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Mathematical theory and applications of multiple wave scattering

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

4th January 2023 to 30th June 2023
Luke Bennetts
Mike Meylan
Malte Peter
Valerie Pinfield
Olga Umnova

Programme theme:

Waves are all around us, as acoustic waves, elastic waves, electromagnetic waves, gravitational waves or water waves. Multiple wave scattering is a vibrant and expanding research area of interest to mathematicians, physicists, engineers and others concerned with the properties of waves in complex materials. The mathematical theory has a long history and was studied by luminaries of 19th-century science, such as Maxwell and Rayleigh. A sound understanding of the field is essential to deliver the potential transformative innovations beyond the realm of sci-fi that can be achieved by manipulating wave behaviour, such as invisibility cloaks, noise cancellation, imaging living cells, and many more. Contemporary mathematical challenges are extensive, ranging from design of metamaterials to numerical difficulties associated with massive scattering simulations. The programme will be an interdisciplinary joining of forces to elucidate the fundamental mathematical aspects of multiple wave scattering in a variety of contexts, aiming for a deep understanding of the commonalities.

The programme will address diverse methodological approaches for multiple wave scattering problems to (a) establish effective communication and mutual understanding across disciplines and application areas; (b) identify correspondences and divergences between the different methods; (c) highlight the most critical challenges for the various methods; (d) identify and progress the approaches most likely to be successful in addressing these challenges; and (e) accelerate innovation in applications such as metamaterial design and medical imaging. The methods that will be covered include (but are not constrained to) homogenisation methods, multipole expansions, addition theorems, spectral methods, eigenfunction methods, plane-wave representations, transfer operators, inverse methods for image reconstruction and semi-analytical techniques.

A number of mathematical questions will be addressed, with common interest across many applications and disciplines. These include closure assumptions, often addressed through historic but unexplored approximations, the effect of structural disorder (e.g. defects or pseudo-randomness), the validity of approximate solutions (Mie, Rayleigh, Born), and approaches to the challenging mid-wavelength regime. Solutions of inverse problems, such as  the design of metamaterials or image reconstruction, will also be a topic of interest.

Consideration of computational methods for multiple scattering problems will go hand-in-hand with the discussion of analytical and semi-analytical methods. Rapid advances in the field have generated new capabilities to investigate high-frequency regimes using homogenisation techniques, direct numerical simulations of vast arrays of scatterers, energy density field modelling using transfer operator methods, etc. Bringing together researchers across disciplines covering analytical methods, computational techniques and scientific/engineering applications will facilitate cross-fertilisation of ideas and promote innovation.

Multiple wave scattering has many applications, which has caused research in the field to be fragmented, despite the known commonalities at a fundamental level. Researchers in medical imaging, electromagnetics, acoustics, quantum electronic structures, atmospheric imaging, water/structure interactions, etc., often adopt or develop related techniques, but communication between the disciplines can be limited and challenging. The programme aims to bring together mathematicians, physicists and engineers across diverse disciplines. We aim to build on shared fundamentals, highlight the most pressing research challenges, and exchange state-of-the-art methodologies and approaches. This all-community approach will promote solutions to the issues facing many practical applications in complex media through the rigorous underpinning of mathematical techniques and the development of effective computational methods.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons