The most important part of the PhD is the student's thesis which is addressed to a forefront research problem or set of problems.** MAC-MIGS ** staff are very diverse in their interests. In their research they study a great variety of methods and models. Because of this diversity, we are able to provide supervision across many areas. We roughly divide the interests of staff into research with primary emphasis on mathematical modelling, on analysis and on computation, although the boundaries between these domains are often indistinct. The common thread of research in the CDT is provided by the emphasis on foundations and desire of all involved staff to find a unified perspective and better solutions for today's challenging problems.

**Mathematical modelling research **: financial models, quantum models , molecular dynamics, interfacing neural networks with physical models, coarse-grained
particle systems, lattice-Boltzmann models, and PDE models of fluids and solids, weather and climate models, interacting agents, biological processes, data-driven model reduction, data-analytic methods including Bayesian inference , optimisation,
gradient flows, and data assimilation.

**Analysis research**: continuum models (fluids and solids), harmonic analysis, propagation of singularities, stochastic analysis (including stochastic processes, stochastic ordinary/partial differential
equations and stochastic approximation algorithms), inverse
problems , data assimilation and filtering.

**Computational research**: numerical analysis (for ordinary and partial differential equations, and stochastic differential equations), finite elements, computational image analysis, medical imaging, parallel computing, high dimensional optimisation, machine learning algorithms, and professional software for scientific applications (molecular simulation, reservoir modelling, gas dynamics, machine learning, neural network models).

There are also practitioners involved in **chemistry** (e.g. for molecular and quantum models, materials science), **engineering** (particle and granular models, processes, materials), **informatics** (machine learning, artificial intelligence), **biology** (systems biology, cell modelling) and **physics** (condensed matter, soft matter, density functional theory).

MAC-MIGS Supervisors are drawn from both institutions (Edinburgh+Heriot-Watt) and from a wide range of departments.

In what follows we note particular skills and interests with the following tags: **COMP**=Computation, **ANAL**=Analysis, **PDE** = Partial Differential Equation, **MAT**=Materials, **BIO**=Biology, **STO** = Stochastic Methods, **STAT** = Statistical Analysis, **MD** = Molecular Dynamics, **FIN** = Financial Modelling and Analysis, **DS** = Data Science (Machine Learning, Image Analysis, AI), **MESO** = Mesoscale Modelling, **UQ** = Uncertainty Quantification