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Resurgence in Mathematics and Physics
from
Tuesday, June 11, 2019 (9:00 AM)
to
Friday, June 14, 2019 (6:00 PM)
Monday, June 10, 2019
Tuesday, June 11, 2019
9:30 AM
Registration of the participants and welcome coffee
Registration of the participants and welcome coffee
9:30 AM  10:00 AM
Room: Centre de Conférences Marilyn et James Simons
10:00 AM
Resurgence’s Two Main Types and Their Signature Complications: Tessellation, Isography, Autarchy

Jean Écalle
(
Université ParisSud Orsay
)
Resurgence’s Two Main Types and Their Signature Complications: Tessellation, Isography, Autarchy
Jean Écalle
(
Université ParisSud Orsay
)
10:00 AM  11:00 AM
Room: Centre de Conférences Marilyn et James Simons
Quite speciﬁc challenges attend the move from equational resurgence (i.e. resurgence in a singular variable – the main type in frequency and importance) to coequational resurgence (i.e. resurgence in a singular parameter – a close second, roughly dual to the ﬁrst) : complexity soars; two Bridge equations are required instead of one; the complex valued Stokes constants make way for discrete tessellation coeﬃcients; the acting alien algebra remains isomorphic to an algebra of ordinary diﬀerential operators, but these are now subject to isographic invariance (meaning that they annihilate some speciﬁc diﬀerential twoform); and lastly, the new resurgence coeﬃcients possess the paradoxical property of autarchy, combining sectorial resurgence with global entireness. We shall attempt a comprehensive, uptodate survey of the ﬁeld, with emphasis on the rather unexpected and quite novel structures spawned by these two regimes of resurgence (a beefedup version of the conference presentation shall be posted on our homepage).
11:00 AM
Coffee break
Coffee break
11:00 AM  11:30 AM
Room: Centre de Conférences Marilyn et James Simons
11:30 AM
Resurgence through Path Integrals

Maxim Kontsevich
(
IHES
)
Resurgence through Path Integrals
Maxim Kontsevich
(
IHES
)
11:30 AM  12:30 PM
Room: Centre de Conférences Marilyn et James Simons
I will review the approach to the resurgence phenomenon via integration over rapid decay cycles (Lefschetz thimbles) in path integrals. Examples include WKB asymptotics, heat kernels, WZW models and ChernSimons theory.
12:30 PM
Lunch
Lunch
12:30 PM  2:00 PM
Room: Centre de Conférences Marilyn et James Simons
2:00 PM
Resurgence, Matrices and Strings

Ricardo Schiappa
(
University of Lisbon
)
Resurgence, Matrices and Strings
Ricardo Schiappa
(
University of Lisbon
)
2:00 PM  3:00 PM
Room: Centre de Conférences Marilyn et James Simons
I will review older work, and report on work in progress, concerning applications of resurgence and trans series within the realms of matrix models and minimal/topological strings.
3:15 PM
Semiclassics, Mixed Anomalies and Resurgence in 2D QFT

Mithat Ünsal
(
NC State University
)
Semiclassics, Mixed Anomalies and Resurgence in 2D QFT
Mithat Ünsal
(
NC State University
)
3:15 PM  4:15 PM
Room: Centre de Conférences Marilyn et James Simons
I will discuss new methods in QFTs which allows us to study the nonperturbative dynamics of (non)supersymmetric theories. The main tools are mixed anomalies, semiclassical methods, resurgence and quantum distillation. As an application, I will describe the charge$q$ Schwinger model and twodimensional sigma models, such as $CP^{N1}$ model compactified on cylinder. Using certain boundary conditions which guarantee persistence of mixed anomaly upon compactification reveals a large set of new saddle points, which play instrumental role in nonperturbative dynamics.
4:15 PM
Coffee break
Coffee break
4:15 PM  4:45 PM
Room: Centre de Conférences Marilyn et James Simons
4:45 PM
Resurgence for Superconductors

