Just a brief note that my second paper on my thesis topic has been accepted by the Journal of Number Theory. The full citation is:
Graeme Taylor, Lehmer’s conjecture for matrices over the ring of integers of some imaginary quadratic fields, Journal of Number Theory, Volume 132, Issue 4, April 2012, Pages 590-607, ISSN 0022-314X, 10.1016/j.jnt.2011.09.006.
(http://www.sciencedirect.com/science/article/pii/S0022314X11002289)
A while ago I became interested in `captured lightning‘ Lichtenberg figures, but without access to megavolt-scale physics gear, I wondered if I could simulate them in software instead. I was reminded of this when I had my JMM art exhibition entry printed on glass, as this would give me something a bit closer to the acrylic blocks. Some initially vague google searches eventually lead me to Diffusion-limited aggregation, a process that generates trees somewhere between ferns and lightning bolts. I set about implementing this in processing, and you can play around with a small version of it here:
Controls: To set things in motion (or pause them), press SPACE. To restart, press R. You can cycle through various colour options with G, and toggle rendering of the random walks with W. The screen is redrawn after a fixed number of points have been tested, which can be decreased with Z or increased with X: if your computer is powerful enough, it can cope with updating the screen more often; even if it can’t, you can decrease this to 1 to watch a step-by-step construction.
How does it work? There is an initial core disc of points which are included in the structure, and its horizon – the distance of the furthest point from the centre – is tracked. New points are launched from a ‘birth’ circle with radius a fixed multiple (until the edge of the screen gets in the way) of the horizon; further out, there is a ‘killing’ circle. Once launched, points take steps in random directions of size large enough to move them within the horizon- although of course they may go the wrong way! If they ever cross the killing circle they are abandoned, and a new point launched; if they move within the horizon, they switch to taking steps of unit distance instead. If at any stage they bump into a point already in the structure, they stick to it: they stop moving, become part of the structure (possibly increasing its horizon, and thus pushing out the birth/killing circles) and a new point is launched. There’s a fixed maximum radius for the horizon, and once this is reached, no more points are launched.
I’ve been trying to extend the results of the work described in the previous post, and following a suggestion of Noam Elkies have changed my search strategy from points corresponding to simple EDS triples to those given by (A,u,c) parametrisations as described here. Experimenting with these revealed some serious deficiencies with the height function in SAGE, so EDS are still involved at a practical level- but with enough magma licenses, one could just test all the points directly.
In good news for maths but perhaps bad news for my would-be paper, this straightforward approach has yielded several new (and record-breaking) examples of small height points, which I’ve added to the tables. A few also match or improve upon the best known values for most, highest, and most consecutive integral multiples. The table below summarises these: for the point [0,0] on the curve E:Y2 + a1XY + a3Y = X3 + a2X2,with P the corresponding point on the minimal model of E, we list the values of n≤50 such that nP is integral.
| w | (A,u,c) | [a1,a2,a3] | n | |
| A | √2 | (w+1,w-1,1) | [-13w - 23, 49w + 70, -1820w - 2576] | 1-10,12,13,15-20,25,35 |
| B | √6 | (w-3,w-3,1) | [-12443w + 30479, -230496005w + 564597600, -7958566915120w + 19494428025840] | 1-15,19,20,21,23,24,26,29 |
| C | √3 | (-2w-4,-w-3,1) | [17298w + 29961, 332452269w + 575824221, 9670381784073w + 16749592578603] | 1-12,14,15,18,24,29 |
| D | √3 | (1,2w-4,1) | [2856w - 4944, 42937344w - 74369664, -746077879296w + 1292244793344] | 1-12,14,15,16,18,27 |
| E | √7 | (2w-6,1-w,1) | [-5922w + 15669, -35749431w + 94584105, -543103643331w + 1436917176387] | 1-11,13,15,17,21,26 |
| F | √3 | (-2w-4,w+1,1) | [1086w + 1881, 716035w + 1240209, 1277410855w + 2212540503] | 1-8,10,11,12,14,15,16,21,22 |
| G | √5/2+1/2 | (w,w-2,1) | [4-w,6w-18,60w-90] | 1-15,18,22 |
Highest integral multiples: Over Q, the record is 31; this is exceeded by point A, at 35.
Most integral multiples: Over Q, the record is 16. All seven examples above match or exceed this: point B has the most, at 22; followed by A at 20; C,D and G at 17; and E and F at 16.
Most consecutive integral multiples: Over Q, the record is 14: points B and G both beat this, with their first 15 multiples being integral.
I have uploaded a preprint of my third paper to the arXiv. In a break from my cyclotomic matrix work, this revisits a project I first became interested in over four years ago: the search for points with small height on elliptic curves over number fields, through the use of elliptic divisibility sequences. There used to be a series of posts on this topic here on Modulo Errors, but I think the paper does a better job of summarising the bits that are right, whilst some of my other claims (on the related question of computing pairings via elliptic nets) I am now dubious about, and a lot of the SAGE code supplied is unusably out of date, so I’ve taken them down for now.
However, I have created a more permanent page that lists all the points/curves I recovered, in fuller detail than summarised in the paper: for each sequence one can easily write down two points on non-isomorphic curves, so in the interests of brevity I gave the recipe and then just one example per sequence. It’s my hope that new entries will be added to this list over time, by the eds method or others: in particular, I’m keen for it to include examples over number fields of higher degree than the quadratic cases it’s currently restricted to. Contributions welcome!
