I am just finishing the final year of a PhD. in Compuational Algebraic Geomoetry (or something like that) at UCL. The problem I have been working on for the past three years involves approximating fundamental domains for SU(n,1;Od) acting on HnC (complex hyperbolic space of dimension n) where Od is the ring of integers of some imaginary quadratic numberfield K. Mathematically the problem is not particularly complicated (at least when the classgroup of K is trivial), but computationally there is a lot of work involved and there are a huge amount of fiddly details which make practically approximating these things hard (to give you an idea of the amount of code I have written it suffices to say that I have developed carpal tunnel syndrome in my left hand). Anyway the good news is that I now have a C++ implementation of an algorithm that works, so I might just get finished before my funding runs out.
In the first two years of my PhD. I ran the postgraduate seminars in the UCL mathematics department. For information on these, including to find out if we have any upcoming talks that you may be interested in please visit the PReSS Talks page. They are now run on a more informal basis by Isidoros Strouthos. In addition to this I co-organised the 2008 UCL Mathematics Postgrad Mini-Conferenceand a Galois Theory One Day Course with my supervisor Andrei Yafaev. For the past three years I have been quite involved with the teaching the department does with local school kids; helping out on all of the GCSE and A-Level revision days (it turns out that I really like doing a bit of teaching). In my free time I have been starting an online graffiti business http://godofgraffiti.co.uk which is close to launch as we now have a deal on the table over content, and I am also in the process of starting an art gallery http://speedcoin.co.uk.
During my PhD. I have become very interested in the C++ programming language. C++ tends not to be used very often in mathematics; people either use C, Fortran or some scripting language such as Mathematica. Mathematica is useful because it does a lot for you and allows you to put something together quickly, but for large scale computation it is far too slow and clunky to be practical. C and Fortran have large pre-existing libraries and are fast, but being procedural languages it can be easy to loose the relationship between the code and the problem itself. C++ is nearly as fast as C and Fortran (and C++ compilers are under constant development) and with object orientation and template metaprogramming techniques it is possible to write code that feels like the structure it is attempting to study.
C++ is definitely not a rapid development tool, it is probably the slowest viable language to write in at the moment since it has a huge learning curve (probably about a year of heavy coding to get profficient) and there aren't any strong mathematical libraries available; there is the Pari C library, but I don't think that there is any real hope of integrating that properly into C++ due to the memory model that they use. That means that you basically have to start from scratch and do everything yourself; the up side is that you learn a lot and have a skill that is useful in the non-academic world, the down side is that it takes 10 times longer than you would like to get your code written. However, the boost library is excellent and eases many woes. My hope is that in time people will move across to C++ so that research students who choose to leave academia (as most do) will take a skill with them that they can apply directly to earn a living rather than needing to retrain for the commercial world.
Before joining UCL I did the Mathematics and the foundations of Computer Science (MFoCS) MSc. course at Oxford University. My dissertation project was entitled "HIGHER RAMIFICATION THEORY: TOTALLY RAMIFIED p-ADIC EXTENSIONS AND THEIR LOWER FILTRATIONS.", on the basis of this they awarded me a distinction. At some point I might get around to picking up the degree certificate from them.
In the dissertation I classify a type of invrariant sequence of rational numbers associated with the field extension of two p-adic fields; this sequence is known as the lower filtration of the extension. The proof of the result is constructive in that it allows you to construct the field extension associated with a sequence whenever such an extension exists. It works using an isomorphism between totally ramified extensions of p-adic fields and the newton polygon (or ramification polygon) of the generating polynomial. The thesis also includes a decent introduction to local fields (handy to read before reading Serre's local fields maybe so you have an idea of what is going on). I don't know if this result is actually in the literature, I haven't published it. I think the Newton polygon is well understood, so my result is something of a corollary of result connecting the polygon with extensions and as such is immediate once you fully understand the polygon. However, the work is entirely my own. You can get a copy of the thesis here lowerfiltrationnocan.dvi (it is a .dvi file since I lost the original images so I am no longer able to create a .pdf - the chances are that if you have any interest in the contents you will know how to open one of these files.)
Please feel free to rip off anything you like on this page. BT MMVI.