Prospective students are encouraged to contact me directly, but may find some inspiration in the following:

Data Assimilation (DA) - the mathematical and statistical question of how to combine an uncertain model forecast with data. I am interested in:

1. Coherent structure DA, that is, employing low-dimensional structures in the DA update in place of the original data. Research questions here may tend towards statistics (how do measurement errors in data create measurement errors in coherent structures?) or applied maths (from the oceanographic/atmospheric sciences literature; how should we employ coherent structures, and on what scales?). 
2. Projected DA, that is, employing projections to split one DA problem into several. The advantage is that one can use a DA method with high accuracy on key parts of the DA problem, and use a DA method suited to high-dimensional inference on the remainder of the DA problem. Some overlap with coherent structures; the key in projected DA is translating dynamical information to statistical information.
3. Non-Gaussian problems; DA algorithms are often founded on the assumption that measurement errors are distributed normally. However there are counter-examples, and work has shown that DA methods constructed for non-Gaussian measurement errors are promising. (This would be a new research area for me - based on work by Craig Bishop.)
4. Surrogate DA, the design of DA methods where a statistical surrogate is trained to empower a large ensemble of forecasts to be approximated from a small number of computationally intensive runs of the physical model.  Many open questions - see my preprint with A/Prof. Elaine Spiller. 

Numerical Multiscale Methods - key focus is on Projective Integration, that accelerates simulation of stiff systems, and patch dynamics, that accelerates simulation of systems with fine and coarse spatial components.

1. Projective Integration for stochastic systems; there is a wealth of literature on this topic, but to my view two questions remain. First is how to implement an accurate, fast PI solver for stochastic systems with unknown slow and fast variables. Second is how to modify such methods to nonstandard slow-fast systems of the sort discussed in http://dx.doi.org/10.1137/19M1242677
2. Patch dynamics; recent work has developed an adaptive moving patch scheme. The scheme can simulate moving fine-scale meshes that come together to form shocks at unknown locations in the spatial domain. A project here might focus on extending and applying the moving patch dynamics to travelling wave problems. 

Thanks for coming this far! Please accept a beautiful picture of adaptive moving patches simulating a problem with heterogeneous advection and diffusion terms; the inset shows details of the little black box.