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Details of Grant 

EPSRC Reference: EP/S004076/1
Title: Resurgence and parametric asymptotics: exact results at all scales
Principal Investigator: Varela Aniceto, Dr I
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Instituto Superior Tecnico University of Edinburgh University of Nottingham
Department: Sch of Mathematical Sciences
Organisation: University of Southampton
Scheme: EPSRC Fellowship
Starts: 01 October 2018 Ends: 30 September 2023 Value (£): 426,226
EPSRC Research Topic Classifications:
Mathematical Analysis Mathematical Physics
Numerical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
18 Jul 2018 EPSRC Mathematical Sciences Fellowship Interviews July 2018 Announced
06 Jun 2018 EPSRC Mathematical Sciences Prioritisation Panel June 2018 Announced
Summary on Grant Application Form
Physical phenomena observed in nature can be modelled by a wide range of mathematical theories. From shock waves in non-linear optics, to turbulence in hydrodynamics, and gravitational ripples caused by collapsing black holes, scientists have a strong grasp on the complex equations governing these phenomena. Nevertheless, solving these equations can be arduous, often done numerically or by taking the parameters of the theory to be small, hence approximating the results as a series of terms. This is a procedure called perturbation theory, and it plays a major role in fields ranging from mathematics to engineering.

When the later terms of this series increase in magnitude, the series diverges. Despite this, summing a small number of such terms often gives a good approximation, and the series is called asymptotic. This behaviour, rather than problematic, is essential to build a complete understanding of physical observables. Hidden in it is a realm of phenomena so small that they disappear from perturbation theory (non-perturbative), but which can grow to eventually dominate our results, changing the physics.

These phenomena are widely found in mathematical descriptions of nature, but the existing methods to study them are mostly problem specific, with varying rigour and practicality. A systematic, unified framework of asymptotics is still missing. A powerful method which studies the intimate relation between asymptotics and non-perturbative phenomena is resurgence theory.

The proposed work aims to use resurgence to unify the different strands of research in asymptotics, bridging methods and disciplines, to obtain a comprehensive and practical theory of asymptotics. Blending the theoretical but powerful aspects of resurgence with practical, numerically driven theories of (exponential) asymptotics, we will be able to effectively construct full non-perturbative solutions to physical problems where only the perturbative series is known.

The research will be conducted in great extend at the host organisation, the Applied Mathematics group at the University of Southampton, in collaboration with Prof C.J. Howls, as well as members of the String Theory group. A strong collaboration with project partners A.B. Olde Daalhuis (Edinburgh), J. King (Nottingham) and R.Schiappa (Lisbon) will be vital to the success of this project. Indeed, by combining the fellow's expertise in the systematic, practical implementations of resurgence theory with the project partners' broad knowledge in the field of exponential asymptotics and its applications, this proposal presents a clear path to achieve this goal.

The physical problems that will be addressed during this project belong to a wide spectrum of research areas in mathematical sciences. Examples of these are: the formation of localised patterns in boundary value problems; the emergence of instabilities from time evolution in non-linear PDEs; asymptotic analysis of free energies of matrix models and gauge theories, and their dependence on coupling constant and rank of the gauge group; calculation of Stokes invariants for linear differential equations, from perturbative data or using integrability tools.

These problems occur in the areas of continuum mechanics, mathematical analysis, non-linear systems and mathematical analysis. The scope and potential inter-disciplinary impact of this research is evident from the universality of the features controlling asymptotic behaviour, which this project is set to address.
Key Findings
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Organisation Website: http://www.soton.ac.uk