Coarse embeddings (maps between metric spaces whose distortion can be controlled by some function) occur naturally in various areas of pure mathematics, most notably in topology and algebra. It may therefore come as a surprise to discover that it is not known whether there is a coarse embedding of three-dimensional real hyperbolic space into the direct product of a real hyperbolic plane and a 3-regular tree. One reason for this is that there are very few invariants which behave monotonically with respect to coarse embeddings, and thus could be used to obstruct coarse embeddings.

Hydrocephalus is a serious medical condition which causes an excess of cerebrospinal fluid (CSF) to build up within the brain. A common treatment for congenital hydrocephalus is to implant a permanent drainage shunt, removing excess CSF to the stomach where it can be safely cleared. However, this treatment carries the risk of vascular brain tissues such as the Choroid Plexus (CP) being dragged into the shunt during drainage, causing it to block, and also preventing the shunt from being easily replaced. In this talk I present results from our fluid-structure interaction model which simulates the deflection of the CP during the operation of the hydrocephalus shunt. We seek to improve the shunt component by optimising the geometry with respect to CP deflection.

We describe a network model for the progression of Alzheimer's disease based on the underlying relationship to toxic proteins. From human patient data we construct a network of a typical brain, and simulate the concentration and build-up of toxic proteins, as well as the clearance, using reaction--diffusion equations. Our results suggest clearance plays an important role in delaying the onset of Alzheimer's disease, and provide a theoretical framework for the growing body of clinical results.