University of Oxford

Soliton like structures called “stable droplets” are found to exist within a paradigm reaction
diffusion model which can be used to describe the patterning in a number of fish species. It is
straightforward to analyse this phenomenon in the case when two non-zero stable steady states are
symmetric, however the asymmetric case is more challenging. We use a recently developed
perturbation technique to investigate the weakly asymmetric case.

We begin by proving the abc theorem for polynomial rings and looking at a couple of its consequences. We then move on to the abc conjecture and its equivalence with the generalized Szpiro conjecture, via Frey polynomials. We look at a couple of consequences of the abc conjecture, and finally consider function fields, where we introduce the abc theorem in that case.

Using the theory of formal groups, Landweber´s exactness theorem provides means to construct interesting invariants of topological spaces out of geometric objects. I will illustrate the resulting connection between algebraic geometry and stable homotopy theory in the special case of elliptic curves.