Lecturer: Brent Pym

Time: Thursdays 10-12

Location: Andrew Wiles Building

Room: C1 (weeks 1-7), C5 (week 8)

Evaluation: written report on a topic to be discussed with the lecturer

Divergent series expansions appear in many branches of mathematics and mathematical physics. In simple cases, these expansions are given by power series in a small parameter, but frequently they also include exponentially small non-analytic corrections, sometimes called "instantons". These corrections make it particularly difficult to assign a meaningful sum to the series.

In the 1970s, J. Écalle developed a theory, called "resurgence", which explains how to deal with these objects using classical techniques of complex analysis. This course is intended as an introduction to the theory of resurgence.

The tools involved are essentially basic, requiring little background beyond a first course in complex analysis (contour integrals, analytic continuation, etc.). But the theory also reveals a hidden complexity: interesting geometric objects, such as infinite-sheeted Riemann surfaces, and new "alien" derivative operations, play an essential role.

The main motivation for the course is the recent interest in resurgence amongst geometers and mathematical physicists, owing to its application in a number of areas:

- Normal forms for dynamical systems
- Gauge theory of singular connections
- Quantization of symplectic and Poisson manifolds
- Floer homology and Fukaya categories
- Knot invariants
- Wall-crossing and stability conditions in algebraic geometry
- Spectral networks
- WKB approximation in quantum mechanics
- Perturbative expansions in quantum field theory (QFT)

This last example is, in some sense, the ultimate one, since many of the others can be interpreted as calculations in low-dimensional field theories. QFT is also where some of the most striking results have been found: there is mounting evidence that, starting only from the perturbative expansion, resurgence can be used to uncover nonperturbative effects by a sort of analytic continuation. We aim to close the course with a discussion of some of these applications, depending on the tastes of the audience.

Lecture notes for the course will be posted here as the term progresses. Please treat these as drafts, and contact me if you find any errors.

- Lecture 1 [PDF]: introduction and motivation
- Lecture 2 [PDF]: real oriented blowups, functions of exponential type and asymptotic expansions; see also Casselman's discussion of rainbows
- Lecture 3 [PDF]: Borel and Laplace transforms; Gevrey series and Watson's theorem
- Lecture 4 [PDF]: convolution products; endless analytic continuation and resurgent series
- Lecture 5 [PDF]: Laplace transforms of endlessly continuable forms; simple singularities; Borel sums and resurgent symbols
- Lecture 6 [PDF]: Real-valuedness of Borel sums; Stokes automorphisms and alien derivatives
- Lecture 7 [PDF]: Application of resurgence to the local classification of planar foliations
- Lecture 8: multi-dimensional integration by steepest descent; cf Delabaere--Howls and references therein, also Witten

This list may be updated from time to time throughout the term.

Candelpergher, Nosmas and Pham (1993), *"Approche de la résurgence"*, Hermann, Paris.

Écalle (1981, 1985), *"Les fonctions résurgentes. Tome I, II et III"*, Publications MathÃ©matiques d'Orsay 81, Vol. 5 and 6. Université de Paris-Sud, Département de Mathématique, Orsay. [URL]

Sternin and Shatalov (1996), *"Borel-Laplace transform and asymptotic theory"*, CRC Press, Boca Raton, FL.

Deligne (1970), *"Équations différentielles à points singuliers réguliers"* Berlin , pp. iii+133. Springer-Verlag.

Deligne, Malgrange and Ramis (2007), *"Singularités irrégulières"* , pp. xii+188. Société Mathématique de France, Paris.

Dingle(1973), *"Asymptotic expansions: their derivation and interpretation"*, Academic Press, London-New York.

Ilyashenko and Yakovenko (2008), *"Lectures on analytic differential equations"* Providence, RI Vol. 86, pp. xiv+625. American Mathematical Society. [URL]

Malgrange B (1991), *"Équations différentielles à coefficients polynomiaux"* Boston, MA Vol. 96, pp. vi+232. Birkhäuser Boston Inc..

Sauzin, "Introduction to 1-summability and resurgence" [arXiv]

Dorigoni, "An Introduction to Resurgence, Trans-Series and Alien Calculus" [arXiv]

Dunne and Unsal (2015), *"What is QFT? Resurgent trans-series, Lefschetz thimbles, and new exact saddles"*, In Proceedings, 33rd International Symposium on Lattice Field Theory (Lattice 2015). [arXiv]

Mariño M (2014), *"Lectures on non-perturbative effects in large $N$ gauge theories, matrix models and strings"*, Fortschr. Phys.. Vol. 62(5-6), pp. 455-540. [DOI] [arXiv]

Bender and Wu (1969), *"Anharmonic oscillator"*, Phys. Rev. (2). Vol. 184, pp. 1231-1260. [DOI]

Berry (1988), *"Stokes' phenomenon; smoothing a Victorian discontinuity"*, Inst. Hautes Études Sci. Publ. Math.. (68), pp. 211-221 (1989). [URL]

Berry and Howls (1990), *"Hyperasymptotics"*, Proc. Roy. Soc. London Ser. A. Vol. 430(1880), pp. 653-668. [DOI]

Delabaere, Dillinger and Pham (1993), *"Résurgence de Voros et périodes des courbes hyperelliptiques"*, Ann. Inst. Fourier (Grenoble). Vol. 43(1), pp. 163-199. [URL]

Delabaere and Howls (2002), *Global asymptotics for multiple integrals with boundaries*, Duke Math. J. Vol. 112(2), pp. 199-264. [DOI]

Delabaere and Pham (1997), *"Unfolding the quartic oscillator"*, Ann. Physics. Vol. 261(2), pp. 180-218. [DOI]

Garay, de Goursac and van Straten (2014), *"Resurgent deformation quantisation"*, Ann. Physics. Vol. 342, pp. 83-102. [DOI] [arXiv]

Getmanenko (2011), *"Resurgent analysis of the Witten Laplacian in one dimension"*, Funkcial. Ekvac.. Vol. 54(3), pp. 383-438. [DOI] [arXiv]

Malgrange (1982), *"Travaux d'Écalle et de Martinet-Ramis sur les systèmes dynamiques"*, In Bourbaki Seminar, Vol. 1981/1982. Vol. 92, pp. 59-73. Soc. Math. France, Paris. [URL]

Martinet and Ramis (1982), *"Problèmes de modules pour des équations différentielles non linéaires du premier ordre"*, Inst. Hautes Études Sci. Publ. Math.. (55), pp. 63-164. [URL]

Stokes (1847), *"On the numerical calculation of a class of definite integrals and infinite series"*, Trans. Camb. Phil. Soc. Vol. 9, pp. 379-407.

Voros (1983), *"The return of the quartic oscillator: the complex WKB method"*, Ann. Inst. H. Poincaré Sect. A (N.S.). Vol. 39(3), pp. 211-338. [URL]

Witten (2008), *"Gauge theory and wild ramification"*, Anal. Appl. (Singap.). Vol. 6(4), pp. 429-501. [DOI] [arXiv]

Witten (2011), *"A new look at the path integral of quantum mechanics"*, In Surveys in differential geometry. Volume XV. Perspectives in mathematics and physics. Vol. 15, pp. 345-419. Int. Press, Somerville, MA. [DOI] [arXiv]

Witten (2011), *"Analytic continuation of Chern-Simons theory"*, In Chern-Simons gauge theory: 20 years after. Providence, RI Vol. 50, pp. 347-446. Amer. Math. Soc. [arXiv]

Casselman, *The Mathematics of Rainbows*