Modulo Errors

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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.

I shall be attending “The Diverse Faces of Arithmetic” in Norwich next week; taking a break from cyclotomic thoughts to revisit some topics I’ve mentioned here in the past, such as elliptic divisibility sequences.

There are now some videos available from the Beyond Part III / Young Researchers in Mathematics conference I attended earlier this year. Of particular note is David Spiegelhalter’s plenary lecture on probability and uncertainty. I summarised one of the ideas from that talk – the micromort – on Everything2, mentioning a comparison between the risks of Ecstasy and horse riding by “the chairman of the Advisory Council on the Misuse of Drugs” which had led to calls for his resignation as early as January. The expert in question was Professor David Nutt, whose sacking in October has sparked controversy and debate over the role of science in policy making. Spiegelhalter’s presentation was highly accessible (and amusing!), so anyone interested in learning a bit more about these often-unintuive subjects should check it out.

There is also video from the panel discussion and some of the accessible talks in the various themed sessions. All of which should help convince you to sign up for next year’s Young Researchers In Mathematics conference, running 25-27 March again at Cambridge.

I’m back from the Fields Institute in Toronto, where I spoke at the above workshop, on my usual topic of cyclotomic and 4-cyclotomic matrices/graphs. During the talk I described my conjecture that a graph is maximal cyclotomic if-and-only if it’s 4-cyclotomic, and after an hour at the blackboards with James McKee I now have a potential approach for proving that. So although I don’t think my talk went especially well, it’s had the desired effect!

You can find my slides here, and an audio recording may become available in the future- the conference was held in Toronto and Vancouver via videoconference (which worked well) so hopefully all the talks will be archived online.

Update 24/x/09: Audio of all talks at Fields (hence, including mine) can now be found on their website.

This week I’ve been at the rather swish Alan Turing building in Manchester for the MAGIC Postgraduate Conference and LMS Northern Regional Meeting.

I spoke at the former, on the subject of Integer Matrices with Constrained Eigenvalues. Here are my slides: it’s a fairly breezy 15 minute overview of the problem (which integer symmetric matrices have all eigenvalues in [-2,2]?) and its solution, covering Mahler measure, cyclotomic matrices, interlacing, and charged signed graphs. For further reading, here is the paper by McKee and Smyth (my supervisor) with their proof of the presented classification; also by Smyth is a survey on Mahler Measure of 1-variable polynomials.

In my own work I’ve generalised the idea of cyclotomicity (all eigenvalues in [-2,2]) to Hermitian matrices with algebraic integer entries from imaginary quadratic extension fields. I think I have a complete classification of these, with an alternative proof of the above rational integer case as a subcase. The results at least will hoepfully appear here at some point, although for the proofs you’ll probably have to wait for my thesis.

This May, I’ll be travelling all the way to Canada for ANTS-VIII, the Eighth Algorithmic Number Theory Symposium; I’m tacking a couple of days holiday on the front as well, so should be good!

Last summer I spent two weeks at the very rewarding Utrecht Summerschool in Mathematics, so I thought I’d spread the word about this year’s course. It’s entitled Topics in Algebra, Analysis and Geometry; analysis is a new inclusion this year (in place of number theory) and will be the main emphasis. Abstracts for the three courses are not yet available, but the titles are QRT and elliptic surfaces, Distributions,and Lie algebras and Integrable Systems.

As last year, the course runs for two weeks in August, with a fairly intensive schedule of lectures and problem classes; when I attended, the students also spent a couple of days preparing a presentation for the final day. The pace is reasonably demanding, and the ideal audience would be students just finishing undergrad and about to enter study for an MSc or PhD (although I went after a year of postgrad study).

There are also social activities organised by both the department and the university – there are fifty courses scheduled across the summer in a wide range of subjects, so you’ll have the opportunity to mix with students from outside of mathematics too. Utrecht itself is a beautiful city – night canoeing through the canals is highly recommended! – and daytrips further afield are also offered.

For further details on the summerschool programme, see here; specifics for the mathematics course are being made available on the department pages. I also took some photos during my stay. Feel free to leave any questions you have in the comments!