Marcos Mariño
(
University of Geneva
)
Resurgence for Superconductors
Marcos Mariño
(
University of Geneva
)
4:45 PM  5:45 PM
Room: Centre de Conférences Marilyn et James Simons
One of the most important nonperturbative effects in Nature is the energy gap of superconductors, which is exponentially small in the coupling constant. A natural question is whether this effect can be incorporated in the theory of resurgence. In this talk I will argue that this is the case. More precisely, I conjecture that the perturbative series for the ground state energy of a superconductor is factorially divergent, and its leading Borel singularity corresponds to the superconducting gap. In the case of the attractive GaudinYang model (a superconductor in one dimension), I develop techniques that make it possible to calculate the exact perturbative series of the ground state energy up to very high order, providing a nontrivial test of this conjecture. For superconductors in three dimensions, evidence for this conjecture can be given by using diagrammatic methods. We also argue that the leading Borel singularity is of the renormalon type, associated to factorially divergent subdiagrams.
Wednesday, June 12, 2019
9:30 AM
Welcome coffee
Welcome coffee
9:30 AM  10:00 AM
Room: Centre de Conférences Marilyn et James Simons
10:00 AM
Resurgent Thetafunctions: a Conjectured Gateway into Dimension D>1 Quantum Mechanics

André Voros
(
IPhT  CEA Saclay  CNRS
)
Resurgent Thetafunctions: a Conjectured Gateway into Dimension D>1 Quantum Mechanics
André Voros
(
IPhT  CEA Saclay  CNRS
)
10:00 AM  11:00 AM
Room: Centre de Conférences Marilyn et James Simons
Resurgent analysis of the stationary Schrödinger equation (exactWKB method) has remained exclusively confined to 1D systems due to its underlying linearODE techniques. Here, building on a solvable 2D case (a Selberg trace formula, as analyzed with P. Cartier), and on a BalianBloch abstract quantum framework in any dimension using complex orbits, we isolate a very special generalizedheattrace function as best candidate to start some resurgent description of quantum mechanics in general dimension. The latter statement is still quite embryonic and speculative  our main hope is to encourage future research.
11:00 AM
Coffee break
Coffee break
11:00 AM  11:30 AM
Room: Centre de Conférences Marilyn et James Simons
11:30 AM
Wallcrossing Formulas and Resurgence

Yan Soibelman
(
Kansas State University
)
Wallcrossing Formulas and Resurgence
Yan Soibelman
(
Kansas State University
)
11:30 AM  12:30 PM
Room: Centre de Conférences Marilyn et James Simons
In the joint paper with M. Kontsevich (arXiv:0811.2435) among other things we introduced the notion of stability data on graded Lie algebras, upgraded later to the notion of wallcrossing structure in arXiv:1303.3253. Both notions turned out to be suitable for spelling out wallcrossing formulas in various circumstances, in particular in DonaldsonThomas theory of 3dimensional CalabiYau categories as well as supersymmetric gauge theories in dimensions two and four. Few years ago, we wrote a paper (yet unpublished) devoted to applications of that to algebraic, analytic and resurgent properties of series arising in wallcrossing formulas. Aim of the talk is to discuss some of these ideas.
12:30 PM
Lunch
Lunch
12:30 PM  2:00 PM
Room: Centre de Conférences Marilyn et James Simons
2:00 PM
Considerations about Resurgence Properties of Topological Recursion

Bertrand Eynard
(
IPhT  CEA  CRM Montreal  IHES
)
Considerations about Resurgence Properties of Topological Recursion
Bertrand Eynard
(
IPhT  CEA  CRM Montreal  IHES
)
2:00 PM  3:00 PM
Room: Centre de Conférences Marilyn et James Simons
To a spectral curve $S$ (e.g. a plane curve with some extra structure), topological recursion associates a sequence of invariants: some numbers $F_g(S)$ and some $n$forms $W_{g,n}(S)$. First we show that $F_g(S)$ grow at most factorially at large $g$, $F_g = O((\beta g)! r^{g})$ with $r>0$ and $\beta\leq 5$. This implies that there is a Borel transform of $\sum_g \hbar^{2g2} F_g$ that is analytic in a disk of radius $r$. The question is whether this is a resurgent series or not? We give arguments for this, and conjecture what are the singularities of the Borel transform, and we show how it works on a number of examples.
3:15 PM
Geometric Recursion with a View Towards Resurgence

Jørgen E. Andersen
(
Aarhus University
)
Geometric Recursion with a View Towards Resurgence
Jørgen E. Andersen
(
Aarhus University
)
3:15 PM  4:15 PM
Room: Centre de Conférences Marilyn et James Simons
We shall review the geometric recursion and its relation to topological recursion. In particular, we shall consider the target theory of continuous functions on Teichmüller spaces and we shall exhibit a number of classes of mapping class group invariant functions, which satisfies the geometric recursion. Many of these classes of functions are integrable over moduli spaces and we prove that there averages over moduli spaces satisfies topological recursion. The talk will end with a discussion of possible resurgence perspectives. The construction of geometric recursion and the results relating it to topological recursion is joint work with Borot and Orantin.
4:15 PM
Coffee break
Coffee break
4:15 PM  4:45 PM
Room: Centre de Conférences Marilyn et James Simons
4:45 PM
On the Resurgent WKB Analysis