I gave the Number Theory Seminar at the Department of Pure Mathematics, University of Waterloo on Thursday, September 8th. My slides are available in presentation or handout form (except the latter is missing the interlacing demo which didn’t render into individual slides correctly).
I used this talk as an opportunity to present some results that were only at the conjectural stage last time I spoke on the topic. I have been working with Gary Greaves on Lehmer’s problem for matrices over the Gaussian and Eisenstein integers; we believe that we have proved the conjecture for those, and are slowly assembling a paper to that effect.
I’d recently ordered Ben Fry‘s Visualizing Data and started reading it this weekend; just a few pages in I learnt how to import data to processing and a project was born… Since New Orleans I’ve been increasingly interested in mathematical art, and whether in particular I could create something interactive. Here’s what I’ve come up with after a couple of rainy afternoons:
So what is it? Each point represents a number up to 10,000, arranged on an Archimedean spiral, and coloured depending on its smoothness: a smooth number is one with only small prime factors. More precisely, N is B-smooth if the largest prime dividing N is at most B (so 2-smooth numbers are powers of 2; 3-smooth numbers are multiples of 2 and/or 3 only; any number shown will obviously be at worst 10,000-smooth). You can adjust the smoothness bound with the slider: in ‘gradient’ mode the brighter a point, the smoother it is; whereas in ‘threshold’ mode a point is simply plotted or not depending on whether it passes the smoothness test (the mode can be toggled by pressing space).
The least smooth numbers are the primes, and it was thinking about prime spirals that lead me in this direction: the Ulam spiral is one of the first examples of computer-aided mathematics visualisation, and I’ve taken the circular layout from its close relative, the Sacks spiral. In fact, my original plan was to use the number of prime divisors, rather than smoothness, for deciding when to plot points, with the Sacks spiral as a special case. But the pictures for larger bounds weren’t particularly interesting- 10,000 just isn’t big enough to allow much of a range of behaviour. So I switched to smoothness, and whilst that means you can’t identify the primes directly, sometimes they’re conspicuous by their absence: in the Sacks spiral there are curves with an unusually high concentration of primes, and in the smoothness spiral there are similarly ‘missing’ curves. There seem to be lots of other features too- if you’d like a closer look, here’s an enormous render of the 101-smooth numbers shown above, created using processing’s PDF mode.
Today’s post by Haggis the Sheep demonstrates how crochet can help understand some topologically-interesting surfaces, so I felt I should mention a similar piece of fibre art I encountered this weekend. The object on the left is a Lorenz Manifold made out of over 25,000 stitches (plus three wires), and took Bristol mathematician Hinke Osinga 85 hours to assemble. Osinga (along with Bernd Krauskopf) had been experimenting with computer visualisation of the manifold, and developed an algorithm which ‘grew’ the image from a small disc, adding layers with additional or fewer points at each step to specify the local features of the surface. This approach conveniently works just as well for wool as pixels – each row of a crochet pattern differs from the last by increasing or decreasing the number of stitches to alter the shape.
But what does it actually represent? Lorenz was one of the founders of chaos theory, discovering the ‘butterfly effect’, the way in which seemingly small changes to a system such as the weather could escalate into major differences in behaviour. The Lorenz oscillator is a set of rules for evolving the position of a point in 3-dimensional space which exhibits this chaotic nature: starting points generally find their way to the Lorentz attractor, a complex pattern that never repeats itself. However, points on the Lorenz manifold manage to avoid this trap, and instead settle at the origin, the ‘central’ point of space.
Some of Hinke and Krauskopf’s computer visualisations, their crochet of the manifold, and a rendition in steel by Benjamin Storch can be viewed for the rest of the month at The Bristol gallery, which can found down by the harbourside. They’re there as part of one of the Changing Perspectives exhibitions, which also includes work from my department’s invaluable Chrystal Cherniwchan: the photographic project Exploring the Valley, and the Mathematical Ethnographies films. As well as maths, there are exhibits inspired by scientific topics from shifting glaciers to high voltage electricity, so if you’re local, why not take a look in person? If not, well, you can get a taste from the links above, or if you’re feeling brave, grab the instructions to crochet your own Lorenz manifold!
…is the less-than-catchy title of my first paper, to appear in the Journal of Algebra. With suitable credentials it can be accessed online through ScienceDirect, otherwise there’s a preprint on the arXiv which is a close approximation. The exact details of the print edition are still being finalised; I should have a limited supply of offprints for the truly keen.
The paper covers the classification of the cyclotomic matrices/graphs for four of the six rings I considered in my thesis, but there have been some improvements to the methods. In particular, the proof that any maximal cyclotomic graph over those rings has all vertices of weighted degree four has been substantially streamlined; and there’s an explicit proof that any cyclotomic graph is contained in a maximal one. A follow-up paper proving Lehmer’s conjecture for polynomials arising from such graphs over the same rings is in preparation.
I’m continuing to tour my Cyclotomic Matrices and Graphs talk; today I presented it at the University of Warwick. Here’s the latest and greatest iteration of the slides, mostly unchanged except for the current state of the computer search for minimal noncyclotomics of at most ten vertices. I’d hoped to finish that this month, but the final round of growing in the most general case over the gaussian integers has progressed much slower than I expected. Given that some batches finished in a twentieth of the wall time others have consumed so far, I’m suspecting the reasons may be non-mathematical. However, I have finished the eisenstein integer case, and there are four new classes with Mahler measure less than 1.3, with representatives given in the slides.
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.
Graeme Taylor
University of Bristol
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