Resurgence and parametric asymptotics: exact results at all scales

Lead Research Organisation: University of Southampton
Department Name: Sch of Mathematical Sciences

Abstract

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.

Planned Impact

The proposed project focuses of the asymptotic analysis of physical problems with multi-parameter phase spaces, and the importance of general non-perturbative features known as the higher-order Stokes phenomena. This research project is rooted in the fields of mathematical analysis and mathematical physics, however the scope and practicality of both the proposed methods and applications span several disciplines, with potential interest in engineering and physics problems where multiple-scale effects are at play. The multidisciplinary nature of the project provides the evident appeal and importance of the proposed research and the resulting achievements. Consequently, the knowledge derived from this project, bridging methods and disciplines, will lead to innovation and renewed interest to the field of asymptotic theory, and its applications.

Direct applications sought during the course of this project will impact a wide range of subjects: the asymptotic the analysis of pattern formation of non-linear ODEs and PDEs appears in continuum mechanics and mathematical analysis; in non-linear systems, phenomena such as bifurcations and chaos can be addressed by asymptotic methods; the development of exponentially accurate numerical methods and associated error bounds impacts the field of numerical analysis; the numerous applications in mathematical physics range from string theory to quantum mechanics.

There is also great potential for applications and technology transfer to more applied fields beyond mathematics and physics. Examples of these are photonic crystals and metamaterials, microstructure growth, aeroacoustic jet noise and asymptotic homogenization of materials of composites. In all these fields, different methods exist to deal with asymptotic divergent series, although they are generally context specific and not systematic. Here lies the natural applicability of the aimed practical, comprehensive framework of exponential asymptotics.

Given the breadth of potential applications of the proposed research methods, the academic publications will be directed to journals of different disciplines (including Phys. Rev. Lett., Proc. Roy. Soc., SIAM). This will allow the research output to have a greater influence and impact, reaching out to the largest audience possible.

In view of this broad, multidisciplinary interest, outreach and dissemination activities will be especially well suited within the framework of the project. For the dissemination targeting early post- graduate students and established academics, the fellow will organise at least one international mini-workshop at leading mathematics centres, and another local workshop at the host organisation. The main objective here will be linking UK research on asymptotic methods in applied mathematics to the mathematical physics research being performed internationally.


Outreach events aimed at non-specialists and a wider audience will also be an important part of dissemination. The importance of the proposed research can be shown in an intriguing way, using phenomena dealing with rainbows, pattern formations and chaotic solutions, and can be of support in explaining why mathematics is present in our everyday lives. This will be achieved through the coordinated activities already set up by the host organisation, university open days, presentations at schools, and other events designed to engage the general public in the scientific research, such as the so-called science cafes.

Publications

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Aniceto I (2019) The large proper-time expansion of Yang-Mills plasma as a resurgent transseries in Journal of High Energy Physics

 
Description Asymptotics and resurgence in 2D Gravity and Painlevé equations 
Organisation University of Amsterdam
Country Netherlands 
Sector Academic/University 
PI Contribution Analytic asymptotic predictions for the non-perturbative behaviour of large parameter perturbative expansions for the tau-function of Painleve ODEs, and partition function in large-N matrix models. Summation of general transseries solutions and prediction of analytic behaviour such as Lee-Yang zeros. Accurate numerical checks.
Collaborator Contribution The work was done fully collaboratively, with each collaborator contributing to the every part of the work.
Impact Four publications so far in collaboration with the university of Lisbon; the first of these, and on-going work, also in collaboration with the University of Amsterdam. DOI: 10.4310/CNTP.2012.v6.n2.a3 ; 10.1007/s00220-014-2165-z ;10.1007/JHEP03(2015)172 ; 10.1016/j.physrep.2019.02.003
Start Year 2010
 
