Research
My fourth year essay at Warwick University was on Transcendence and Irrationality Proofs, and my initial research at Bristol continued this theme.
One of the results proved in the aforementioned essay is the Lindemann–Weierstraß theorem. A special case of this theorem is the Hermite–Lindemann theorem which says that, provided x isn't zero, at most one of x and ex can be algebraic. One way of thinking about this theorem is that it tells us about the algebraic points on the curve y = ex, to wit that there is only one, the point (0,1).
What is the natural way to generalise this? Initially my research focused on replacing the exponential function by a general Pfaffian function.
After some twisty turns and turny twists, my research finally settled on an open problem living on the cusp of number theory and model theory. This problem — known as Wilkie's conjecture since it's a conjecture, by Wilkie — says that if we define a set in Rn using addition, multiplication, the less-than relation, and the exponential function, and maybe project it down to a lower dimension, then this set's algebraic points will be very scarce indeed. The actual conjecture sans hand waving can be found in the link on the right.
