Painlevé equations were discovered at the beginning of XX-th century by Paul Painlevé for purely mathematical reasons. Their solutions, the Painlevé transcendents, are, in general, "new transcendental functions" and, as the classical special functions, they appear in many problems of mathematics and physics. Applications "exploded" at the end of XX-th century: Einstein metrics, Frobenius manifolds, correlation function of the $2$-dimensional Ising model, reduction of integrable PDEs, reduction of self-dual Yang-Mills equations, random matrix theory, $2$-dimensional CFT (conformal blocks), non perturbative effects in strings theory ($2$d quantum gravity)...

First Lecture

Firstly, we will recall basics about Painlevé equations. Each Painlevé vector field initially defined on a trivial bundle of fibre $C^2$ (the naive phase space) can be extended to a fiber bundle (the Okamoto bundle) whose fiber is the Okamoto space of initial conditions: an affine rational surface endowed with a canonical symplectic structure. On this extension the Painlevé vector field is complete.

We will detail the "simplest case": Painlevé VI. The basis of the Okamoto bundle is $C\setminus \{0,1\}$ and we have a non-linear monodromy (with two generators) which induces a dynamics on each Okamoto space.

Painlevé VI can be interpreted as a traduction of the isomonodromic deformations of some linear second order equations (the linearized equations). The space of monodromy data of the linearized equations is an affine cubic surface: the character variety of Painlevé VI. It is endowed with a canonical symplectic structure.

For generic values of the parameters the character variety is non-singular and there exists an analytic (symplectic) diffeomorphism, the Riemann-Hilbert map (RH) between each Okamoto variety and the character variety. The dynamics on the Okamoto variety (non linear monodromy) is conjugated by RH to a dynamics on the character variety. An essential result is that this last dynamics is algebraic and can be explicitely calculated. As a consequence it is possible to prove that (generically) the dynamics of PVI is "rich" (chaotic...) and that the Galois-Malgrange differential groupoid is "as big as possible" (in particular PVI is not integrable!).

The main purpose of the minicourse is to describe a generalization of this picture to the others Painlevé equations (it is a work in progress...).

For PVI the linearized equation is Fuchsian. The character variety is the set of (classical) monodromy representations up to equivalence.

In the other cases the linearized equation has (as the Painlevé equation itself) irregular singularities.

There are Stokes phenomena and it is necessary to "add" in some sense these phenomena to the classical monodromy data. The corresponding character variety is the set of the extended (wild) monodromy data up to equivalence. It is also a cubic surface.

The non-linear monodromy remains but it is a "poor information". The induced dynamics on the Okamoto variety is "too small". Some years ago (2012), I proposed to define a "better dynamics", the wild dynamics. The idea is to generalize the wild dynamics I defined before in the linear case: this dynamics is generated by the classical dynamics, the Stokes phenomena and some continuous tori actions (exponential tori actions).

In the Painlevé case the definition of the non-linear Stokes phenomena and of the exponential tori actions are far to be trivial... In 2012 they were conjectural.The rigorous approach is due to A. Bittmann and quite recent: 2016.

The wild dynamics induces, via RH, a (symplectic) dynamics on the character variety. This dynamics is a priori local. I conjectured that it is in fact rational (2012).

In the second part of the first lecture I will detail the case of PII. I will give a simple and totally explicit description (partly conjectural...) of the wild dynamics on the character variety. An essential point is that it is rational.

The $9$ lines on the character variety of PII play an essential role in the wild dynamics and they are related to some special solutions (Boutroux truncated, tritrucated, bitruncated solutions). There are also related to the resurgence (in Ecalle spirit) of PII. An heuristic principle follows: the lines on the character varieties (affine cubic surfaces) play a central role in the theory of Painlevé equations.

The moduli space of flat $C^\infty$ connections on rank $n$ vector bundles on a smooth complex algebraic curve $\Sigma$ has an explicit description as the space of representations of the fundamental group: $${\mathcal M}_B = {\rm Hom}(\pi_1(\Sigma),G)/G,\qquad G={\rm GL}_n({\mathbb C})$$ often called the character variety (or Betti moduli space). It has an algebraic Poisson structure with symplectic leaves ${\mathcal M}_B({\bf{\mathcal C}})\subset {\mathcal M}_B$ given by fixing the conjugacy classes of monodromy around each puncture. Choosing suitable generators of $\pi_1(\Sigma)$ this becomes the quotient by $G$ of a space of matrices satisfying a relation of the form $$[A_1,B_1]\cdots[A_g,B_g]M_1\cdots M_m=1$$ where $[a,b]=aba^{-1}b^{-1}$. Thus ${\mathcal M}_B$ “looks like” a multiplicative version of a symplectic (Marsden-Weinstein) quotient, with $G$-valued moment map $\mu$ given by the left-hand side of the relation, so that ${\mathcal M}_B=\mu^{-1}(1)/G$. This “quasi-Hamiltonian” theory was set-up by Alekseev-Malkin-Meinrenken, and leads to the construction of the symplectic manifolds ${\mathcal M}_B({\bf \mathcal C})$ as the multiplicative symplectic quotient of the fusion product of two basic types pieces (conjugacy classes ${\mathcal C}$ and doubles ${\rm \bf D}$).

However this story is just the tip of the iceberg: In algebraic geometry the space ${\mathcal M}_B$ just parametrises the special class of algebraic connections on rank $n$ algebraic vector bundles on $\Sigma$ with tame/regular singular behaviour at the punctures. In this course I’ll first review the topological description of the full category of algebraic connections in terms of Stokes local systems, and the resulting explicit presentations of the wild character varieties (many of which go back to Birkhoff 1913). In turn I’ll describe the extension of the above story, constructing the wild character varieties as algebraic symplectic/Poisson varieties, by fusing together and reducing some new basic pieces (the fission spaces ${\mathcal A}$ and ${\mathcal B}$).