Description Asymptotics and resurgence in 2D Gravity and Painlevé equations 
Organisation University of Lisbon
Country Portugal 
Sector Academic/University 
PI Contribution Analytic asymptotic predictions for the non-perturbative behaviour of large parameter perturbative expansions for the tau-function of Painleve ODEs, and partition function in large-N matrix models. Summation of general transseries solutions and prediction of analytic behaviour such as Lee-Yang zeros. Accurate numerical checks.
Collaborator Contribution The work was done fully collaboratively, with each collaborator contributing to the every part of the work.
Impact Four publications so far in collaboration with the university of Lisbon; the first of these, and on-going work, also in collaboration with the University of Amsterdam. DOI: 10.4310/CNTP.2012.v6.n2.a3 ; 10.1007/s00220-014-2165-z ;10.1007/JHEP03(2015)172 ; 10.1016/j.physrep.2019.02.003
Start Year 2010
 
Description Late time asymptotics in hydro 
Organisation University of Warsaw
Country Poland 
Sector Academic/University 
PI Contribution Asymptotic analysis of non-perturbative behaviour associated non-hydrodynamic modes of strongly coupled fluids. Prediction of associated coupled quasi-normal mode solutions as perturbations to Einstein PDEs.
Collaborator Contribution Numerical solutions of Einstein PDEs, and expertise in the fluid/gravity correspondence. Analysis of gradient expansion causal hydrodynamics systems.
Impact two publications so far: DOI: 10.1103/PhysRevD.93.085008 ; 10.1007/JHEP02(2019)073
Start Year 2015
 
Description Contributed talk at the BAMC 2019, University of Bath 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Other audiences
Results and Impact Contributed talk at the BAMC 2019. Title of the talk "From asymptotics to exact results: Panleve I and 2d gravity", on 24th April 2019. Work presented was on the summation of asymptotic series to describe the analytic properties of Painlevé transcendents (Painlevé I ODE) to a general audience of applied mathematicians.

Discussions followed, regarding applications of methods to problems with boundary layers and finite difference equations.
Year(s) Of Engagement Activity 2019
 
Description Invited Speaker at the Non-Perturbative Methods in Quantum Field Theory Workshop, ICTP, Trieste, Italy 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact Invited speaker at the workshop on Non-Perturbative Methods in Quantum Field Theory. Title of the talk "A resurgent transseries for N=4 SUSY Yang-Mills", on 4th September 2019. Work presented was on the late time asymptotic behaviour of the energy density for a conformal strongly coupled fluid, and its relation to exponentially small non-hydrodynamic modes. Relation to coupled quasi-normal modes around a black brane when solving for the dual Einstein solution. The audience consisted of field and string theory researchers and students.

discussions followed with different researchers on the different summations methods used to obtain analytic results, relations between tau functions and conformal blocks, and successful numerical approximations of asymptotic series.
Year(s) Of Engagement Activity 2019
URL http://indico.ictp.it/event/8716/speakers
 
Description Invited seminar at the Applied and Interdisciplinary Mathematics seminar series, University of Bath 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other audiences
Results and Impact Invited seminar speaker, title of the talk "Unraveling the analytic structure of observables: from local asymptotics to global properties", on 12th November 2019. Work presented was on the summation of asymptotic series to describe the analytic properties of observables, in particular obtaining the position of Lee-Yang zeros signalling a phase transitions from asymptotic data alone. The general audience was of applied mathematicians.

Discussions followed on possible applications fo transseries in simple asymptotic analysis of water waves.
Year(s) Of Engagement Activity 2019
URL https://wiki.bath.ac.uk/display/cnmseminar/2019-20+Schedule
 
Description Invited seminar at the Geometry and Mathematical Physics Seminar Series, University of Birmingham 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other audiences
Results and Impact Invited seminar, title of the talk "Asymptotics and resurgence in string and gauge theories", on 8th October 2019. Work presented was on the general applications fo resurgence and asymptotics in different physical problems of relevance, from Painleve equations and matrix models to hydrodynamics and gauge theories. The audience was composed of geometry and topology researchers.

Discussions followed on general results obtained from asymptotics and summation for matrix models.
Year(s) Of Engagement Activity 2019
URL http://talks.bham.ac.uk/talk/index/3911
 
Description Invited seminar at the String Theory Seminar Series, DAMTP, University of Cambridge 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other audiences
Results and Impact Invited seminar, title of the talk "A resurgent transseries for N=4 SUSY Yang-Mills", on 7th November 2019. Work presented was about the late time asymptotic behaviour of the energy density for a conformal strongly coupled fluid, and its relation to exponentially small non-hydrodynamic modes. Relation to coupled quasi-normal modes around a black brane when solving for the dual Einstein solution. The audience consisted of field and string theory researchers and students.

