We will discuss basic properties of ultrametric spaces. Well-known examples of complete ultrametric spaces are p-adic numbers as well as p-adic integers. Also, it is known that the boundary at infinity of metric trees as well as more general Gromov 0-hyperbolic spaces is a complete bounded ultrametric space when equipped with a visual metric. We will discuss this result in details and show that the converse statement also holds. Namely, we show that every complete ultrametric space arises as the boundary at infinity of both a Gromov 0-hyperbolic space as well as a metric tree.

Fourth order diffusion PDEs have recently been proposed for use for noise removal in image processing, as a way to overcome some shortcoming associated with some well known second order methods. However, before now little mathematical analysis had been performed on fourth order models, and in experiments they exhibited their own artifacts, a kind of splotchiness which appears in flat areas of an image. I will discuss the existence of unique solutions to a class of fourth order PDEs proposed for image denoising, and present a newly proposed fourth order model which, along with being well-posed, overcomes the splotchiness exhibited by other models.