The 51st annual meeting of the Australian Mathematical Society (AustMS) was held this week, and ran from Tuesday until today. It was held at Latrobe (Bundoora Campus) northeast of Melbourne, and luckily close (30 mins by car) to my house. The meeting included some plenary talks by some notable figures including the delightful Paul Baum (of K-theory fame), Tony Chan (the Assistant Director for Mathematics and Physical Sciences (MPS) at the NSF), Joe Monaghan (of SPH fame), and Mark Kisin (the 2007 Mahler Lecturer [1]).
There were many other talks on a range of topics, including: Algebra, Combinatorics, Computational Mathematics, Number Theory, and Dynamical Systems, just to name a few. The different sessions all ran concurrently, which made it hard to select which talks to attend. Being interested in many of these disciplines I made a broad selection and have included some (non-technical) comments from the interesting talks here. Graham, Kerry, Rebecca, Judy, Doug, and Ian from the Combinatorics group at Monash (of which I lurk around) also gave talks – however I saw some previous versions of these talks and so did not attend their final versions.
Tuesday 12.00pm
Subtle Sums and Curious Cubes
Alfred van der Poorten
Alfred van der Poorten is an extremely jovial number theoretician and is inspiring be around. His talk [2] was on the problem of finding numbers X and Y such that X^2 + Y^2 has digits that are the concatenation of the digits of X and Y. A simple example is 12^2 + 33^2 = 1233. He noted a few ways of constructing infinite families of these numbers, and commented on work by some students of his on numbers such as {1,5,3} (where 1^2+5^3+3^3 = 153).
An amusing comment of Alf’s was the “oBdA” should replace “w.l.o.g” in the mathematical literature, as the latter contains ambiguity (w = “with” or “without”) and is too similar to “log”, whereas the former is quite precise in what it means. What precisely it means … I have no idea, but it’s German. :)
Another amusing anecdote of his was a conversation he had with John Conway about Conway’s extraordinary ability to quickly factorise 3 digit numbers.
Alf: “How do you factorise so quickly…? I mean, there are 168 primes less than a thousand…”
John (with a devious smile): “Come on Alf, you wouldn’t have any trouble memorising 168 numbers would you?”
Anyway, it turns out that John simply remembers the 100 non-trivial composites less than one thousand, where a non-trivial composite is a composite not divisible by 2,3,5,10, etc. (i.e., eliminating all the cases where it is quick to test compositeness.) Alf then recommended that a good exercise (during a boring lecture for example) is to work out these non-trivial composites, remember them, and then factorise away, annoying your friends in the process!
11am Wednesday
Geometrical Numerical Integration of Differential Equations
Reinout Quispel
This talk by Quispel was interesting and covered the so-called “Geometric Methods” (formerly known as “Structure-Preserving Methods” of numerical integration of DEs. These methods are for particular classes of equations, and preserve some important property or properties of the system.
A simple example is the autonomous equation:
dx/dt = f(x),
which (if f(x) is divergence free) has an exact solution which is volume-preserving. Hence it is ideal to use a volume-preserving integrator to solve it rather than a general purpose integrator. Later on Quispel gave three examples of geometric methods: the familiar implicit mid-point method preserves symplectic(?) structure, a variant of the mid-point rule (a so-called B-Series method) preserved energy, and finally a splitting method preserved volume.
An interesting theorem he stated was that, in general, there is no method that preserves energy AND symplectic structure. Great, another one of those interesting, yet ultimately harmful, (meta-)mathematical results…!
Reinout also pointed to the theory of representing elementary differentials by rooted trees (by Cayley) … something that seems quite interesting, I will have to follow this up.
12pm Wednesday
Particle Methods in Fluid Dynamics
Joe Monaghan
Joe’s been at Monash for quite a while apparently. I think I have seen him around occasionally, though I never realised that he was one of the first people involved in the formulation of Smoothed Particle Hydrodynamics (SPH), which I was exposed to a bit during my work at CSIRO. I’m unsure of the details, but basically it is a particle method for solving PDE’s for fluid, and granulated material. It is used for various simulations (e.g., testing the sloshing of liquid in an accelerating oil sump at CSIRO), for visual effects [3], and I guess for many other engineering/scientific research domains.
Joe is involved in much research with SPH, the latest of which involves the investigation of articulated bodies swimming in simulated fluid environments. He presented some interesting movies of fish-like creatures swimming around a sea of particles, one of which involved a fish surfacing and splashing around.
12pm Thursday
An introduction to K-Theory
Paul Baum
The title of this talk is misleading and by the 6th slide or so I was completely lost. If you don’t know what a fibre bundle, sheaf or germ is, then you would have been lost too. :) I have come across fibre bundles back in my complex analysis / differential geometry education, but I never made it to germs and sheaves. K-theory is a framework for understanding … wait … I actually have no idea what it is. I know what it can be used for though…
Imagine you have a sphere covered in hair. It is impossible to comb the hair of the ball such that there are no discontinuities. If the ball was a 4-dimensional sphere however, then it IS possible. K-theory helps us prove (and understand) this phenomenon.
The talk was too technical for me to follow, however Paul was amusing, and it was great to see him in person.
5pm Thursday
Discrete & Continuous Dynamical Systems
Chris Tisdell
Chris’ talk was on a generalised calculus that can deal with continuous and discrete time domains. This means the differential equations and difference equations can be treated simultaneously in the same system. He motivated the development of such a calculus with the problem from population biology of non-overlapping generations. This is a problem that involves continuous time scales (the dynamics of a population) with periodic discrete jumps (the example given was an organism that lays eggs that don’t hatch for a length of time, resulting in discontinuous jumps in the population).
The theory goes back to Stefan Hilger in 1988 and has actively being developed only recently. Chris commented on the existence of a fundamental theorem of calculus for this new calculus, as well as the existence of a generalised LaPlace-type transform that has as special cases the ordinary LaPlace transform and the z-transform. I’m interesting in keeping an eye on the development of this.
I attended many more talks, however they weren’t as interesting/accessible as these that I have commented on. My notes for these talks are covered in little sketches — a tell-tale sign of my wandering mind.
Overall the week went well. The Bundoora campus is very nice, and is surrounding by two moats … one water-based, and the other car-park-based. The river seems well tended and there seemed to be families of ducks and ducklings everywhere. I should now really get back to my own, non-mathematical, study….
References and Further Links
1. http://www.austms.org.au/AMSInfo/Medal/mahler.html
2. Which can be found here: http://www.maths.mq.edu.au/~alf/ as the document “Richard Guy is 9 plus 0”
3. Apparently the NextLimit engine uses SPH … http://www.nextlimit.com/realflow/
