PhD projects

• Real world problem solving using dynamical systems, stochastic modelling and queueing theory for stochastic transport and signalling in cells. 

• Real & Computational Algebraic Geometry: Possible subjects include nonnegativity of real polynomials, polynomial system solving, semialgebraic sets, and algorithmic aspects of real algebraic & convex geometry.
  
• Polynomial & Convex Optimization: Potential topics include convex relaxations, designing algorithms, exploiting structure (e.g. sparsity), and applications in science & engineering.

• Dynamical Systems and Ergodic Theory: Projects that combine techniques from nonlinear dynamics, ergodic theory, functional analysis, differential geometry, or machine learning and can range from pure mathematical theory through to numerical techniques and applications (including ocean/atmosphere/fluids/blood flow), depending on the student.
• Optimisation: Projects are occasionally available in optimisation, mainly using either techniques from mixed integer programming to solve applied problems (e.g. transport, medicine,…) or mathematical problems arising from dynamics.

• Modelling and analysis of ocean biogeochemical cycles including isotope dynamics, inverse modelling of hydrographic data to detect climate-driven circulation changes, and analysis of large-scale ocean transport. PhD students should be highly motivated, have a strong background in applied mathematics and/or theoretical physics, and will have the opportunity to contribute to shaping their project.

• Data-Driven Multi-stage Robust Optimization:
  The aim of this study is to develop mathematical principles for multi-stage robust optimization problems, which can identify true optimal solutions and can readily be validated by common computer algorithms, to design associated  data-driven numerical methods to locate these solutions and to provide an advanced optimization framework to solve a wide range of real-life optimization models of multi-stage technical decision-making under evolving uncertainty.
• Semi-algebraic Global Optimization:
  The goal of this study is to examine classes of semi-algebraic global optimization problems, where the constraints are defined by polynomial equations and inequalities. These problems have numerous locally best solutions that are not globally best. We develop mathematical principles and numerical methods which can identify and locate the globally best solutions.

• Detection and cloaking of surface water waves created by submerged objects
• Decomposition of ocean currents into wave-like and eddy-like components

• Theory and application of Quasi-Monte Carlo methods:
  for high dimensional integration, approximation, and related problems.

• Computational Mathematics:
  with specialised topics in radial basis functions, random fields, uncertainty quantification, partial differential equations on spheres and manifolds, stochastic partial differential equations. 

• Discrete Integrable Systems:
  These are birational maps with particularly ordered dynamics and their study is a nice motivation for using algebraic geometry, symmetry, ideal theory and number theory in the study of dynamical systems.
• Arithmetic Dynamics: 
  This field is the study of iterated rational maps over the integers or rationals or over finite fields, rather than the complex or real numbers. I am particularly interested in how the usual structures present in dynamical systems over the continuum manifest themselves over discrete spaces.

• Convex geometry:
  Focused on the study of the facial structure of convex sets and the relations between the geometry of convex optimisation problems and performance of numerical methods. The project can be oriented towards convex algebraic geometry, experimental mathematics or classical convex analysis.

• Algebraic and Geometric Aspects of Integrable Systems:
  The ubiquitous nature of integrable systems is reflected in their (apparent or disguised) presence in a wide range of areas in both mathematics and (mathematical) physics. Projects focus on the algebraic and/or geometric aspects of discrete and/or continuous integrable systems, depending on the individual student's background and preferences. 

Hello how are you. Goodbye :)

• Fluid flow in channels with porous walls
• Mathematics education
• Nonlinear differential equations
• Difference equations
• Dynamic equations on time scales

• How many oceans are there? Using novel statistical and machine learning techniques to characterise oceanic zones and provide a blueprint for quantifying the ocean's role in a changing climate.
• How does heat get into the ocean? An investigation of the physical mechanisms that control the ocean's uptake of heat and its effect on climate.
• Making climate models work better: Developing new methods to validate and improve the inner workings of numerical climate models and improve their projections of global warming and its impacts.
• Will it mix? New perspectives on turbulence in rotating fluid flows and how we estimate mixing from observations.