inner sep=0.1cm]
\tikzstyle{point2}=[ball color=green, circle, draw=black, inner sep=0.1cm]
\tikzstyle{point3}=[ball color=red, circle, draw=black, inner sep=0.1cm]
\node (v1) at (0,8) [ball color=blue, circle, draw=black, inner sep=0.1cm] {};
\node (v2) at (0,6) [point1] {};
\node (v3) at (2,5.5) [point1] {};
\node (v4) at (1.5,4) [point1] {};
\node (v5) at (0,0) [point2] {};
\node (v6) at (2.75*2,8-2.75*2.5) [point2] {};
\node (v7) at (1.5*1.5,8-1.5*4) [point2] {};
\draw (v1) -- (v2) -- (v5);
\draw (v1) -- (v3) -- (v6);
\draw (v1) -- (v4) -- (v7);
\draw (v2) -- (v3) -- (v4) -- (v2);
\draw (v5) -- (v6) -- (v7) -- (v5);
\node (v8) at (intersection of v2--v3 and v5--v6) [point3] {};
\node (v9) at (intersection of v2--v4 and v5--v7) [point3] {};
\node (v10) at (intersection of v3--v4 and v6--v7) [point3] {};
\draw (v3) -- (v8) -- (v6);
\draw (v4) -- (v9) -- (v7);
\draw (v4) -- (v10) -- (v7);
\draw (v8) -- (v9) -- (v10);

… and the second one:

\tikzstyle{point} = [ball color=black, circle,  draw=black, inner sep=0.1cm]
\foreach\x in {0, 72, 144, 216, 288}{
      \coordinate (o1) at (-0.588, -0.809);
      \coordinate (o2) at (0.588, -0.809);
      \coordinate (c1) at (-1.1, 4.6);
      \coordinate (c2) at (1.1, 4.6);
      \coordinate (o3) at (0, 3.236);
      \draw[color=black] (o3) -- (3.236*-0.588, 3.236*-0.809);
      \draw[color=blue] (o1) ..  controls (c1) and (c2) ..  (o2);
\foreach\x in {0, 72, 144, 216, 288}{
      \coordinate (o2) at (0.588, -0.809);
      \coordinate (o3) at (0, 3.236);
      \fill[point] (o2) circle (2pt);
      \fill[point] (o3) circle (2pt);

The generalised quadrangle of order 2

I think Gordon gave me the original tikz code for this and then I tweaked it.

\tikzstyle{point}=[ball color=magenta, circle, draw=black, inner sep=0.1cm]
\foreach \x in {18,90,...,306}{
\node [point] (t\x) at (\x:2.65){};
\foreach \x in {54,126,...,342}{
\draw [color=blue, double=green](\x:1cm) circle (1.17557cm);
\fill [white] (0,0) circle (1cm);
\foreach \x in {54,126,...,342}{
\node[point] (i\x) at (\x:1cm) {};
\node[point] (o\x) at (\x:2.17557cm) {};

\draw [color=blue,double=green] (t90)--(o126)--(t162)--(o198)--(t234)--(o270)--(t306)--(o342)--(t18)--(o54)--(t90);
\draw (t90)--(i270)--(o270);
\draw (t162)--(i342)--(o342);
\draw (t234)--(i54)--(o54);
\draw (t306)--(i126)--(o126);
\draw (t18)--(i198)--(o198);

The two generalised hexagons of order 2

These pictures were originally drawn by Schroth in his 1999 paper, and then appeared in Burkard Polster’s book “A geometrical picture book”.


