Background

Complex analysis is a branch of mathematics that studies analytical properties of functions of complex variables. It lies on the intersection of several areas of mathematics, both pure and applied, and has important connections to asymptotic, harmonic and numerical analysis. Techniques based on complex variables are very powerful, with a large number of applications to the solution of physical problems.
 
The discipline covers a wide range of different techniques including solution methods to free-boundary problems such as Hele-Shaw and Stokes flow, conformal mappings, Fourier and other transform methods and Riemann-Hilbert problems. In practice many problems that may be difficult to solve in the real domain can be more easily solved when transformed into complex variables due to a number of special properties of the complex domain.
 
Importantly, there has been a surge of activity in the advancement of complex analysis methods in recent years, driven by applications in engineering, biology and medicine. The application of these methods to real world problems include propagation of acoustic waves relevant for the design of jet engines, development of boundary-integral techniques useful for solution of many problems arising in solid and fluid mechanics as well as conformal geometry in imaging, shape analysis and computer vision.
 
This knowledge exchange day was part of a four month research programme at the Isaac Newton Institute on Complex Analysis: techniques applications and computations. It formed day three of the week long workshop Complex analysis in mathematical physics and applications. The research programme brings together researchers from mathematics, physics and engineering communities, whose research shares a common theme of using complex analysis to attack real-world problems.
 

Aims and Objectives

This knowledge exchange event showcased the state of the art application of complex analysis methods to solve industrial driven problems, as well as where mathematical advances in this area are most needed. Another key aim was to identify techniques most commonly used by end-users and where further improvements would be most beneficial. This will help focus the areas of investigation undertaken in the remainder of the research programme.
 
The programme for the day reflected the breadth of application areas where complex analysis methods are important and included talks representing both academic research and end-user perspectives from a range of different industrial areas. These talks also highlighted how complex analysis methods have the potential to tackle challenging problems in a number of areas including understanding of aeroacoustics, medical imaging methods, tissue engineering approaches and radar signal processing. Furthermore, the increasing difficulty of many problems in these fields will help inform the agenda for complex analysis research.
 
The four sessions that were covered are:

• Acoustics
• Continuum mechanics
• Life sciences
• Radar

 
This event brought together mathematicians and scientists working at the forefront of complex variable theory and their applications, with end users from industry to further investigate opportunities for the use of complex variable methods in the solution of applied problems.
 

Registration and Venue

A registration fee was charged to cover direct costs of attendance at this event. This was £25 for academic and public sector attendees and £50 for industrial attendees. 

There was no fee for registered participants of the INI Complex Analysis: techniques applications and computations Programme. 

The workshop took place at the Isaac Newton Institute for Mathematical Sciences in Cambridge. Please see the Isaac Newton Institute website for further information about the venue.