David Sauzin
(
CNRS  IMCCE  Paris Observatory  PSL University
)
On the Resurgent WKB Analysis
David Sauzin
(
CNRS  IMCCE  Paris Observatory  PSL University
)
4:45 PM  5:45 PM
Room: Centre de Conférences Marilyn et James Simons
I'll report on a work in progress with F. Fauvet (Strasbourg University) and R. Schiappa (IST Lisbon) about the WKB formal expansions solutions to the 1D stationary Schrödinger equation with polynomial coefficients. Our emphasis is on the coequational resurgent structure, in the sense of J. Écalle, which is a way of using the singularity structure in the Borel plane to access the connection formulae for the WKB solutions and the Voros data.
Thursday, June 13, 2019
9:30 AM
Welcome coffee
Welcome coffee
9:30 AM  10:00 AM
Room: Centre de Conférences Marilyn et James Simons
10:00 AM
Resurgence, Topology and Modularity

Sergei Gukov
(
Caltech Pasadena
)
Resurgence, Topology and Modularity
Sergei Gukov
(
Caltech Pasadena
)
10:00 AM  11:00 AM
Room: Centre de Conférences Marilyn et James Simons
11:00 AM
Coffee break
Coffee break
11:00 AM  11:30 AM
Room: Centre de Conférences Marilyn et James Simons
11:30 AM
Arithmetic Resurgence of Quantum Invariants

Stavros Garoufalidis
(
MaxPlanck Institute for Mathematics
)
Arithmetic Resurgence of Quantum Invariants
Stavros Garoufalidis
(
MaxPlanck Institute for Mathematics
)
11:30 AM  12:30 PM
Room: Centre de Conférences Marilyn et James Simons
I will explain some conjectures concerning arithmetic resurgence of quantum knot and 3manifold invariants formulated in an earlier work of mine in 2008, as well as numerical tests of those conjectures and their relations to quantum modular forms, state integrals and their qseries. Joint work (in parts) with R. Kashaev and D. Zagier.
12:30 PM
Lunch
Lunch
12:30 PM  2:00 PM
Room: Centre de Conférences Marilyn et James Simons
2:00 PM
Topology of the Stokes Phenomenon

Philip Boalch
(
CNRS  Université ParisSud
)
Topology of the Stokes Phenomenon
Philip Boalch
(
CNRS  Université ParisSud
)
2:00 PM  3:00 PM
Room: Centre de Conférences Marilyn et James Simons
3:15 PM
Nonabelian Hodge Theory for Monopoles with Periodicity

Takuro Mochizuki
(
RIMS  Kyoto University
)
Nonabelian Hodge Theory for Monopoles with Periodicity
Takuro Mochizuki
(
RIMS  Kyoto University
)
3:15 PM  4:15 PM
Room: Centre de Conférences Marilyn et James Simons
Recently, we obtained equivalences between monopoles with periodicity and difference modules of various types, i.e., periodic monopoles and difference modules, doubly periodic monopoles and $q$difference modules, and triply periodic monopoles and difference modules on elliptic curves. In this talk, we shall give a review on the equivalences. If possible, we would like to discuss some deeper aspects of the correspondences from nonabelian Hodge theoretic viewpoints.
4:15 PM
Coffee break
Coffee break
4:15 PM  4:45 PM
Room: Centre de Conférences Marilyn et James Simons
4:45 PM
The Mano Decompositions and the Space of Monodromy Data of the $q$Painlevé V I Equation

