Prospects in Mathematics (Durham 2005)

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).

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