# PhD opportunities in Mathematics

Funding may be available for the projects below, if there is no specific funding mentioned with the project details please check the 'Funding' tab for further options.

We also welcome applications from students who already have funding in place.

For further information contact the relevant supervisor.

## Projects

### Eigenvalue computation

This PhD project is about computing certified bounds for eigenvalues in regimes where the classical methods fail catastrophically. The main idea of the PhD as a whole will be to give the student a broad view on the state-of-the-art in computational spectral theory.

Supervisor: Dr Lyonell Boulton

### Non-selfadjoint Schrödinger operators, wild spectral behaviour

The purpose of this PhD project is to study properties of the spectrum of non-selfadjoint Schrödinger operators. This includes pseudospectra, one-parameter semigroup and eigenvalue asymptotics properties.

Supervisor: Dr Lyonell Boulton

### The motion of Non-Newtonian fluids

Non-Newtonian fluids are liquids whose viscosity is dependent on the shear rate and hence has the ability to vary. The time evolution of an incompressible Non-Newtonian is governed by a generalized Navier-Stokes system of Partial Differential Equations (PDE) for the velocity field and the hydrodynamic pressure. The aim of this project is to analyse these PDEs concerning existence and qualitative properties of solutions.

Supervisor: Dr Dominic Breit

### Numerical Approximation of Stochastic Navier-Stokes equations

The dynamics of liquids and gases can be modeled by systems of partial differential equations (PDEs) describing the balance of mass, momentum and energy in the fluid flow. In recent years their has been an increasing interest in random influences on their solutions modeled by Stochastic partial differential equations (SPDEs). The aim of this project is to analyse the numerical approximation of certain SPDEs for viscous fluids.

Supervisor: Dr Dominic Breit

### Formation in finite time of geometric singularities in moving boundary problems

Moving boundary problems are nonlinear PDE problems set on a domain evolving in time (the moving boundary being part of the unknowns of the problem). The project will concentrate on establishing (by proving) some situations where finite time-singularity happens, for either the PDE or the domain (with the formation of a cusp singularity for instance).

Supervisor: Dr Daniel Coutand

### Twisted yangians and reflection algebras in integrable models

The main aim of the proposed project is the investigation of twisted Yangians and reflection algebras at both classical and quantum level. The aforementioned algebras describe integrable models in the presence of non trivial boundary conditions. At the classical level such studies involve algebraic and analytical tools via the underlying Poisson structure and the relevant Lax pair formulation. At the quantum level the Bethe Ansatz methodology or the vertex operator approach can be implemented for the computation of the associated physical quantities.

Supervisor: Dr Anastasia Doikou

### Geometric analysis of BPS equations

The aim of this project is to understand the geometric analysis of gauge fields in theoretical physics, in particular nonlinear Laplace equations arising in string and M-theory. We intend to start from recent work by Mazzeo and Witten on monopoles and the singular behaviour of solutions and study applications in the topology of knots.

Supervisor: Dr Heiko Gimperlein

### Numerical analysis of concert halls

Our group considers accurate and efficient numerical methods for the wave equation, e.g. the sound emitted by an instrument in a concert hall. Some key challenges: Oscillatory solutions at high frequencies, rigorous error estimates for adaptive mesh refinement procedures, complex modelling tasks and the pure analysis of nonlinear equations.

Supervisor: Dr Heiko Gimperlein

### Knot homology and supersymmetric gauge theory

Physical questions in supersymmetric gauge theories often turn out to reveal deep connections with mathematics. One particularly interesting theme in this respect is knot theory. In this PhD project we will study how knot homologies make their entrance into supersymmetric gauge theory and string theory.

Supervisor: Dr Lotte Hollands

### Quantum integrability in supersymmetric gauge theories

Supersymmetric gauge theories are toy models for describing QCD. In recent years much progress has been made in understanding their low energy regime. Exact computations of low energy quantities have revealed interesting links with quantum integrable systems. In this PhD project we will deepen this connection and employ it to study the quantum Hitchin system.

Supervisor: Dr Lotte Hollands

### Degeneracy and finite -size scaling at first order quantum phase transitions

It was initially thought that "all first order transitions looked alike" - in that they displayed similar finite size scaling properties, unlike the rich variety of critical exponents arising in the different universality classes for continuous transitions. It was first observed that boundary conditions could have a radical effect on finite size corrections and then more recently noted that macroscopic degeneracy could also affect scaling. The project will investigate the role of macroscopic degeneracy in the scaling behaviour of first order quantum phase transitions, which have only very recently begun to be studied.

Supervisor: Professor Des Johnston

### SUSY and the ASEP

Supersymmetry (SUSY), a symmetry between bosons and fermions, has been conspicuous by its absence in particle physics where it was originally formulated. However, an exact lattice supersymmetry has been found in various 1D spin chains where it relates chains of different lengths. The asymmetric exclusion process (ASEP) is known to be closely related to the XXZ spin chain and it has already been observed that a transfer matrix ansatz (TMA) for the ASEP exists which satisfies a relation that takes a similar form to the defining relation of lattice supersymmetry. The project will investigate the implications of this for the solvability and spectrum of the ASEP.

