**Stokes
phenomenon, Resurgence and Physics**

**Titles
and abstracts**

**(Click
on the title to get the slides)**

**Ines
Aniceto** (Univ.
of krakow) : **Applications
of resurgence in gauge and string theories**

The perturbative study of physical observables often leads to asymptotic expansions, and resurgence has played a major role in understanding the nonperturbative phenomena associated with this asymptotic behaviour. In the context of gauge/stringdualities (AdS/CFT), perturbation theory in different regimes of the coupling and rank of the gauge group can lead to both convergent or asymptotic expansions. In this talk, I will analyse the asymptotic regime of different observables in the context ofAdS/CFT. As a warm-up, I will start by reviewing different asymptotic regimes for localisable observables in supersymmetric gauge theories. I will then focus on N=4 supersymmetric Yang-Mills gauge theory and study a particular observable calledcusp anomalous dimension. This observable is determined by an integral equation which is obtained from the integrable properties of the gauge theory. At strong coupling it leads to an asymptotic expansion, and the corresponding transseries solution is shown to be resurgent from large-order checks. A resummation of this transseries allows us to then interpolate between the strong and weak coupling regimes, and connect to the well know (convergent) results at weak coupling.

**Ovidiu
Costin** (Ohio
State Univ) : **Ecalle
Resurgence of Hyperasymptotics expansions and Factorial Series**

Hyperasymptotics is one of the most used tools in obtaining precise numerical results from divergent series. In recent work with MV Berry, RD Costin and C Howls we found that Ecalle resurgence analysis leads to important improvements: already the corrected second stage re-expansion is more accurate than infinitely many stages of usual hyperasymptotics. Another classical technique of re-summation, factorial series, also benefits substantially from Borel plane analysis. The factorial series limitations, ranging from slow convergence, small domain of validity and inability to describe the Stokes sector are removed when resurgence tools are used.

**Jean
Ecalle** (CNRS
– Univ. Paris-Sud) : **Resurgence
theory’s algebraic side**

Though rooted in Analysis, Resurgence theory relies on a rather elaborate algebraic apparatus, distinguished by utter naturalness (there exist no ,sensible alternatives to it), great elegance (as bets nature-imposed notions) and extreme ease of use. Prominent in the tool kit are alien derivations, convolution averages, resurgence monomials and monics, acceleration operators, `displays', `bridge equations', mould calculus etc. Rather than attempt a systematic survey, which would be self-defeating, we shall highlight a few of these tools, sketch some of their most convincing applications, and review along the way the main intra-mathematical sources of resurgence known to this day.

**Jean-Pierre
Ramis** (Univ.
De Toulouse) : **Dynamics
on the wild character varieties of the Painlevé equations and
resurgence**

**Norisuke
Sakai** (Keio
University) : **Non-perturbative
Contributions from Complexified Solutions in CP^1 Quantum Mechanics**

I will describe our recent study of nonperturbative contributions in the CP^1 quantum mechanics with fermionic degrees of freedom. Nonperturbative contributions come from a composite soliton called bion, which is usually an approximate solution. Recently it has been observed in simple quantum mechanical systems like sine-Gordon model that there exist exact saddle point solutions corresponding to the bion when one introduces fermions and complexifies the theory. The CP^{N-1} quantum mechanics instead of the sine-Gordon model correctly describes the small radius limit of the asymptotically free CP^{N-1} field theory on RxS^1. We take the simplest of these, CP^1 quantum mechanics and find bion solutions, which correspond to (complexified) instanton-antiinstanton configurations stabilized in the presence of the fermonic degrees of freedom. By computing the one-loop determinants in the bion backgrounds, we obtain the leading order contributions from both the real and complex bion solutions. The quasi zero modes are evaluated using the Lefschetz thimble formalism. The non perturbative contributions from the real and complex bions are shown to cancel out in the supersymmetric case. They also give an (expected) ambiguity in the non-supersymmetric case, which plays a vital role in the resurgent trans-series. For nearly supersymmetric situation, evaluation of the Lefschetz thimble gives results in precise agreement with those of the direct evaluation of the Schroedinger equation. We also perform the same analysis for the sine-Gordon quantum mechanics and point out some important differences showing that the sine-Gordon quantum mechanics does not correctly describe the small radius limit of the CP^{N-1} field theory on RxS^1. (arXiv:1607.04205, T.Fujimori, S.Kamata, T.Misumi, M.Nitta and N.Sakai).

**David
Sauzin** (IMCCE) :
**Iterated
convolutions and endless Riemann surfaces**

I'll discuss a version of Écalle's definition of resurgence, based on the notion of endless continuability in the Borel plane. I'll relate this with the notion of Ω-continuability, where Ω=(Ω_L)_{L>0} is a discrete filtered set, and show how to construct a universal Riemann surface X_Ω whose holomorphic functions are in one-to-one correspondence with Ω-continuable functions. Then, I'll discuss the Ω-continuability of convolution products and give estimates for iterated convolutions of the form f_1*...*f_n. This allows one to handle nonlinear operations with resurgent series, e.g. substitution into a convergent power series. Joint work with Shingo KAMIMOTO (Hiroshima univ.).

**Ricardo
Schiappa** (Univ.
of Lisbon) : **Asymptotics
and Resurgence in Physics … for Mathematicians**

Asymptotic series are ubiquitous in Fundamental Physics, from quantum mechanics to quantum field theory, from gauge theory to string theory. This talk will be aimed at lightly describing such Physics problems for an audience of Mathematicians, with an emphasis towards problems in gauge and string theory.

Ricardo Vaz(DESY, Hamburg):The Quartic Matrix Model: Transseries, Resurgence and Resummation

In this talk I will discuss some recent developments in the study of resurgence in the context of the quartic matrix model (QMM). Matrix models are very interesting as prototypical gauge theories, as well as dualities (e.g. to topological string theory), and their advantage is that they are very tractable in the so-called 1/N expansion. Transseries solutions emerge as natural objects in these contexts, since there is a clear physical interpretation of the nonperturbative effects. Our computational power allows us to explore the resurgent structure of these solutions, in particular the large-order relations that connect the different components of the transseries. This has been done in the 1- and 2-cut phases of the QMM, as well as their well-known double-scaling limits. Finally, we can also make use of our large-N transseries to generate results at finite N via resummation, and thus make predictions far away from the domain of validity of the original expansion. The work presented is contained in references 1106.5922, 1302.5138 and 1501.01007, and if there is still time I will discuss some future directions and work in progress.

Marcel Vonk(Univ. Amsterdam):Two-parameter transseries for Painlevé I: what's new?

The Painlevé I equation is one of the most studied non-linear ordinary differential equations. Its solutions have been constructed in many different frameworks, including numerics, perturbative series, resurgent transseries and transasymptotic expansions. Several of these approaches have been related to each other in the literature, but mostly for a particular one-parameter subfamily of solutions, and often only in restricted domains of the complex plane. Using resurgent transseries as a starting point, I show how this domain can be extended, which allows to study the famous pole fields in the Painlevé I solutions in much more detail. I will give an updated status report on how our techniques can be extended to the full two-parameter family of Painlevé transseries solutions, and how they uncover interesting relations to modularity.