In this talk I will review mathematical models used to describe the dynamics of ice sheets, and highlight some current areas of active research.  Melting of glaciers and ice sheets causes an increase in global sea level, and provides many other feedbacks on isostatic adjustment, the dynamics of the ocean, and broader climate patterns.  The rate of melting has increased in recent years, but there is still considerable uncertainty over  why this is, and whether the increase will continue.  Central to these questions is understanding the physics of how the ice intereacts with the atmosphere, the ground on which it rests, and with the ocean at its margins.  I will given an overview of the fluid mechanical problems involved and the current state of mathematical/computational modelling.  I will focus particularly on the issue of changing lubrication due to water flowing underneath the ice, and discuss how we can use models to rationalise observations of ice speed-up and slow-down.

I will discuss some reaction-diffusion equations of bistable type motivated by biology and medicine. The aim is to understand the effect of the shape of the domain on propagation or on blocking of advancing waves. I will first describe the motivations of these questions and present a result about the existence of generalized “transition waves”. I will then discuss various geometric conditions that lead to either blocking, or partial propagation, or complete propagation. These questions involve new qualitative results for some non-linear elliptic and parabolic partial differential equations. I report here on joint work with Juliette Bouhours and Guillemette Chapuisat.

Fang, Lu and Yoshikawa conjectured a few years ago that a certain string-theoretic invariant (originally introduced by the physicists M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa) of Calabi-Yau threefolds is a birational invariant. This conjecture can be viewed as a "secondary" analog (in dimension three) of the birational invariance of Hodge numbers of Calabi-Yau varieties established by Batyrev and Kontsevich. Using the arithmetic Riemann-Roch theorem, we prove a weak form of this conjecture. 

I will discuss how \zeta(3) occurs in quantum corrections to the Einstein action, and how this causes \zeta(3) to be seen in the moduli space of CY manifolds. I will also draw attention to the fact that the dependence of the moduli space on \zeta(3) has a p-adic analogue.