Next week is the 25th Journées Arithmétiques, my first proper conference, conveniently held right here at the University of Edinburgh. Should be an interesting experience even if I find myself unable to follow much of the content. More at my level is Topics in Algebra, Geometry and Number Theory, a two week summerschool for beginning masters/postgraduates in Utrecht, The Netherlands. I’m hoping that’ll help me shore up the foundations in a few areas connected to my studies and allow me to finally understand what the other geometers are on about! Then I’ll be in Dublin for a few days of September, at ECC2007, the 11th Workshop on Elliptic Curve Cryptography. Again, much could be beyond me, but it’s pretty much the field I’ve settled in to and the whole area interests me, so it seems worth trying nonetheless.

So, I attended my first mathematics conference last week; two days of pure mathematics talks to lure us into postgraduate study. There are very few ‘pure’ topics I wouldn’t enjoy a lecture on, and I’ve been attending my own university’s staff/postgrad colloquia series this semester simply out of mathematical curiosity and enthusiasm. But beyond this entertainment value, the Durham lectures helped confirm/deny some opinions on potential research areas, so the event was certainly worthwhile.

Michael Drischel (Nottingham) gave the first talk, on Sums of Squares, which you can find online, so I won’t discuss the content too much.

Bill Jackson (Queen Mary London) presented a talk on Rigidity of Graphs concerning combinatorics and graph theory. The first section was presented using the geometry package Cinderella with which I was working for my summer research, demonstrating its many applications. This isn’t a field I’ve studied at all, but the ideas are both accessible and interesting so the talk was one of my favourites. There were even some connections to organic chemistry, which I haven’t thought about for a long time!

Patrick Dorey (Durham) gave a talk entitled Surprises in Quantum Mechanics; sadly I doubt I can ever get to grips with this topic (I can only remember abandoning two books partly read, and both were on Quantum Physics). However (ignoring a talk on funding) the next talk managed to overcome even my general dislike of Physics- Nina Snaith (Bristol)’s talk Every moment brings a treasure: how physicists came to the rescue of number theory. This was one of the more entertaining presentations anyway, but the central result was genuinely intriguing- how random matrix theory, a topic developed in the context of mathematical physics, was able to back up conjectures related to the Riemann zeta function arrived at by traditional number theoretic approaches. The method has turned out to have applications in other areas, and even features as a plot device in the film Proof!

The first day closed with a traditional talk-and-chalk on Geometry and integrability by David Calderbank. Due to a quirk of the MMath structure, I wasn’t allowed to take our differential geometry course. So this was a topic I knew very little about; the talk itself was interesting but I don’t think the field holds much appeal for me. Playing with surfaces is fun, but I prefer my analysis to be more topological rather than heavily connected to calculus.

Some of the ideas of the previous talk were picked up in the first of day two; Michael Singer (Edinburgh) giving an outline of a popular example of an integrable systems in a talk entitled The geometry of nonlinear waves. I’m hoping to track down the Maple worksheet for this one; you really have to see the graphs (or perform experiments with canals!) to appreciate what’s going on.

The most influential talk for me was John Cremona (Nottingham)’s Explicit methods in Number Theory. This was more accurately subtitled Rational points on curves and has cemented my interest in Algebraic Number Theory. For some time I’ve been deliberating between algebraic geometry and algebraic number theory; hindered by our lack of a number theory course at Bath! Based on this talk (and fortunate discussions with John at breakfast) it seems that the aspects of the algebraic geometry course I particularly liked more naturally fall within the remit of number theory; as do the bits of computer algebra that I enjoyed.

Norbert Peyerimhoff (Durham) spoke on averaging and equidistribution problems in geometry; I think this was another talk-and-chalk but I didn’t make notes because the content didn’t really appeal (didn’t help that it got very difficult very quickly!). Similarly Cobordism and groups of formal power series by Neil Strickland (Sheffield) confirmed that algebraic topology, whilst utterly fascinating, is really really difficult. Maple users can find the talk itself at this address, a non-interactive pdf version is also available if you can’t read Maple10 (it seems that, with Maple9.5, I can’t).

The final talk, given by Farid Tari (Durham) Singularities and the imagination offered more diffgeo, and an opportunity for me to demonstrate my complete lack of spatial awareness during the ‘practical’ component where we attempted to build some surface out of a piece of paper! Again, interesting as a talk but not as a career (although Farid was a very good speaker and well suited to holding our attention at the end of a demanding two days).