Discussions followed regarding the applications of resurgence and asymptotics to chaotic observables with large -N expansions.
Year(s) Of Engagement Activity 2019
URL http://www.damtp.cam.ac.uk/research/hep/string-theory-seminars
 
Description Invited seminar speaker at the 8th LMS OPSFOTA Meeting, University of Reading 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other audiences
Results and Impact Invited speaker at the 8th LMS meeting OPSFOTA, title of the talk "Unraveling the analytic structure of Painleve trancendents: from local asymptotics to global properties", on 20th November 2019. Work presented was on the summation of asymptotic series to describe the analytic properties of Painlevé transcendents (Painlevé I ODE and discrete version) to a general audience of experts on orthogonal polynomials and special functions.

Further discussions followed on the relation between zeros of tau functions and conformal blocks.
Year(s) Of Engagement Activity 2019
URL https://janivirtanen.wordpress.com/lms-otw2019/
 
Description Invited seminar speaker at the Mathematical Physics Seminar Series, University of Bristol 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other audiences
Results and Impact Invited seminar, title of the talk "From asymptotics to exact results: Painleve I and large-N phase transitions", on 20th September 2019. Work presented was on the summation of asymptotic series to describe the analytic properties of Painlevé transcendents (Painlevé I ODE) and large-N expansions of matrix models, in particular how different summation techniques allow us to analyse beyond the asymptotic behaviour. The audience was of experts on matrix models and asymptotic methods.

Discussions followed with local researchers on asymptotic observables in different mathematical models and expected behaviour beyond asymptotics.
Year(s) Of Engagement Activity 2019
URL https://www.bristolmathsresearch.org/seminar/ines-aniceto/
 
Description Invited seminar speaker, University of Plymouth 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other audiences
Results and Impact Invited seminar, title of the talk "A resurgent transseries for N=4 supersymmetric Yang-Mills plasma", on 12th December 2018. Work presented was on the late time asymptotic behaviour of the energy density for a conformal strongly coupled fluid, and its relation to exponentially small non-hydrodynamic modes. Relation to coupled quasi-normal modes around a black brane when solving for the dual Einstein solution. The audience consisted of field and string theory researchers, mathematicians working on geometry and topology and students.

Discussions followed on different applications of the asymptotic methods to observables in field theories.
Year(s) Of Engagement Activity 2018
 
Description Invited speaker at mini-symposium on Recent trends in asymptotics, OPSFA 2019 meeting, Hagenberg, Austria 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact Invited speaker at a mini-symposium on Recent trends in asymptotics, at the OPSFA 2019 meeting. Title of the talk "From asymptotics to exact results: unraveling the analytic structure of Painleve I", on 22nd July 2019. Work presented was on the summation of asymptotic series to describe the analytic properties of Painlevé transcendents (Painlevé I ODE) to a general audience of experts on orthogonal polynomials and special functions.

Many discussions followed, regarding error bounds for the predictions of positions of zeros.
Year(s) Of Engagement Activity 2019
URL https://www3.risc.jku.at/conferences/opsfa2019/?content=schedule
 
Description Invited speaker at the Asymptotic analysis and special functions workshop, ICMS, Edinburgh 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact Invited speaker at the Asymptotic analysis and special functions workshop. Title of the talk "Borel-Pade analysis of asymptotic behaviour: predictions beyond the leading singularities", on 17th April 2019. The work presented analysed how to obtain the exponential small terms and their respective asymptotic series directly from a Borel -Pade perspective. The audience consisted of experts on exponential asymptotics methods, as well as special functions.