Supervisor: Professor Des Johnston

### Flash Sintering

In "flash sintering" a powdery material quickly bonds into a solid object through the action of an electric current. The project will mathematically model the interaction of the electric current and changing temperature during the process.

Supervisor: Professor Andrew Lacey

### Homeless Modelling

Earlier work has had success in modelling sizes of waiting lists for public-sector housing and of the local homeless population. The research is to be extended to take into account possible causes of homelessness, such as alcohol dependency.

Supervisor: Professor Andrew Lacey

### Efficient numerical methods for stochastic differential equations.

Many applications give rise to differential equations which include some time dependent noise term. One such example is the stochastic filtering problem others arise in mathematical biology or flow through porous media. Typically these are stochastic PDEs but may also be stochastic nonlocal integral equations or stochastic differential equations coupled to PDEs. The aim of this project is to develop novel solution techniques and that can be used to estimate the uncertainty in computed results and examine convergence. One area that has not received much attention are methods that are adaptive in space and time and may take account of different scales in the problem.

Supervisor: Professor Gabriel Lord

### Computing spectra

A typical problem in quantum mechanics or in determining the stability of travelling waves or coherent structures is to compute the eigenvalues of systems of Schrodinger operators; in general non-self-adjoint. There are many methods for determining the pure point spectra (eigenvalues) of such an operator. Three prominent ones are the Evans function, the transmission coefficient and a Fredholm determinant. The Evans function and transmission coefficient are finite determinants and you might therefore be surprised (or not!) that they are both directly related to the Fredholm determinant. Another straightforward method for computing the pure point spectra is simply to discretize the underlying differential operators and find the eigenvalues of the resulting algebraic eigenvalue problem. These different methods have their own advantages and disadvantages. In particular some are readily extended to multi-dimensional domains. For those that are not such as the Evans function, there has been a lot of recent research activity trying to elicit their natural extensions and implment them. The project has two parts. One part is to look at how to extend the Evans function approach to compute eigenvalues in multi-dimensional domains, and further how this might also be sued to compute the Maslov index in such cases. The second part would be to study the different methods, implement some of the methods for some key representative challenging problems and discuss how the different methods could be compared via a complexity measure (if that is possible). This is an open research problem of big interest...

Supervisor: Dr Simon J.A. Malham

### Stochastic partial differential equations, their algebraic structure and simulation

Stochastic partial differential equations appear naturally in physics (for example the famous KPZ equation), biology (neuroscience and axon firing) and engineering (extracting oil from difficult resevoirs). Understanding SPDEs analytically and then simulating them accurately was an important future research objective highlighted by the US Department of Energy in 2008. This project has three parts. First, it is important to undertand how to interpret stochastic partial differential equations and the analytical properties of their solutions. What class of solutions can we establish existence for, and for what length of time? Second, as has been achieved for stochastic differential equations, how can we encode solution expansions for the flowmap combinatorially, algebraically, and then consider functions of the flow map to try to generate efficient integrators? Third, can we implement any such efficient integrators? This will involve efficient simulation techniques for the Levy areas associated with a high dimensional Wiener process (if possible). These are open research problems of very broad interest.

Supervisor: Dr Simon J.A. Malham

### Modelling the self-organisation of ants into trails

Ants coordinate socially through their secretion of pheromones, exemplified by their well known tendency to spontaneously organise into trails during behaviour such as food searching. This PhD project will employ individual and continuous mathematical modelling to explore the precise mechanisms behind these emergent dynamics.

Supervisor: Dr Kevin Painter

### Pattern formation during embryonic development

The transformation of the embryo into the adult organism is amongst the most astonishing examples of self-organisation in the natural world. This project will investigate the mechanisms that coordinate this patterning process through mathematical modelling.

Supervisor: Dr Kevin Painter

### Modelling, analysis, and computations of inverse problems for random heterogeneous systems

Random heterogeneous systems are ubiquitous in our world. Examples include natural porous media such as soil/rocks, biological membranes such as plasma membranes, synthetic membranes for desalination/filtration, or growth of neural networks in brains. We will start with fundamental statistical, thermodynamic and physical formulations describing transport in random heterogeneous systems on the microscale. Further details can be found here.

Supervisor: Dr Markus Schmuck

### Analytical and computational methods for the efficient and reliable approximation of complex, high-dimensional systems

We shall investigate simple methods for approximating complex systems with functions and networks and analyse them for efficiency and reliability based on the size of the input/available data. Further details can be found here.

Supervisor: Dr Markus Schmuck

### Developing effective macroscopic equations for reactive multiphase flow in porous media using thermodynamic and rigorous mathematical principles

Important energy technologies such as fuel cells and batteries crucially depend on the complex interplay of multiple phases on different scales. The mathematical optimization of complex multiphase systems, e.g. reducing weight and size, increasing storage capacity and life time, crucially depend on reliable, effective macroscopic equations which we shall systematically and rigorously derive and validate by computational experiments. Further details can be found here.

