The Earth's mantle is almost entirely solid, but on geological timescales it convects vigorously, the well-known surface expression of this being plate tectonics. At depths up to ~100 km beneath plate-tectonic boundaries (mid-ocean ridges and subduction zones), and beneath ocean islands such as Hawaii, the mantle melts, and that melt rises to the surface to feed volcanism and form new crust. Such magmatism plays a key role in the chemical evolution and dynamics of our planet. Although the basic thermodynamics of melt generation in these settings is well understood, how the melt is transported to the surface is not, despite several decades of work on the problem. Furthermore, recent observational evidence suggests that mantle melting is not restricted to the near surface (top 100 km): it may occur within the mantle transition zone (410-660 km depth) and above the core-mantle boundary (2900 km). For these deeper instances of melting, an understanding of the dynamical and thermochemical characteristics is currently lacking.

Understanding the formation and migration of melt in the mantle presents a formidable scientific and mathematical challenge. One key challenge is in bridging diverse length scales - melt lies along grain boundaries at micron scales, may focus into channels at metre scales, and migrates over 100 km. Sophisticated mathematical techniques, such as homogenisation theory, are needed to map an understanding of physics at the smallest scales to plate-tectonic scales. Seismology offers a way to image melt in the mantle, but the development of new tools in inverse theory are required to extract that information. Models of melt transport are eventually cast as a series of coupled non-linear partial differential equations, which require advanced numerical techniques to solve. This programme will bring together a broad spectrum of mathematicians and solid Earth scientists to tackle these and other fundamental challenges of melt in the mantle.