Applied mathematics research
Pāngarau tono | Applied mathematics is the branch of mathematics led by a drive to solve real-world problems through the development of theory and numerical methods. We study a wide range of applications to other fields of science, from mathematical biology and climate change to industrial mathematics and laser physics.
In mathematics, dynamical systems theory is the study of systems in which a function describes the time dependence of a point in a geometrical space.
Our research in this area covers a range of topics including heteroclinic cycles, intrinsic excitability, systems with multiple timescales, volume diffusion and the geometry of chaos.
Mathematical biology applies mathematical principles to biological questions. For example, using multiscale modelling of the human lung and the salivary secretion system, we can apply mathematics to the field of health.
The lung model incorporates microscopic cellular events across the entire organ, while the model of the salivary secretion system highlights the role of molecular and cellular elements in the secretion process.
Many real-world problems cannot be solved explicitly and require computational techniques to generate approximate solutions that are sufficiently close to make predictions or recommendations. In numerical analysis, we develop and study properties of such computational techniques, for example, to improve the efficiency or accuracy of the method.
Our research in this area includes the development of new numerical methods for use in dynamical systems, fluid mechanics and wave propagation phenomena. When starting from a real-world application, we employ control theory as well as parameter estimations to help gain insight in system behavior and parameter sensitivity.
Pattern formation and emergent dynamics
Patterns emerge as ordered outcomes of self-assembly/organisation while emergent dynamics occurs when the sum of multiple interacting units in a network display new behaviours beyond that of each individual unit. Both these situations naturally arise in diverse physical systems, from aircraft engines to powergrids and even neurons firing in our brain.
We look to understand the mechanisms that create such complex behaviours by studying 'minimal recipe' models that retain enough detail of the inspiring application but are amenable to analysis. We use methods developed in diverse mathematical areas such as dynamical systems, numerical analysis, asymptotic methods, numerical algebraic geometry, and topological data analysis.
- Mathematical biology and physiology
- Rare event and Monte Carlo simulation
- Inverse problems
- Numerical analysis and partial differential equations
- Dynamical systems and nonlinear ordinary differential equations
- Local and global bifurcations
- Models of intracellular calcium dynamics
- Dynamical systems theory and its applications to real-world problems
- Effects of time delays and different time scales
- Dynamical systems theory and applications
- Numerical methods for global invariant manifolds
- Transient dynamics in systems with timescales
- Dynamical systems and applications
- Dynamics of networks
- Coupled oscillator dynamics and model reduction
- Chaotic fluid mixing
- Mixing by cutting-and-shuffling, with application to granular materials
- Mathematical medicine and physiology
- Dynamical systems and the theory of calcium waves and oscillations
- Spatio-temporal pattern formation
- Soft matter quasicrystals
- Emergent dynamics
- Control theory of partial differential equations
- Differential equations and applications to real-world problems
- Nonlinear functional analysis and partial differential equations
- Fluid dynamics and hydrodynamic stability