MSI Weekly Bulletin - Week starting Monday 18 September, 2006

Talk 1: In this talk I will describe work done by myself and my supervisor, Ben Andrews, on the Ricci flow of four manifolds with a certain curvature condition. We have shown that the class of four manifolds with pointwise quarter pinched flag curvature, which includes quarter pinched sectional curvature, is preserved by Ricci flow. Furthermore, as the flow approaches a singularity the metric is asymptotic to a space form, $\mathbb{S}^{4} \backslash \Gamma$. In particular this gives a different proof of the quarter pinching sphere theorem in dimension four. Unlike previous methods, we do not use any special structure of the curvature tensor in dimension four. Talk 2: $V$-variable fractals were recently introduced and developed by Barnsley, Hutchinson and Stenflo. They provide a link between (deterministic) fractals and random fractals. The integer parameter $V$ controls the degree of variability and $V \rightarrow \infinity$ corresponds to the random case. There is a natural ``superfractal'' whose elements are each a $V$-tuple of $V$-variable fractals. This enables the rapid construction of $V$-variable fractals by means of a certain high level "chaos game". In this talk I will review the concept of $V$-variable fractals and provide motivation for their construction. I will demonstrate a subclass of $V$-variable fractals for which almost sure box counting dimensions may be computed.

The Allen-Cahn equation arises from several applied fields, such as material science and mathematical biology. Recent research has gained deeper understanding of its solutions in the one dimensional case. However, the high dimensional case is much more difficult to understand. It is known that there are solutions with layers, and solutions with spikes. We establish for the high dimensional case the existence of solutions with layers as well as spikes. Our approach relies on sharp estimates of small eigenvalues and some new techniques in the so-called reduction methods.