Tom's mathematical things
As I said on the previous page, I'm a second-year post-graduate mathematician at the University of Nottingham, supervised by John Cremona, and working on computational number theory, concentrating on extending the well-known and effective algorithms for computing the ranks of elliptic curves over the rationals so that they might work over other number fields..
I've once or twice read requests for elliptic curves of given rank and small conductor, so, having discovered a good example by a sieving procedure, I compiled a page about them. While compiling that page, I found several thousand curves of reasonably high rank (at least six): if you can think of anything for which they'd be useful, feel free to contact me. Jesper Petersen suggested that I also look for Mordell curves of large rank, for which my sieving procedure was quite effective. At the moment I'm thinking of how to extend my sieving procedure to other number fields, for which the results will be probably more novel, if even more devoid of practical application.
I don't know of any use for curves with interesting torsion groups, but I've constructed a table of them anyway as a test for my analytic-rank code.
I recently discovered the mathematical programming language Pari, and use it for any of my mathematical programs which require non-trivial library functions. I've written a small collection of routines for computing analytic ranks (and a set of test examples for it), which people might want to download: analytic-rank functionality is not obviously available in the other standard computational-mathematics environments.
To find ECs of really high rank, sieving is prohibitatively time-consuming and a better approach is to use parameterised families of curves known to have high rank over Q(t); I've written up something about this here.
I did my undergraduate degree at Merton College, Oxford, writing a dissertation about equal sums of like powers (that link is to a PDF version; you might want to read the abstract first) under Professor Heath-Brown: in my last year, I was secretary of the Invariant Society there.
I have a deep distrust of uncountable things, or of anything much more complicated than an explicitly-presented finite group, a finite-dimensional vector space, a finitely-generated Z-module or perhaps a finite field; I much prefer working with objects which can be considered as arrays of integers.
Accordingly, my main interest is Diophantine equations, where, whilst the machinery required to obtain a result like958004+2175194+4145604=4224814 or 22204229323 - 2830599653 - 22188885173 = 30 can be quite complicated - the first result comes from an existence proof by Elkies using elliptic curves and extending a result of Demjanenko, followed by a search on a massively parallel supercomputer by Frye; the second was found by a fairly straightforward search by four graduate students at the University of Georgia and separately by Elkies&Bernstein by a more sophisticated method involving approximating the surface x3=y3+z3 by a series of cuboids and using lattice reduction - the result can be explained to anyone capable of handling multiplication and addition.
Because of my interests in computing, my personal value of 'intractable' is quite high: it's really not that difficult to organise a CPU-decade of computation over a month or so (ie 150 fairly modern computers working on your problem in parallel), especially if you can convince people who've organised such things before that your problem is worth their using their contacts. With current values of 'fairly modern computer', that translates to about 10^17 cycles: enough to perform an exhaustive search over 245 cases (say, the region 0 < x < y < 8000000 or 0 < x < y < z < 60000) if each case takes a few thousand instructions.
Another thing I'm interested in is how to use the amazingly powerful computers available nowadays to provide help in maths teaching. Maple is an excellent tool for doing algebra too complicated or too boring to want to do by hand, but is rather bad at visualisation.
One thing that modern computers, and particularly modern printers, can help with is illustration. It seems a pity that the modular group is introduced with a couple of sketches drawn in thick chalk, wigglily wielded, on a blackboard when a few seconds of computer time can produce beautiful colourful diagrams to project with an OHP: the complaint may be that students won't be able to take notes reliant on pretty coloured diagrams, but a few seconds in Photoshop produces an outline version which can be printed out and handed round.
If there are lots of computers around, and especially if video projection is available, we can do interesting things with animation. Complex analysis is plagued with the problem that functions from C to C appear to be fundamentally four-dimensional objects, and so rather hard to draw on a flat board. But there aren't really four dimensions going on: there's a function with value a two-element vector defined at every point in the plane. And our eyes are capable of perceiving even three-vectors directly if we just set the red, green and blue components of a pixel to the values of the three-vector there.
With complex numbers, I noticed that colours are often plotted on a colour wheel, so used a spectrum of colours to represent arg z to get illustrations like this one (that represents tan z). You can also use intensity of colour to represent |z|, but it's rather harder to get the scaling right and avoid images consisting of huge areas of black or white: log |z| is often a sensible thing to work with.
Using carefully-written software on a sufficiently fast Pentium-3 PC, it ought to be possible to render one of these illustrations full-screen (albeit on a rather low-resolution screen) in a thirtieth of a second: this means that you can allow students to, for example, manipulate the coefficients of a function interactively and watch how the structure in the plane changes. On slower PCs it's possible to pre-render videos to illustrate interesting things.
Note-taking becomes harder, but I'm optimistic enough to expect all students to have PCs capable at least of replaying videos by the time I start lecturing; CDs are cheap enough to produce that, if you have a very media-intensive course, you could hand out one CD per student per week of lecturing for less cost than photocopying twenty sheets of notes per student.