Modulo Errors

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).