I spent last week in New Orleans for the Joint Mathematics Meetings 2011. I’d made a rather last minute booking after noticing a couple of sessions could be useful, and hadn’t quite grasped the scale of the event. I’d normally think of 200 mathematicians as a large gathering, but the JMM had over six thousand participants and at peak more than thirty parallel sessions to choose between… the densely typed book of abstracts runs to 450 pages! Hence, as well as the content that justifies dipping into my travel budget, I was able to see a wide range of talks purely out of curiosity. So, partly for my own future convenience, and partly to give some indication of the range available, I thought I’d note down everything I attended. As that was 42 talks – plus an art exhibition and a film – this post got rather long, so the rest is beneath the cut.
I’m speaking this afternoon at the Heilbronn Seminar in Bristol: my slides are available here. This is essentially (up to permutation, and modulo errors!) the talk I gave at Royal Holloway in October, although the last few slides have been replaced with a result I’ve found since then.
I try to avoid technical details of proofs in my talks, and to make the slides intelligible even if you weren’t there, so if you just want the motivation for, or results of, my PhD work then this is probably the best place to look. For all the proofs in tedious detail, there’s my thesis itself. I’ve since come up with a much more compact proof of the results in Chapter 5, which has lead to this draft paper classifying all cyclotomic L-graphs for d=-15,-11,-7 and -2: it’s far more compact than the corresponding sections of my thesis, but perhaps at the price of readability!
If you’d just like to know more about Mahler measure and Lehmer’s problem in general, then I’d recommend this survey by Chris Smyth. For various records related to small Mahler measure, see Mossinghof’s tables.
The church of St Mary Redcliffe is a stone’s throw from my new home in Bristol, and with its soaring gothic architecture is well worth including on any visit to the city. But for a mathematical tourist there’s an extra treat: the ‘chaotic pendulum’ in the north transept.
The pendulum could be considered the symbol of predictability and regularity, serving as the essential component of timekeeping devices for hundreds of years. Yet simple modifications yield devices that instead exhibit chaotic behaviour, the typical example being the double pendulum, where one pendulum is attached to the end of another. The chaotic pendulum at St Mary’s is believed to be unique in design: water is continually pumped into the crossbeam, causing it to tip one way or the other to let it flow out again; but it’s not generally possible to predict which way it’ll go next!
It’s important to note that this isn’t because the motion is random: there are rules governing its behaviour, and if you could return the chaotic pendulum to exactly the same state as it was at the start of the video, then it’d do exactly the same again. However, in chaotic systems tiny changes can be amplified into disproportionate effects, whilst large changes might have no impact. This leads to what is known as sensitive dependence on initial conditions. Think of catching a train, where if you’re a second early then you arrive at your destination on time, but if you’re a second late you’ll have to wait for the next one, so your two second delay grows into half an hour. On the other hand, whether you arrive at the station ten minutes or ten seconds early makes no difference to when you arrive at the other end. Thus your arrival time at the destination depends on your arrival time at the station in a highly non-linear way.
So if your attempt to reset the chaotic pendulum were even slightly flawed, then the video would be useless for making predictions. Similarly, even if you could build a completely accurate mathematical model, any readings taken from the real pendulum would contain errors, and so after a while the predictions of the model would diverge from the real behaviour. Fortunately for a pure mathematician like myself it’s more interesting to build that model, capturing the essence of all chaotic pendulums, than to be able to predict the activities of a single example!
The Clifton scientific trust has this to say about the lessons that can be drawn from the chaotic pendulum:
Some people look to science for certainties on which to base their lives. Increasingly we realise our knowledge can never provide certainty, even for this simple machine. The world is a more wonderful and a more surprising place than we could have imagined.