Discussions followed on how to introduce error bounds to the results obtain via the Borel-Pade method.
Year(s) Of Engagement Activity 2019
URL https://www.maths.ed.ac.uk/~adri/Workshop2019.pdf
 
Description Invited speaker at the StringMath Conference 2019, University of Uppsala, Sweden 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact Invited speaker at the StringMath Conference. Title of the talk "From asymptotics to exact results in String and Gauge Theories", on 4th July 2019. The presentation included a general introduction to resurgence and asymptotics, and its relation to non-perturbative phenomena. Further work presented was on the late time asymptotic behaviour of the energy density for a conformal strongly coupled fluid, and its relation to exponentially small non-hydrodynamic modes. Relation to coupled quasi-normal modes around a black brane when solving for the dual Einstein solution. The audience consisted of field and string theory researchers, mathematicians working on geometric and topological models relevant for mathematical physics, and students.

Discussions followed with different researchers on the different summations methods used to obtain analytic results, and possibility of applications to different models.
Year(s) Of Engagement Activity 2019
URL https://www.stringmath2019.se
 
Description Invited speaker at the stringtheory.pl conference, University of Warsaw, Poland 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Other audiences
Results and Impact Invited speaker at the stringtheory.pl conference. Title of the talk "A resurgent transseries for N=4 supersymmetric Yang-Mills plasma", on 20th October 2018. Work presented was on the late time asymptotic behaviour of the energy density for a conformal strongly coupled fluid, and its relation to exponentially small non-hydrodynamic modes. Relation to coupled quasi-normal modes around a black brane when solving for the dual Einstein solution. The audience consisted of field and string theory researchers and students.

Discussions followed on possible follow-up works on this subject, including summation of non-hydrodynamic modes and consequences at early times.
Year(s) Of Engagement Activity 2018
URL https://stringtheory.pl/stringtheory-pl-2018/
 
Description Local seminar speaker at the Gravity Seminar Series, University of Southampton 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Postgraduate students
Results and Impact Local seminar, title of the talk "Asymptotics and resurgence in string and gauge theories", on 31st January 2019. Work presented was on the general applications fo resurgence and asymptotics in different physical problems of relevance, from Painleve equations and matrix models to hydrodynamics and gauge theories. The audience consisted of field theory, string theory and classical gravity researchers and students.

Discussions followed on general applications of the methods to understand non-perturbative field configurations.
Year(s) Of Engagement Activity 2019
URL https://www.southampton.ac.uk/maths/news/seminars/2019/01/gravity-aniceto-seminar.page
 
Description Local seminar speaker at the String Theory Seminar Series, University of Southampton 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other audiences
Results and Impact Local seminar speaker, title of the talk "A resurgent transseries for N=4 supersymmetric Yang-Mills plasma", on 31st October 2018. Work presented was on the late time asymptotic behaviour of the energy density for a conformal strongly coupled fluid, and its relation to exponentially small non-hydrodynamic modes. Relation to coupled quasi-normal modes around a black brane when solving for the dual Einstein solution. The audience consisted of field and string theory researchers and students.

Discussions followed on possible application in more general strongly coupled fluids.
Year(s) Of Engagement Activity 2018
URL https://sites.google.com/site/stringtheoryseminars/past-years/string-seminars-2018---2019
 
Description Seminar Invitation at LPTHE in Paris, France 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact Invited seminar speaker, title of the talk " Unraveling the analytic structure of observables: from local asymptotics to phase transitions", on 31st January 2020. Work presented was on the analytic structure of observables from asymptotic perturbation theory, and its relation to phase transitions. The audience of 10-20 researchers was largely composed of condensed matter/field theory physicists.

Discussions followed on the role of acceleration methods in asymptotics and higher order Stokes phenomena.
Year(s) Of Engagement Activity 2020
URL https://www.lpthe.jussieu.fr/~ikhlef/seminaire.html
 
Description Southampton masterclass for high school students 2019, associated to the annual STAG lecture 
Form Of Engagement Activity Participation in an open day or visit at my research institution
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Schools
Results and Impact around 100 pupils from local and regional schools attended a "masterclass", where they explored classic groundbreaking experiments in the development of the current quantum particle physics: Rutherford scattering and quarks and their bound states. These experiments were chosen in preparation for the annual STAG lecture, given by Prof. Nathan Seiberg. The main goal of these experiments was to show two of the major experience that opened the words of particle physics as it is today: the existence of atoms mostly made of empty space, and the existence of quarks and the fundamental particles.

Many discussions arose with the students, in relation to statistics and bias in experiments.
Year(s) Of Engagement Activity 2019