# Applied Mathematics Group

The Applied Mathematics group applies mathematical methods to increase our understanding of biological and physical phenomena.

### Group members

- Professor Mark Broom (Professor of Mathematics, Head of Group)
- Dr Andrea Baronchelli (Reader in Mathematics)
- Dr Oliver Kerr (Reader in Mathematics)
- Dr Robert Noble (Lecturer in Mathematics)
- Dr Lara Silvers (Reader in Mathematics)
- Dr Soumya Dasgupta (Postdoctoral Researcher)
- Dr Matthieu Nadini (Postdoctoral Researcher)
- Raneem Aizouk (PhD student, 2018-)
- Johann Bauer (PhD student, 2017-)
- Gianluca Boni (PhD student, 2020-)
- Alberto Bracci (PhD student, 2019-)
- Veselin Manojlovic (PhD student, 2020-)
- Alan Scaramangas (PhD student, 2018-)
- Pedro Schimit (Honorary Visiting Fellow)

Former members:

- Laura Alessandretti (PhD student, 2015-2018)
- Abeer Ebahrawy (PhD student, 2016-2019)
- Christoforos Hadjichrysanthou (PhD student, 2010-2012)
- Anne Kandler (Lecturer in Mathematics, 2012-2016)
- Klodeta Kura (PhD student, 2012-2016)
- Karan Pattni (PhD student, 2013-2017)
- Mahdi Raza (PhD student, 2013-2016)
- Fabiano Ribeiro (Visiting Researcher 2017-18)
- Jan Teichmann (PhD student, 2011-2015)

The group also hosts regular short term research visitors.

There are three main areas of research: evolutionary game theory, network science and fluid dynamics.

## Evolutionary game theory

Since its inception in the 1960s, evolutionary game theory has become increasingly influential in the modelling of biology. Important biological phenomena, such as the fact that the sex ratio of so many species is close to one half, the evolution of cooperative behaviour and the existence of costly ornaments like the peacock's tail, have been explained with ideas underpinned by game theoretical modelling.

Our work involves the development of the general mathematical theory of evolution, including an extension of the standard two player models to the multiplayer case, the integration of evolutionary games and life history theory, as well as the consideration of evolutionary games on networks and involving structured populations more generally.

We model specific biological behaviour. In particular we consider parasitic behaviour such as food stealing (kleptoparasitism) and brood parasitism, where birds lay eggs in the nests of others. We also investigate biological signalling, where animals signal invisible properties in order to attract mates or ward off predators.

We also use mathematical and computational models to investigate the evolution and ecology of cancer, in collaboration with experimental biologists and clinicians. A common thread in his research is investigating how aspects of ecology shape evolutionary dynamics. The overarching vision is to establish general mathematical principles of cancer evolution. Methods include agent-based models, analysis of dynamical systems, stochastic processes, matrix population models, and Bayesian data analysis.

## Human behaviour, collective dynamics and network science

Human cognition is the product of the interaction of tens of billions of neurons, societies are constituted by millions of individuals, and ecosystems consist of many interacting species. However, understanding the system-scale emerging properties of such complex systems starting from the knowledge of their constituents is unfeasible. A powerful framework to overcome this problem is provided by network science. Thanks to the network approach, where the units of the system are described as nodes and their interaction patterns as links, network science has provided in the last 15 years a unifying framework to study different systems under the same conceptual lens, with important practical consequences.

Research in the Mathematical Biology group includes fundamental investigations on the behaviour of dynamical processes on complex networks (e.g., random walks, reaction-diffusion processes, etc.), and more interdisciplinary research efforts. Among the latter, issues in language dynamics (consensus problems, emergence of conventions, etc.), evolution (biological conditions for language diversity, evolution in a changing environment, etc.), social sciences (the study of information and opinion spreading in online and offline social networks), and cognitive science (emergence of shared categorisation systems).

## Fluid Dynamics

Dr Silvers and Dr Kerr consider modern problems in fluid dynamics. Dr Silvers' central theme of research area is to extend our understanding of instabilities in fluids. She is primarily motivated by understanding astrophysical and geophysical regions. One current research theme focuses on understanding shear flow instabilities, the turbulent state that arises and how these flows interact with a magnetic field. A second theme focuses on deepening our understanding of how buoyant magnetic structures interact with various flows. Finally she is working on addressing fundamental questions related to convection, especially double-diffusive convection.

Dr Kerr's research interests lies in the area of theoretical fluid dynamics, and in particular the study of the onset and evolution of thermal and double-diffusive convection. More recently the focus has been on the stability of evolving systems, such as the heating of a body at an isolated boundary (horizontal or vertical, with or without salinity gradients). In such situations the fluid can be initially stable, and then becoming unstable sometime after the initiation of the heating. Dr Kerr's research involves developing methods to find the optimal growth as the background state evolves, and identifying the most appropriate measure for this growth.

If you are interested in finding out more about research in Mathematical Biology at City, University of London, please contact Professor Mark Broom.