I’d agree with their conclusion, but not for their reasons… A truly random world, for example, would be constantly surprising. What’s wonderful about ours is that seemingly complicated phenomena can be described by simple rules – complexity can emerge from the interaction of otherwise easy-to-understand parts, rather than requiring some irreducibly complex explanation (such as a creator). Whilst the existence of chaos might restrict our ability to predict, it shouldn’t discourage us from trying to explain.
I shall soon be delivering the final copies of my thesis to the University of Edinburgh, who will in time make it available in both hard copy and electronically. But for convenience, you can already obtain the pdf version here.
Today I successfully defended my PhD thesis, Cyclotomic Matrices and Graphs. There are of course numerous corrections to be made, but I hope to have those done within the next couple of weeks and to make the final version available online. Until then, here is the abstract:
We generalise the study of cyclotomic matrices – those with all eigenvalues in the interval [-2,2] – from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers of imaginary quadratic fields. As in the rational integer case, a corresponding graph-like structure is defined.
We introduce the notion of `4-cyclotomic’ matrices and graphs, prove that they are necessarily maximal cyclotomic, and classify all such objects up to equivalence. The six rings of integers for Q(√(d)) for d=-1, -2, -3, -7, -11, -15 give rise to examples not found in the rational-integer case; in four (d=-1, -2, -3, -7) we recover infinite families as well as sporadic cases.
For d=-15, -11, -7, -2, we demonstrate that a maximal cyclotomic graph is necessarily 4-cyclotomic and thus the presented classification determines all cyclotomic matrices/graphs for those fields. For the same values of d we then identify the minimal noncyclotomic graphs and determine their Mahler measures; no such graph has Mahler measure less than 1.35 unless it admits a rational-integer representative.
I’ve recently returned from my second ‘Young Researchers in Mathematics’ event in Cambridge, a city I never tire of visiting. At over eight hundred years old, Cambridge University has more history than some countries, so there are plenty of mathematical connections to be found as a result- I thought I’d share just a few of them today.
The Mathematical Bridge
As I wrote last time about a Mathematical River, it seems sensible to start with this ‘Mathematical Bridge’! Popular myth asserts that the original design was so clever that it was held together simply by gravity, a feat attributed to anyone from undergraduate students to Isaac Newton, depending on who’s telling the tale. But (the story goes) when perplexed Fellows dismantled the bridge in an attempt to understand its construction, they were unable to put it back together again without resorting to using the nuts and bolts which can be seen today. Sadly, the bolts have always been needed, although as first built (in 1749) they were invisible to those crossing the bridge – and it is at least true that it didn’t require any nails!
Trinity College, Cambridge
It’s unlikely, then, that Newton had a hand in the mathematical bridge, since he died twenty years before it’s construction, but that won’t dent his mathematical legacy too much. Trinity College has always had a strong mathematical reputation, admitting almost a fifth of undergraduates in the subject (of 31 colleges in total), with Newton easily their most famous predecessor. He measured the speed of sound in one of the courtyards; the library holds his annotated first edition of Principia Mathematica (and a lock of his hair!); and the tree outside the main gate is claimed to be a descendant of his famous apple tree.
Centre for Mathematical Sciences
Until the mid-20th century, there was no central mathematics department, and instead work was done in the colleges. Today, however, research activity is concentrated at the Centre for Mathematical Sciences. It’s often joked that to find the mathematics department at a university you should look for the ugliest building, but the CMS, which opened in 2003, is a world away from 60s brutalist concrete. The ‘low energy’ design exploits natural ventilation to control the internal environment, and it’s green in a more literal sense too, with the main core having a grass roof. But it’s clearly a building meant for mathematicians, as the first things the architects were asked to design were the coffee rooms. Like the city it resides in, it’s a wonderful place to contemplate mathematics, and I hope I’ll be back again next year.
(First published on the SoSauce travel blog.)
(Cross-posted to the frontpage)
Just a quick note to mention two talks from the Edinburgh International Science Festival, which my flatmate chaired and I took some photos at: Marcus du Sautoy’s The Num8er My5teries and Ian Stewart’s Cows in the Maze. Summaries, courtesy of Haggis the Sheep, can be found here and here respectively.