Supervisor: Dr Markus Schmuck

### Pattern Formation in Desert Plants

In desert regions, plants tend to cluster together, with areas of bare ground between the clusters. This type of pattern formation may be an indicator of future regime shifts for example a transition to total desert. This project concerns the mathematical modelling of such patterns, with the objective of predicting the key ecological and environmental parameters that control the wavelength and other features of the pattern, and the resilience of the pattern to future climate change.

Supervisor: Professor Jonathan Sherratt

### Discrete holomorphicity and quantum groups

This project aims to construct operators in 2D solvable lattice models that obey a discrete version of the cauchy integral theorem around lattice plaquettes. Such operators have played a key role in the rigorous proof of the scaling limit of solvable lattice models to conformal field theories. In this project you will use the technology of quantum groups to construct such operators in range of lattice models, and to identify their conformal limits. This approach has been developed by R. Weston and collaborators in the last two years. You will extend the approach to both dynamical quantum groups and to lattice models with elliptic Boltzmann weights.

Supervisor: Dr Robert Weston

### Boundary States in solvable lattice models

Integrable quantum field theories and solvable lattice models with boundaries have been a topic of major interest for over 20 years. Applications arise in both string theory and condensed matter physics. Exact expressions for boundary states (eigenstates of the Hamiltonian acting parallel to the boundary) can sometimes be obtained via the vertex operator approach to solvable lattice models developed by Jimbo, Miwa and collaborators in the 1990s. In this project you will construct such boundary states in a new class of models - those associated with 'twisted Yangian' algebraic symmetries. You will analyse the scaling limit of these models and compare your results with calculations for the corresponding boundary scattering matrices of integrable quantum field theories.

Supervisor: Dr Robert Weston

### Quantum Integrability in the lab

In the last decade there have been have rapid, major developments in the study of experimental systems related to 1D integrable quantum spin chains. These experimental systems include both cold atom systems and optical lattices. There are range of experiments such as neutron scattering, electron spin resonance, and quantum quenches, who outcomes may be computed exactly by exploiting the mathematical machinery of quantum integrability developed over the last 20 years. In this project, you will work to bridge the gap between the hard mathematical physics language of integrability, quantum groups, vertex operators, and conformal field theory and this recent, pioneering experimental work.

Supervisor: Dr Robert Weston

### Mathematical modelling tools for conservation ecology

The introduction and invasion of non-native species is recognised as a major threat to native biodiversity. Understanding the mechanisms and processes that determine successful invasion are challenges that can be addressed by mathematical modelling with results used to address the key issues of conservation agencies.

Supervisor: Professor Andy White

##### Other projects may be available, please check information about research activities in the relevant areas which can be found here.

## Funding

### MIGSAA studentship

EPSRC / SFC Centre for Doctoral Training: Maxwell Institute Graduate School in Analysis and its Applications (MIGSAA) studentships. Find out more here

### EPSRC-funded PhD studentship

Applications are invited from eligible students for EPSRC-funded PhD studentship for up to 42 months to study any area of Mathematics. Candidates should have a strong mathematical background, including previous study of their chosen area.

British and Irish applicants are eligible for the full scholarship. Other EU students are eligible for a fees only scholarship, i.e. no maintenance grant, and may be eligible for the full scholarship if they have completed their UG/PG studies in the UK in the past 3 or 4 years.

The full scholarship covers all tuition fees and an annual tax-free stipend which is currently around £14,000.

Information about research activities in the relevant areas can be found on the departmental website.

Further enquiries should be addressed to Dr Anastasia Doikou

### James Watt scholarship

As part of an ambitious expansion programme to intensify further our world-leading research programmes, Heriot-Watt University is currently offering James Watt Scholarships in the School of Mathematical & Computer Sciences for the next academic year, opportunities may be available to both UK/EU and Overseas students. For more information see here or contact Dr Anastasia Doikou.

### STFC-funded PhD studentship

The Mathematical Physics Group has an STFC studentship for PhD positions starting 1st October 2017 with a duration of 4 years. Projects are available in the general areas of non-geometric flux compactifications, geometry and D-branes in orientifold backgrounds, higher gauge theory and quantization, and topological phases of matter. The studentship is open to all EU nationals. Informal enquiries and applications should be forwarded to Professor Richard Szabo.

### AWE-funded PhD studentship

The Applied and Numerical Analysis groups have two studentships for PhD positions funded by AWE plc. Projects are available in the areas of nonlinear heat equations and their numerical analysis. The studentships are open to all EU nationals. Informal enquiries and applications should be forwarded to Professor Andrew Lacey or Dr Heiko Gimperlein.

### MACS alumni scholarship

A 20% discount is available for all Heriot-Watt School of Mathematical and Computer Sciences graduates, with a 10% discount available for all other Heriot-Watt graduates. This discount will be applied automatically - there is no requirement to apply.

## How to apply

Informal enquiries can be made to Dr Anastasia Doikou

Full details of how to apply can be found here.

If applicable, please specify for which funding option you are applying.