Vincent Ardourel – "Intertheory Relations in Fluid Mechanics"
This talk focuses on intertheory relations in fluid mechanics. I examine two cases in which I discuss whether there is a failure of limiting reduction due to singular limits. First, I focus on the relation between classical mechanics and fluid mechanics. Specifically, I analyze the derivation of the Navier-Stokes equations from Newton's equations that involve the Boltzman-Grad limit and the hydrodynamic limit. Then, I focus on the relation between viscous and ideal (inviscid) fluid mechanics. I discuss the conditions under which the inviscid limit (or high Reynolds limit) between the Navier-Stokes equation and the Euler equation is singular and the consequences we should draw from this. In both cases, I analyze the role of asymptotic expansions in dealing with singular limits.
Michael Berry – "Divergence, resurgence and borderland asymptotics in physics"
Starting with the 1747 discovery by Bayes that Stirling’s series for the gamma functions is divergent, and Euler’s more substantial investigations shortly afterwards, I explain the seminal 1847 discovery by Stokes that in his divergent series for Airy’s rainbow function a small exponential appears when hidden behind a large one. In the modern period, this jump was understood as universal and smooth, and originating in the resummed divergent tail of the large exponential. Underlying this calculation is the resurgence discovered by Dingle and Écalle, in which the high and low orders of the different divergent series describing the same function are related. Recursive application of resurgence leads to exponentially improved asymptotics. Relations between different theories describing physical phenomena at different levels involve singular limits, naturally signalled by divergent series. This will be illustrated by waves near caustics (Airy’s rainbow) and slowly-induced quantum transitions, including geometric phases.
Pierre Clavier –" Borel-Ecalle resummation for quantum field theory"
In this talk, I will explain how Ecalle's theory of resurgence includes a generalisation of the Borel-Laplace resummation procedure of resummation of divergent series. As much as possible, I will introduce in non-technical ways the crucial ingredient of this procedure: resurgent functions, well-behaved averages... I will then present how this method is relevant for physics and in particular for QFT. Some results will be mentioned but I will also discuss some open questions and conjectures regarding this program.
Brenda Davison – "Divergent Series and Asymptotic Expansions, 1850-1900"
Divergent series enjoyed a period of free and easy use in the eighteenth century. This was somewhat curtailed in the early nineteenth century even though interesting applied problems were solved using asymptotic expansions of divergent series during that time. In the second half of the nineteenth century asymptotic expansions were placed on a firm and rigorous foundation such that their use become standard practice by early in the twentieth century. I elucidate how and why that happened by examining the work on asymptotic expansions of George Gabriel Stokes, Jules Henri Poincaré and Thomas Jan Stieltjes.
Laurent Desvillettes – "From Kinetic Theory to Fluid Mechanics: the Chapman-Enskog Asymptotics"
We present some of the ideas leading to the establishment of the Chapman-Enskog expansion in fluid mechanics. This expansion constitutes a bridge between kinetic theory (more precisely, the Boltzmann equation), and macroscopic modeling, represented by the Navier-Stokes system of compressible gases. The presentation tries to follow the history of the concepts and computations related to the expansion, and concludes with remarks about some contemporary research issues on the topic.
John Dougherty, James Fraser and Mike Miller – "What is the Renormalon Problem?"
The Feynman diagram expansion of quantum field theory (QFT) is divergent even after the ultraviolet and infrared divergences at each order have been dealt with. What does this imply for the foundational status of the theory? We approach this question by comparing the QFT case to better understood divergent expansions that occur in non-relativistic quantum mechanics. A key difference with quantum mechanics is the appearance of so-called renormalons in QFT perturbation theory: patterns of factorial growth in the series coefficients associated with logarithms in the renormalization scale. The implications of these renormalon divergences are not yet understood but we discuss some possible scenarios.
Nicolas Fillion – "Approaches to Evaluating Perturbative Solutions: The Case of the WKB Method"
Although perturbation methods are very widespread, being used to find approximate solutions to all kinds of mathematical problems, there are profound discrepancies between how different people think about them. This paper will elaborate on differences between the mathematicians' and the physicists' way of thinking about perturbation methods, and I will use the WKB method as my running example. After a brief historical discussion of the method, I will suggest that what distinguishes the two modes of thinking is not only that one is more mathematically rigorous whereas the other rests on physical intuitions, but rather that they embody different perspectives on approximation. I will show how a simple error-theoretic framework common among numerical analysts--backward-error analysis--can beneficially be applied to the assessment of perturbative solutions.
Samuel Fletcher – "The Limits of Approximation"
Since the 1970s, philosophers have recognized that not all reductive explanations in the sciences are deductions, in the logical sense, of an explanandum. Some involve instead "limit relations" between theories or models. More recently, some have suggested that some explanations involving approximation, such as asymptotic expansions, involve no limits at all or cannot be reductive. I will argue that limit relations do represent a special class of approximations, but that their success in reductive explanations accrues to their properties as good approximations, not their special properties as limits.
Pierre-Yves Lagrée – "Asymptotic Expansions in Fluid Mechanics: example of Matched Asymptotic Expansions, some classical results and application to boundary layer separation"
The method of Matched Asymptotic Expansions (MAE) is one of the classical tools to look at singular problems in fluid mechanics. WKB or multiple scale give the same result, but more or less tractable depending on the problem. MAE has been used intensively from the 50’ to solve problems depending on a small parameter in the case where the problem becomes singular when the parameter is zero.
Singular problems arise at small Reynolds number, we need MAE to obtain the viscous Oseen flow around a cylinder 1957.
Singular problems arise at small inverse of Reynolds number, Navier Stokes equations give Euler/Boundary Layer decomposition 1905. We will discuss the order two of Boundary Layer 1962 and how it creates a perturbation of Euler at next order. We will apply MAE to boundary layer separation (wich is a singularity of the Boundary Layer which has to be solved by the "triple deck" 1969: a boundary layer in the boundary layer).
More recently other problems like pinching, drop impact, thin films… present some singularities and are solved with asymptotic methods together with numeric simulations showing the continuous need of some asymptotics to understand flows.
Étienne Ligout – "Asymptotic Expansions in General Relativity"
This talk focuses on asymptotic expansions used in General Relativity (GR), and more specifically the so-called Post-Newtonian (PN) expansion scheme. I first explain the motivation for such approximation by explaining why the two-body problem has no exact solution in GR. I then detail how the PN expansion is carried out in practice, and its features. Those developments lead to a comparison with asymptotic expansions used in other classical and quantum field theories. As an outlook, I resort to a rather recent result of the theory of distributions to explain why asymptotic methods appear, most often than not, in field theories.
Patricia Palacios – "Long-range interactions, equilibrium and the role of infinite limits"
Kasia Rejzner – Dealing with divergences in perturbative algebraic quantum field theory
In this talk I will introduce the framework of perturbative algebraic quantum field theory (pAQFT), which is a was to axiomatise perturbative QFT using the idea of locality and the approach to renormalisation due to Epstein and Glaser. This approach allows one to handle the UV divergences of the theory, but there remains some freedom in how physical observables are defined, which is the inherent renormalisation freedom.