# Function theory in multiply-connected domains & applications to physical systems

Lead Research Organisation:
Imperial College London

Department Name: Dept of Mathematics

### Abstract

Multiply-connected domains are what mathematicians call regions with holes . In physics, the holes can correspond to lots of different things such as air bubbles in fluids or regions of swirling motion (for example, storm systems or hurricanes) in the atmosphere or different clusters of bacteria competing for a common food supply. Thus, the mathematical concept of a multiply-connected domain occurs in many different places in the study of everyday phenomena. To understand such phenomena, it is necessary to study and understand mathematical models of them. This requires a knowledge of mathematical functions and techniques specially tailored to the multiply-connected domains in which these phenomena are taking place. Unfortunately, mathematicians in the past who have developed the mathematics of functions in multiply-connected domains have not done a very good job of translating the significance of their results to scientists interested in describing and studying everyday phenomena such as bubbles in fluids or the motion of storm systems. Yet, recent work by the PI has shown that if one can successfully translate these mathematical results and demonstrate their applicability to these various everyday phenomena, powerful new techniques become available to those scientists who study them, making their jobs much easier and leading to new Insights. This research proposes to continue in this crusade to develop and apply the mathematical results of classicalfunction theory and complex analysis to real-life problems.

## People |
## ORCID iD |

Darren Gregory Crowdy (Principal Investigator) |

### Publications

Antipov Y
(2007)

*Riemann-Hilbert Problem for Automorphic Functions and the Schottky-Klein Prime Function*in Complex Analysis and Operator Theory*Riemann-Hilbert Problem for Automorphic Functions and the Schottky-Klein Prime Function*in Complex Analysis and Operator Theory

Baddoo PJ
(2019)

*Periodic Schwarz-Christoffel mappings with multiple boundaries per period.*in Proceedings. Mathematical, physical, and engineering sciences
Buchak P
(2008)

*Exact solutions for the evolution of ellipsoidal inclusions in porous media*in The Quarterly Journal of Mechanics and Applied Mathematics
Choi J
(2010)

*Diffusion-limited aggregation on curved surfaces*in EPL (Europhysics Letters)
Crowdy D
(2010)

*Stokes flows past gaps in a wall*in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Crowdy D
(2010)

*The Schottky-Klein Prime Function on the Schottky Double of Planar Domains*in Computational Methods and Function Theory
Crowdy D
(2008)

*Conformal mappings from annuli to canonical doubly connected Bell representations*in Journal of Mathematical Analysis and Applications
Crowdy D
(2008)

*Multiply Connected Quadrature Domains and the Bergman Kernel Function*in Complex Analysis and Operator Theory
Crowdy D
(2009)

*An assembly of steadily translating bubbles in a Hele-Shaw channel*in NonlinearityDescription | Many problems in the applied sciences involve interacting entities: these can be anything from aerofoils, vortices, elastic inclusions, swimming microorganisms or electrodes. These entities interact by communicating information through the ambient medium surrounding them. Mathematically, this ambient medium is "multiply connected'' because of the presence of "holes'' caused by the presence of the interacting entities. Until recently, there has been very little theoretical technology specifically aimed at solving the mathematical problems that arise in such cases. The principal focus of this Fellowship project has been to devise and develop a mathematical framework for solving problems in multiply connected domains. The applicability of this framework is very broad and the new theoretical approach makes available a new armoury of techniques for the applied scientist. This project has demonstrated the versatility of the new methods in a variety of circumstances. |

Exploitation Route | The collected results of this programme of research has provided a new mathematical framework, of some versatility, for the solution of physical problems in holey, or multiply connected, domains. The generality of the approach lends it broad potential for future ongoing development. There is already scope for translation of the ideas into new software algorithms. This will render the techniques more accessible to the wider scientific community. |

Sectors | Aerospace, Defence and Marine,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology,Other |

URL | http://wwwf.imperial.ac.uk/~dgcrowdy/ |

Description | The mathematical framework for multiple connectivity has already found its footing and is now being adopted by a variety of researchers around the world. Groups in Japan, Brazil and the United States have espoused the methods and are developing them in their own applications and scientific contexts. For example, in Japan, the Japan Science and Technology agency (JST) has recently funded long-term Fellowship grants (within their CREST and PRESTO schemes) whose scientific purview is essentially based on building on the developments of this grant. |

First Year Of Impact | 2011 |

Sector | Aerospace, Defence and Marine,Education,Manufacturing, including Industrial Biotechology |

Impact Types | Societal |

Description | ARC Research Grant |

Amount | $360,000 (AUD) |

Organisation | Australian Research Council |

Sector | Public |

Country | Australia |

Start | 09/2013 |

End | 09/2015 |

Description | EPSRC |

Amount | £7,165 (GBP) |

Funding ID | EP/I004920/1 |

Organisation | Engineering and Physical Sciences Research Council (EPSRC) |

Sector | Public |

Country | United Kingdom |

Start | 06/2010 |

End | 12/2010 |

Description | Leverhulme Trust Research Grant |

Amount | £242,000 (GBP) |

Funding ID | RPG-358 |

Organisation | The Leverhulme Trust |

Sector | Charity/Non Profit |

Country | United Kingdom |

Start | 10/2012 |

End | 10/2015 |

Description | National Science Foundation |

Amount | £152,000 (GBP) |

Funding ID | 970113 |

Organisation | National Science Foundation (NSF) |

Sector | Public |

Country | United States |

Start | 10/2010 |

End | 10/2013 |

Description | Public Lecture in Melbourne, Australia |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | Yes |

Geographic Reach | Local |

Primary Audience | Public/other audiences |

Results and Impact | The talk engaged local community in mathematics research. The talk engaged local community in mathematics research and increased awareness of what applied mathematicians do. |

Year(s) Of Engagement Activity | 2011 |

URL | http://amsi.org.au/ |