About the EPSRC EXE-MATH Doctoral Training Partnership
The University of Exeter’s Mathematical Sciences Doctoral Training Partnership (EXE-MATH) aims to deliver advanced training in the mathematical sciences across a wide spectrum of topics. These include fundamentals such as number theory, algebra, and geometry in addition to training in mathematics capable of transforming the economy and society. Achieving this is through collaboration with its multidisciplinary institutes (the Living Systems Institute, Institute for Data Science and Artificial Intelligence, Global Systems Institute, Environmental Sustainability Institute) and external collaborators, such as those at the Heilbronn Institute, Met Office, the Alan Turing Institute, and the World Health Organization.
Having a wide scope of research activities is important, as all branches of mathematics are relevant to contemporary life, especially in the development of new technology. For example, number theory plays a key role in cryptography, which is important for cyber security; graph theory is one of the foundation concepts of the data science revolution and theorems of differential geometry have led to new numerical methods being built in weather and climate models that are currently under development.
A principal goal of EXE-MATH is to create a new generation mathematics researcher who has experienced a rich culture that includes both fundamental and impactful mathematics research.
The EXE-MATH Doctoral Training Partnership has eight fully funded studentships available for September 2021 entry.
A fully funded EXE-MATH DTP studentship will cover,
- A stipend for 4 years (currently £15,609 for 2021-22 entry) in line with the UK Research and Innovation rates
- Payment of the university tuition fees
- Research and Training costs
The project below is one of a number that are in competition for funding from the EXE-MATH Doctoral Training Partnership
Project Information
Alemi Ardakani (2019) developed Euler–Poincare variational principles with advected parameters, based on the Euler–Poincare reduction theorem in geometric mechanics, for the motion of a rigid-body, free to undergo three-dimensional rotational and translational motions, dynamically coupled to its interior fluid motion described by the Euler equations. The PhD research project aims to:
(1) Derive new (non-canonical) Lie–Poisson brackets for the Eulerian variant of the Lagrangian functional presented in Alemi Ardakani (2019), and hence develop new Lie–Poisson variational principles and Lie–Poisson equations with symmetry breaking parameters for coupled geometric ideal fluid and rigid-body motion in three dimensions.
(2) The Noether theory relates symmetry of a Lagrangian or Hamiltonian to a conserved quantity. The milestone in this part of the project is to develop new Kelvin-Noether theorems or conservation laws for the variational principles of part (1).
(3) Develop new shallow-water Lie–Poisson variational formulations of the coupled fluid and rigid-body dynamics, starting from the reduced shallow-water variant of the Lagrangian functional in part (1), which is presented in Alemi Ardakani (2021), and hence derive Casimir invariants of the coupled formulation in Eulerian coordinates.
(4) Extend the 2016 Stewart–Dellar structure-preserving finite difference Poisson-bracket discretisations, the Poisson integration of rigid-body systems by Austin et al. (1993), and the symplectic discretisation of rigid bodies as constrained systems to develop new constraint-preserving, and energy- and potential-enstrophy-conserving (Casimirs preserving) numerical integrators for the shallow-water Lie–Poisson bracket formulation of coupled nonlinear rigid-body and weakly dispersive shallow-water equations.
Entry requirements:
Applicants should have obtained, or be about to obtain, a First or Upper Second Class UK Honours degree, or the equivalent qualifications gained outside the UK, in an appropriate area of mathematics, science or technology. Applicants with a Lower Second Class degree will be considered if they also have Master’s degree or have significant relevant non-academic experience.
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