Department of Mathematics

Our researchers develop methods for deriving low-dimensional models of various physical phenomena that take the form of systems of ordinary differential equations and investigate the properties of solutions of such systems employing analytical and computational methods. Our projects are focused on systems as diverse as atmospheric flows and flows of complex non-isothermal fluids, multi-phase gas-fluid systems, chemical reactors, DNA regulatory networks, epidemiology and interactions between tumours and the immune system.

We are also interested in large scale numerical solution of nonlinear differential equations with applications in modelling material failure, fracture and rupture propagation. The projects we run also involve optimal control of stochastic Volterra integral equations in Hilbert spaces, with applications in EEG modelling of brain wave activity and mathematical finance.

Finally, we use bifurcation theory and numerical continuation methods to study stationary solutions of nonlinear PDEs for the dynamics of ultra-thin layers of liquid polymers, with applications in coating and painting processes.

Contact: Associate Professor Tonghua Zhang, Associate Professor Federico Frascoli, Dr Louise Olsen-Kettle, Dr Andriy Pototskyy, Professor Sergey Suslov, Professor Vo Anh. 

Mathematical biology is an active and fast growing inter-disciplinary area in which mathematical concepts and techniques are being applied to a variety of problems in the biological sciences. Our group is very active in the mathematical modelling of a number of fundamental biological processes including:

• predator-prey models for the control of insect populations and pests
• predictive modelling for the development and spread of infectious diseases (pertussis) and epidemics (influenza, malaria and AIDS)
• stochastic and computational models of bacteria and active particles on thin films
• models of millimetre-sized Marangoni surfers that propel themselves at fluid interfaces, by creating gradients of surface tension.
• investigating the effect of external mechanical vibration on the formation of Faraday-like body waves in living earthworms.
   

We are also investigating mathematical immunology, with a focus on tumour-immune interactions, immunotherapy and oncolytic virotherapy.

Contact: Associate Professor Tonghua Zhang, Associate Professor Federico Frascoli, Dr Andriy Pototskyy

Our research focus is on the teaching and learning of mathematics at tertiary level. We use cognitive and socio-cultural approaches to investigate the effects of different pedagogies and technologies at university. Examples of our work and interests include:

• the place of mathematics in science, technology and engineering education
• the role of emotions in the learning of mathematics
• mathematical modelling as a way to engage students in conceptual understanding
• practices that include students as partners, and their effect on students in their development as mathematics learners (identity)
• students’ use of examples in tertiary mathematical reasoning
• student engagement with learning resources
• approaches to measuring the impact of mathematics and statistics support.
   

Contact: Dr Ant Sowards, Dr Paul Hernandez Martinez, Dr Emily Cook

Fluid dynamics modelling and computational mechanics in material sciences

Our group develops models of material failure and damage evolution in anisotropic and composite materials, with the goal of forecasting risk and damage in a range of applications in oil and gas, mining, construction and manufacturing industries.

We are also interested in a range of problems at the frontier of fluid dynamics and material science, including drag reduction in fluids, properties of colloidal suspensions, separation of granular and powder materials, cavitation, electrochemical dynamics and Leidenfrost effects.

Contact: Professor Derek Chan, Dr Louise Olsen-Kettle

Hydrodynamics stability theory 

Hydrodynamic Stability theory (HST) aims at finding multiple solutions of partial differential equations describing fluid motion, for example Navier-Stokes, thermal energy and constitutive equations describing fluid's physical properties and determining which of them correspond to physically observable situations. It uses ideas of Bifurcation and Dynamical Systems theories and relies on Asymptotic Analysis and high-accuracy computational methods for solving partial differential equations. 

Our current interests include theoretical developments of general aspects of HST related to its validity and accuracy, as well as its applications to flows of complex fluids with unusual physical properties found in a wide range of modern technological and physical applications. These include magnetic nano-fluids, piezo-viscous and electrically conducting fluids and fluids that exhibit strong variation of their transport properties with temperature.

We are also interested in Faraday instabilities arising in vibrated multi-fluid systems such as drops and films as well as the study of bifurcations observed in non-linearly oscillating gas bubbles. Collaboration with experimental researchers in other disciplines is an integral part of many of our projects.

Contact: Professor Sergey Suslov, Dr Andriy Pototskyy

Optimisation and operation research 

The goal of optimisation in mathematical programming is to obtain "best available" values of an objective function subject to certain constraints. Many real-life applications, including scheduling, outputs and quality-monitoring in manufacturing and resource planning require the optimisation of different types of functions.

Our main interests include:

• the design of efficient and reliable computational algorithms for the solutions of complex optimisation problems
• the use of polynomial spline approximation through optimisation
• the limits of approximating methods for a continuous function using a piecewise polynomial. The case of knots joining the polynomials being also variable is still an open problem.
   

Contact: Dr Nadezda Sukhorukova

Statistical mechanics, Molecular Dynamics and Nanofluidics

Our work in this area is primarily devoted to understanding the behaviour of soft matter, e.g. water, ionic fluids, solutions, polymer melts and mixtures, etc. We are interested in a range of properties at equilibrium and out of equilibrium, where systems may be confined at nano-scales and driven by external fields such as temperature gradients, mechanical stresses and rotating electrodynamic fields.

Equilibrium systems of polymers  

We have developed radically efficient Monte Carlo sampling algorithms and novel enumeration algorithms for studying combinatorial versions of these models. Current research examples include:

• efficient implementation of the pivot algorithm for dilute polymer systems allowing simulations of unprecedented accuracy of polymers with up to one billion monomers.
• efficient implementation of connectivity changing moves for dense polymers allowing rapid sampling of systems with millions of monomers
• novel enumeration algorithms for 3d self-avoiding walks and polygons
• novel enumeration algorithms for 2d self-avoiding walks and self-avoiding polygons. These algorithms are the current world record holders for enumeration of these important combinatorial objects. For example, there are exactly 17 076 613 429 289 025 223 970 687 974 244 417 384 681 143 572 320 self-avoiding polygons of 130 steps on the square lattice.
   

Non-equilibrium systems

We are interested in developing simulation algorithms that are fully compatible with the principles of statistical mechanics and have a firm theoretical foundation. The application of these algorithms to significant technological problems in soft matter is also actively undertaken.

Current research examples include:

• the fundamental problem of how one modifies the highly successful Navier-Stokes equations for fluid flow at the nanometer length scale
• the use of non-equilibrium molecular dynamics (NEMD) methods in systems of alkanes and dense polymer melts to study the foundations of rheology
• the role of microscopic chaos in determining the transport coefficients and the ergodic behaviour of atomic systems at the nano-scale
• the flow of molecular liquids such as water or solutions in highly confined geometry such as carbon nanotubes or graphene sheets
• the development and application of response theory (both linear and non-linear) to systems of atoms and molecules driven out of equilibrium by external fields
• novel methods of fluid actuation at the nano-scale, such as electropumping
• the application of NEMD methods to study molecular tribology and model new lubricants with significantly lower resistance to friction and improved energy efficiency.
   

Contact: Professor Billy Todd, Associate Professor Federico Frascoli, Dr Andrey Pototskyy, Professor Derek Chan, Dr Nathan Clisby