The following is a list of projects that I am interested in supervising in 2011-2012. I am open to suggestions for related projects. I will take no more than two MSci project students, so if you're interested, please arrange to see me as soon as possible.
1. Nevanlinna theory - the value distribution of meromorphic functions
Picard's theorem says that any non-rational meromorphic function takes every value in the extended complex plane an infinite number of times, with at most two exceptions. Nevanliina theory provides a framework in which such value distribution results can be vastly generalised.
Possible directions include
2. The ABC conjecture
Let n be a positive integer. The radical of n (denoted "rad(n)'') is the product of all the prime divisors of n. For example, rad(24)=rad(3 × 23)=2 × 3=6. The ABC conjecture was first proposed by Joseph Oesterlvé and David Masser in 1985. It says that for every ε > 0, there are only finitely many triples of coprime positive integers a + b = c such that c > (rad(abc))1 + ε. This conjecture has many simple consequences, including Fermat's Last Theorem. Based on the close analogue with Nevanlinna theory above, some of the proofs of weak versions of the ABC conjecture could be extended.
3. Singularities of solutions of differential equations
This project is about understanding the kinds of singularities that solutions can develop in the complex domain.
4. p-adic analysis
The rational numbers are not complete with respect to the usual absolute value (i.e., there are Cauchy sequences of rational numbers that do not converge to rational numbers). The completion of the rationals with respect to this absolute value leads to the real numbers. The real numbers are not algebraically closed (there a polynomials with real coefficients that do not have real roots). This in turn leads to the complex numbers and complex analysis.
For each prime number p, the field of rational numbers also admits a so-called p-adic absolute value. The completion of the rationals with respect to this absolute value leads to the p-adic numbers, which are very important in number theory. In this project we will explore p-adic analogues of complex analysis. One eventual aim would be to better understand certain "special functions" and solutions of differential and discrete equations in this setting.
5. General relativity
Topics could include
6. Tropical geometry
For some background, see Tropical Mathematics by David Speyer and Bernd Sturmfels.