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Resurgence and Mould Calculus

David Sauzin

Abstract

Resurgence Theory and Mould Calculus were invented by J. Ecalle around 1980 in the context of analytic dynamical systems and are increasingly more used in the mathematical physics community, especially since the 2010s. We review the mathematical formalism and touch on the applications. This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics.

Resurgence and Mould Calculus

Abstract

Resurgence Theory and Mould Calculus were invented by J. Ecalle around 1980 in the context of analytic dynamical systems and are increasingly more used in the mathematical physics community, especially since the 2010s. We review the mathematical formalism and touch on the applications. This is an invited contribution to the 2nd edition of the Encyclopedia of Mathematical Physics.
Paper Structure (19 sections, 4 theorems, 73 equations, 5 figures)

This paper contains 19 sections, 4 theorems, 73 equations, 5 figures.

Table of Contents

  1. Introduction
  2. 'Simple' version of Resurgence Theory
  3. Simple $\Omega$-resurgent series
  4. Stability under product and nonlinear operations
  5. Alien operators. Alien derivations
  6. Freeness of the alien derivations and a first glimpse of mould calculus
  7. Simple resurgent functions obtained by Borel-Laplace summation
  8. Borel-Laplace summation
  9. Transseries, symbolic Stokes automorphism
  10. The algebra $\raisebox{.1ex}{${\stackrel{ \raisebox{-.23ex}{$\bigtriangledown$} }{\mathcal{R}}}$}$ of general resurgent singularities
  11. Relaxing the constraint on the location of singularities
  12. Relaxing the constraint on the nature of singularities
  13. Examples from mathematical physics
  14. Mould Calculus
  15. The mould algebra
  16. ...and 4 more sections

Key Result

Theorem 3

For each $\omega\in\Omega^+$, the operator $\Delta_\omega$ is a derivation of the algebra $\widetilde{\mathcal{R}}\,\!^{\mathrm{s}}_{\Omega}$.

Figures (5)

  • Figure 1: The path $\gamma$ avoids $\Omega$ and ends at ${\zeta}_1$, near $\omega$.
  • Figure 2: An example of path $\gamma({\varepsilon})$, here with $r=4$ and ${\varepsilon}=(+,-,+)$.
  • Figure 3: Above: Directions for Laplace integration with $J=({\theta}_1,{\theta}_2)$. Below: the union of half-planes $\mathscr D^J$.
  • Figure 4: Decomposition of the contour for the difference of two Laplace transforms as a sum of contours.
  • Figure 5: ${\theta}$-rotated Hankel contour.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Theorem 3: Eca81
  • Theorem 4: Eca81
  • Proposition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Theorem 10: Eca81
  • ...and 1 more