JeanPierre Ramis
(
Université Paul Sabatier
)
The Mano Decompositions and the Space of Monodromy Data of the $q$Painlevé V I Equation
JeanPierre Ramis
(
Université Paul Sabatier
)
4:45 PM  5:45 PM
Room: Centre de Conférences Marilyn et James Simons
The talk is based upon a joint work with Y. OHYAMA and J. SAULOY. Classically the space of Monodromy data (or character variety) of PV I (the sixth Painlevé diﬀerential equation) is the space of linear representations of the fundamental group of a 4punctured sphere up to equivalence of representations. If one ﬁxes the local representation data it “is” a cubic surface. We will describe a $q$analog: the space of $q$Monodromy data of the $q$Painlevé V I equation. For the $q$analogs of the Painlevé equations (which are nonlinear $q$diﬀerence equations), according to H. SAKAI work, “everything” is well known on the “left side” of the ($q$analog of the) RiemannHilbert map (the varieties of “initial conditions”), but the “right side” (the $q$analogs of the spaces of Monodromy data or character varieties) remained quite mysterious. We will present a complete description of the space of Monodromy data of $q$−PV I (some local data being ﬁxed). It is a “modiﬁcation” of an elliptic surface and we will explicit some “natural” parametrizations. This surface is analytically, but not algebraically isomorphic to the Sakai surface of ”initial conditions”. Our description uses a new tool, the Mano decompositions, which are a $q$analog of the classical pants decompositions of surfaces. We conjecture that our constructions can be extended to the others $q$Painlevé equations. This involves $q$Stokes phenomena.
Friday, June 14, 2019
9:30 AM
Welcome coffee
Welcome coffee
9:30 AM  10:00 AM
Room: Centre de Conférences Marilyn et James Simons
10:00 AM
Resurgence and Phase Transitions

Gerald V. Dunne
(
University of Connecticut
)
Resurgence and Phase Transitions
Gerald V. Dunne
(
University of Connecticut
)
10:00 AM  11:00 AM
Room: Centre de Conférences Marilyn et James Simons
11:00 AM
Coffee break
Coffee break
11:00 AM  11:30 AM
Room: Centre de Conférences Marilyn et James Simons
11:30 AM
On the Laplace Transform of the Monodromy as a Function of the Pertubation Parameter in WKBVoros Resurgence

Carlos Simpson
(
Université NiceSophiaAntipolis
)
On the Laplace Transform of the Monodromy as a Function of the Pertubation Parameter in WKBVoros Resurgence
Carlos Simpson
(
Université NiceSophiaAntipolis
)
11:30 AM  12:30 PM
Room: Centre de Conférences Marilyn et James Simons
We consider the Voros resurgence or WKB problem of monodromy for a family of connections of the form $\nabla + t.\Phi$, and look at the transport along a path as a function of $t$. Taking the Laplace transform, we discuss the analytic continuation properties leading to asymptotic estimates for the monodromy as $t \to \infty$. If time permits we'll discuss possible relations with spectral networks and harmonic mappings to buildings.
12:30 PM
Lunch
Lunch
12:30 PM  2:00 PM
Room: Centre de Conférences Marilyn et James Simons
2:00 PM
Lagrangian Fibration of the de Rham Moduli Space and Gaiotto Correspondence

Olivia Dumitrescu
(
University Central Michigan
)
Lagrangian Fibration of the de Rham Moduli Space and Gaiotto Correspondence
Olivia Dumitrescu
(
University Central Michigan
)
2:00 PM  3:00 PM
Room: Centre de Conférences Marilyn et James Simons
There have been new developments in understanding Lagrangian fibrations of the de Rham moduli space in connection to Lagrangian stratifications of the Dolbeault moduli space through biholomorphic isomorphisms of the Lagrangian fibers. I will report recent results by different groups of authors.
3:15 PM
Bion Saddle Points and Resurgence in $CP^N$ Model

Toshiaki Fujimori
(
Keio University
)
Bion Saddle Points and Resurgence in $CP^N$ Model
Toshiaki Fujimori
(
Keio University
)
3:15 PM  4:15 PM
Room: Centre de Conférences Marilyn et James Simons
Perturbation series in quantum field theory are generically divergent asymptotic series. Resurgence theory relates such perturbation series and nonperturbative effects which cannot be captured by the perturbative expansion. It has been shown that the socalled bion saddle points play an important role in resurgence theory in a certain class of quantum systems. In this talk, I will overview the recent studies on the bion saddle points in the $CP^{N −1}$ models based on the complexified path integral and the bion saddle points.
4:15 PM
Coffee break
Coffee break
4:15 PM  4:45 PM
Room: Centre de Conférences Marilyn et James Simons
4:45 PM
Algebra of the Infrared and Perverse Schobers

Mikhail Kapranov
(
Kavli  IPMU
)
Algebra of the Infrared and Perverse Schobers
Mikhail Kapranov
(
Kavli  IPMU
)
4:45 PM  5:45 PM
Room: Centre de Conférences Marilyn et James Simons