    \foreach\n in {0, 1,..., 6}{
        \coordinate (a0) at (10,0);
        \coordinate (b0) at (7,0);
        \coordinate (c0) at (1.45,0);
        \coordinate (d0) at (4.878,-0.4878);
        \coordinate (e0) at (2.1729,0.37976);
        \coordinate (f0) at (1.45,0.612);
        \coordinate (g0) at (2.78,-0.585);
        \coordinate (h0) at (4.074,0.7846);
        \coordinate (i0) at (6.0976,2.9268);
        \foreach\k in {1, 2,..., 6}{
          \foreach\p in {a,b,c,d,e,f,g,h,i}{
            \coordinate (\p\k) at ($ (0,0)!1! \k*51.4286:(\p0) $);
        \draw[thick,blue] (a0)--(b0)--(c0);
        \draw[thick,blue] (d0)--(e0)--(f0);
        \draw[thick,black] (g0)--(h0)--(i0);
        \draw[thick,black] (a0)--(g1)--(a3);
        \draw[thick,green] (i0)--(d1)--(i2);
        \draw[thick,black] (c0)--(g3)--(d3);
        \draw[thick,color=purple] (b0) .. controls (5.7,-1.8) and (4.6,-2.2)
           .. (h6) .. controls (1.5,-3.2) and (-1,-2.4) .. (e4);
        \draw[thick,black] (b0)--(h0)--(e2);
        \draw[thick,purple] (f6)--(c0)--(f0);
        \foreach\k in {1, 2,..., 6}{
          \foreach\p in {a,b,c,d,e,f,g,h,i}{
            \draw[fill] (\p\k) circle [radius=0.12];



  \foreach\n in {0, 1,..., 6}{
        \coordinate (a0) at (85,0);
        \coordinate (b0) at (55,0);
        \coordinate (c0) at (12.5,0);
        \coordinate (d0) at (8.5,1.3);
        \coordinate (e0) at (16.2,9);
        \coordinate (f0) at (30,14.3);
        \coordinate (g0) at (26.6,17.0);
        \coordinate (h0) at (26.3,22.4);
        \coordinate (i0) at (29.5,28);
        \foreach\k in {1, 2,..., 6}{
          \foreach\p in {a,b,c,d,e,f,g,h,i}{
            \coordinate (\p\k) at ($ (0,0)!1! \k*51.4286:(\p0) $);
        \draw[thick,black] (a0)--(e1)--(a3);
        \draw[thick,green] (b0)--(h0)--(b2);
        \draw[thick,purple] (f0)--(g0)--(f1);
        \draw[thick,blue] (h0)--(i0)--(a1);
        \draw[thick,purple] (h6)--(c0)--(d0);
        \draw[thick,purple] (f0)--(e0)--(i3);
        \draw[thick,black] (b0)--(i6)--(g6);
        \draw[thick,color=blue] (g0) .. controls (27,2) and (17,-5)
           .. (d6) .. controls (-7,-5) and (-5,-5) .. (c3);
        \draw[thick,color=blue] (c0) .. controls (16,3) and (17,5)
           .. (e0) .. controls (15,13) and (8,15) .. (d1);
        \foreach\k in {1, 2,..., 6}{
          \foreach\p in {a,b,c,d,e,f,g,h,i}{
            \draw[fill] (\p\k) circle [radius=1.1];

The first is the Split Cayley hexagon as it is usually given, whilst the second is its dual.


The projective plane of order 2


\tikzstyle{point}=[ball color=cyan, circle, draw=black, inner sep=0.1cm]
\node (v7) at (0,0) [point] {};
\draw (0,0) circle (1cm);
\node (v1) at (90:2cm) [point] {};
\node (v2) at (210:2cm) [point] {};
\node (v4) at (330:2cm) [point] {};
\node (v3) at (150:1cm) [point] {};
\node (v6) at (270:1cm) [point] {};
\node (v5) at (30:1cm) [point] {};
\draw (v1) -- (v3) -- (v2);
\draw (v2) -- (v6) -- (v4);
\draw (v4) -- (v5) -- (v1);
\draw (v3) -- (v7) -- (v4);
\draw (v5) -- (v7) -- (v2);
\draw (v6) -- (v7) -- (v1);

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