Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications

Created: 2019-08-07 13:30
Institution: Isaac Newton Institute for Mathematical Sciences
Description: Programme Theme
The Wiener-Hopf technique enables us to solve numerous physical problems motivated by real world applications modelled, for example, by partial differential equations and stochastic processes. The Wiener-Hopf technique is currently used in a wide range of disciplines including acoustics, finance, Lèvi processes, hydrodynamics, elasticity, potential theory and electromagnetism.

The theory of scalar Wiener-Hopf equations is now very rich and well developed. In contrast, much less is known about matrix Wiener-Hopf equations. These are a natural extension of the scalar case, and enable us to model more advanced problems. As it stands, solutions to matrix Wiener-Hopf problems have to be constructed on a case-by-case basis, or in an approximate fashion. The focus of this programme will be on matrix Wiener-Hopf equations, constructive numerical methods and applications. The three main aims will be:

Matrix factorisation, approximate methods and their numerical implementation.
There are numerous approximate methods for Wiener-Hopf matrix factorisation. However, there is no clear picture for deciding when a method is appropriate or numerically efficient. Consequently, approximate Wiener-Hopf factorisation is highly specialised and a difficult subject for non-experts. An important outcome of the programme will be to make this more routine by filling the outstanding gaps in the literature. There is also a lot of scope for the theoretical study of methods motivated by applications. This requires a close collaboration between mathematicians from the pure and applied communities. Developing existing methods into working algorithms and making them freely available in a toolbox will make the Wiener-Hopf technique more easily accessible for applications than is currently the case.

Establishing links between different applications of the Wiener-Hopf method.
There are several disjoint communities who rely on the Wiener-Hopf method, and developments in one field often go unnoticed in another. There is much to be gained by generalising the developments in one area to solve open problems in other areas.

Consolidating existing knowledge and developing a set of promising new directions.
We will consider the area as a whole, pose open problems and map out promising directions. This will highlight new developments and applications, and make the area more attractive to young researchers.

The above aims will be achieved by bringing together internationally leading experts in the Wiener-Hopf technique from diverse areas and catalyse new interactions between them. There will be an emphasis on enabling young researchers to meet and interact with established experts.

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