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Applicable resurgent asymptotics: towards a universal theory

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.

“An interpretation of the tronquée poles of Painlevé 1 with rain drops.  In a mathematical analogy the region between calm and disturbed water is delineated by an anti-Stokes line, across which exponentially small terms may grow to dominate the behaviour of a function. (Credit: iStock.com/Photo-Max)”
Programme
4th January 2021 to 25th June 2021

“An interpretation of the tronquée poles of Painlevé 1 with rain drops.  In a mathematical analogy the region between calm and disturbed water is delineated by an anti-Stokes line, across which exponentially small terms may grow to dominate the behaviour of a function. (Credit: iStock.com/Photo-Max)”

Programme Theme

Asymptotic analysis and perturbation methods can provide approximate solutions and analytical properties to a broad range of problems where an exact solution cannot be found. They are therefore some of the most critically important tools in mathematics and theoretical physics. Nevertheless, the existing approaches to study asymptotic problems are often context specific, varying in rigour or practicality. A key challenge, which this programme will seek to address, is to unify these approaches in asymptotics into techniques of enhanced efficacy and broader applicability.

The role of previously neglected exponentially small terms in asymptotics has been formalised, understood and subsequently exploited to deliver a radical change to the century-old, but ambiguous, approach of Poincaré asymptotic analysis.  Significant mathematical breakthroughs have been achieved in a number of areas including rigorous bounds, PDEs, discrete systems and eigenvalue problems. These have wide-ranging applications to, amongst others, fluid dynamics, aero-acoustics, pattern formation, dynamical systems, optics and biomathematics.

Recently, remarkable progress has also been made in theoretical physics in the applications of the comprehensive theory of resurgent asymptotic analysis.  This approach has revealed new and deeper insights into the non-perturbative structure and dynamics of quantum field theories, string theory, random matrix and knot theories, as well as computationally efficient techniques for path integral evaluation.   Simultaneously this has opened up developments in Riemann Hilbert problems, integrable nonlinear systems and orthogonal polynomials with the potential for applications to wide classes of nonlinear multidimensional problems. 

Although overlapping, these advances have developed largely in parallel. However, there is increasing realisation from those working in these distinct areas that there is significant potential for mathematical technology transfer. One ambitious goal of this programme is to bring these communities together to develop a unified set of comprehensive, yet practical, advanced asymptotic approaches, widely applicable not only in mathematics and physics, but also in rapidly emerging areas such as in engineering, data science and systems biology.

The work on unified approaches to asymptotics envisaged during this programme is broad in scope and currently includes transseries and their practical implementation; parametric resurgence and higher order Stokes phenomena for multidimensional systems; analysis of Stokes coefficients; realistic sharp error bounds for highly accurate numerics (e.g.,  Borel-Padé); complex singularity dynamics in finite and late time phenomena; Riemann-Hilbert methods; exact WKB analysis; practical implementation of Lefschetz thimbles in high-dimensional integrals; nonlinear uniform asymptotics; Painlevé analysis and Picard-Lefschetz theory for novel computational methods.

The applications of these approaches under study during the programme include resurgence and non-perturbative physics in gauge theory, matrix models, string theory, AdS/CFT, supersymmetry, and localizable QFTs; highly correlated systems and relativistic hydrodynamics; metastability, free boundary and late time behaviour of nonlinear PDEs; homogenisation and other multiple scales problems; discrete to continuum limits in biological systems; interplay between integrability and asymptotics.

The programme will bring together applied mathematicians, mathematical analysts, theoretical physicists and subject specialists  working on asymptotic analysis to enable significant technology transfer and to inaugurate the next generation of interdisciplinary researchers within these fields.  Given the breadth of activity, and the diverse disciplines involved, the stage is set for further major advances and for unforeseen new directions.

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