At the end of March I was in Cambridge for Young Researchers in Mathematics. Personal highlights include Gowers’ keynote, the plenary by Michael Atiyah, and having my own work mentioned in Gary Greaves’ talk. Having spent a lot of time recently thinking about a very small section of number theory, it was good to be able to attend something multidisciplinary, giving me the opportunity to hear about some algebraic geometry, combinatorics and string theory too. That broader diet looks set to continue this week- I’m back in Edinburgh for the British Mathematical Colloquium and British Applied Mathematics Colloquium, featuring up to a dozen splinter sessions at a time (this afternoon I opted for history of mathematics). The Edinburgh International Science Festival is also running in April, and as part of that I’ll be at the Royal Society tomorrow for Meet the Mathematicians, where I’m part of the careers panel. Busy times!
Sometimes you don’t have to go far to find travel inspiration and a change of scenery. In my search of the world for sites of mathematical significance, it turned out I’d been overlooking one practically on my doorstep!
The Union Canal, near Falkirk
In 1822 the Union Canal opened, providing (with the Forth and Clyde Canal) a link between Scotland’s two major cities, Edinburgh and Glasgow. It became known locally as ‘the mathematical river’- by following a natural contour line, the Union Canal maintained a fixed height for its 31 mile course from Falkirk to Edinburgh, removing the need for time-consuming locks. Nor is this its only mathematical claim to fame- in 1834, the scientist John Scott Russell discovered what are now known as soliton waves whilst experimenting on the canal:
“I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.”
As Scott Russell described, such waves are unusual in that they can travel long distances whilst preserving their shape, rather than toppling over or simply flattening out with time. Named in his honour in 1995, The Scott Russell Aqueduct carries the Union Canal over the Edinburgh city bypass, yet the thousands of people who drive underneath it every day have probably never heard of his work- many have probably not even heard of the canal! Yet as well as having added to our understanding of physics, electronics and biology, soliton waves are of great practical importance today for their role in long distance communication with fibre-optics.
It seems that a waterside stroll is often of benefit to the advance of mathematics. Nine years after Scott Russell’s discovery – and several hundred miles away, in Dublin – the Irish mathematician Sir William Rowan Hamilton had a ‘flash of genius’ whilst walking along the Royal Canal. He had realized the equations for the quaternion group and, fearful that he might forget them just as suddenly, promptly carved them into the nearby Broom bridge. The original carving did not survive, but there is now a stone plaque in its place, which has been described as “the least visited tourist attraction in Dublin.”
Despite its clever design, the Union Canal’s importance would be short-lived: within twenty years, trains had overtaken barges as the fastest way to travel. The banks became overgrown and the canal filled with rubbish, and the decline continued after its eventual closure in 1965, as the construction of housing and the M8 motorway caused sections to be cut or filled in. Fortunately, an £85-million project – the millennium link – came to the rescue. The two canals had originally been joined by a series of 11 locks in Falkirk, but as these had not survived, a more spectacular solution was found- the Falkirk Wheel.
This engineering marvel is the world’s only rotating boat lift, capable of transferring boats between the two waterways in minutes – and, thanks to physics, using only as much energy to do so as boiling 8 kettles! The wheel opened in 2002, providing the final piece to restore the link between the two cities, providing ideal opportunities for walking, cycling or boating. I can’t wait to explore it further in the spring!
(First published on the SoSauce travel blog.)
At The Diverse Faces of Arithmetic there were a pair of (early morning!) overview lectures for postgraduates. I’ve finally got around to typesetting my notes from the first, Tom Ward’s session on recurrence sequences, available as pdf via the above link. The topics included are divisibilty sequences and primitive divisors; linear recurrences; elliptic divisibility sequences; integrability/ Laurent phenomena; growth rates and Lehmer’s problem.