If time permits I'll discuss other topics such as

1. the fact the Drinfeld-Jimbo quantum group quantises a quite simple wild character variety,
2. the notion of wild Riemann surface and the resulting wild mapping group action on the wild character varieties,
3. the upgrading of the symplectic structure to a hyperkahler structure, yielding the link to meromorphic Higgs bundles,
4. the quasi-Hamiltonian fusion approach to complex WKB.

Some references/sources:

Quasi-Hamiltonian geometry (for compact Lie groups) was defined in:

• Alekseev, Malkin, Meinrenken, Lie group valued moment maps, math.DG/9707021

and the extension of this formalism to complex groups that we use is in:

• Alekseev, Bursztyn, Meinrenken, Pure spinors on Lie groups, arXiv:0709.1452

The wild extension is in the sequence of papers:

1. Quasi-Hamiltonian Geometry of Meromorphic Connections, arXiv:math/0203161, published in Duke 2007 [[the generic case, for any reductive group $G$]].
2. Through the analytic halo: Fission via irregular singularities, arXiv:1305.6465, Ann. Inst. Fourier 59, 7 (2009) 2669-2684 [[the simplest non-generic case]
3. Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Math. 179 (2014) 301-365 (arXiv:1111.6228) [[the general untwisted case]]
4. P. Boalch and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091 [[the general case]]

An earlier analytic approach to these symplectic manifolds is in P.Boalch, Symplectic Manifolds and Isomonodromic Deformations, Adv. in Math. 163 (2001) 137-205, and the hyperkahler upgrade is in:

• P. Boalch and O. Biquard, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004) no. 1, 179-204

A “simple as possible” description of several intrinsic approaches to Stokes data for general linear groups is in

• arXiv:1903.12612 (Topology of the Stokes phenomenon).

The introduction of this paper aims to give a good guide to the Stokes data literature.

Several explicit descriptions of some of the simplest examples is in arXiv:1501.00930 (Wild Character Varieties, points on the Riemann sphere and Calabi's examples), and reviews of several different directions related to this story are in arXiv:1305.6593, arXiv:1703.10376.

Second Lecture

The main topic of this lecture is the heuristic principle stated at the end of the first lecture:

the lines on the character varieties (affine cubic surfaces) play a central role in the theory of Painlevé equations.

This principle works very well for PVI. Surprisingly it seems that this remains unnoticed until very recently.

In a first part (mainly in XIX-th algebraic geometry style), we will describe the lines on the cubic surfaces, in particular the 27 lines in the case of nonsingular (complete) cubic surfaces.

In the second part of the lecture I will explain some relations between the 24 lines on the character variety of PVI and some partial reducibility properties of the monodromy representations of the linearized equations.

Afterwards I will sketch a generalization for the other Painlevé equations.

The moduli space of flat $C^\infty$ connections on rank $n$ vector bundles on a smooth complex algebraic curve $\Sigma$ has an explicit description as the space of representations of the fundamental group: $${\mathcal M}_B = {\rm Hom}(\pi_1(\Sigma),G)/G,\qquad G={\rm GL}_n({\mathbb C})$$ often called the character variety (or Betti moduli space). It has an algebraic Poisson structure with symplectic leaves ${\mathcal M}_B({\bf{\mathcal C}})\subset {\mathcal M}_B$ given by fixing the conjugacy classes of monodromy around each puncture. Choosing suitable generators of $\pi_1(\Sigma)$ this becomes the quotient by $G$ of a space of matrices satisfying a relation of the form $$[A_1,B_1]\cdots[A_g,B_g]M_1\cdots M_m=1$$ where $[a,b]=aba^{-1}b^{-1}$. Thus ${\mathcal M}_B$ “looks like” a multiplicative version of a symplectic (Marsden-Weinstein) quotient, with $G$-valued moment map $\mu$ given by the left-hand side of the relation, so that ${\mathcal M}_B=\mu^{-1}(1)/G$. This “quasi-Hamiltonian” theory was set-up by Alekseev-Malkin-Meinrenken, and leads to the construction of the symplectic manifolds ${\mathcal M}_B({\bf \mathcal C})$ as the multiplicative symplectic quotient of the fusion product of two basic types pieces (conjugacy classes ${\mathcal C}$ and doubles ${\rm \bf D}$).

However this story is just the tip of the iceberg: In algebraic geometry the space ${\mathcal M}_B$ just parametrises the special class of algebraic connections on rank $n$ algebraic vector bundles on $\Sigma$ with tame/regular singular behaviour at the punctures. In this course I’ll first review the topological description of the full category of algebraic connections in terms of Stokes local systems, and the resulting explicit presentations of the wild character varieties (many of which go back to Birkhoff 1913). In turn I’ll describe the extension of the above story, constructing the wild character varieties as algebraic symplectic/Poisson varieties, by fusing together and reducing some new basic pieces (the fission spaces ${\mathcal A}$ and ${\mathcal B}$).

If time permits I'll discuss other topics such as

1. the fact the Drinfeld-Jimbo quantum group quantises a quite simple wild character variety,
2. the notion of wild Riemann surface and the resulting wild mapping group action on the wild character varieties,
3. the upgrading of the symplectic structure to a hyperkahler structure, yielding the link to meromorphic Higgs bundles,
4. the quasi-Hamiltonian fusion approach to complex WKB.

Some references/sources:

Quasi-Hamiltonian geometry (for compact Lie groups) was defined in:

• Alekseev, Malkin, Meinrenken, Lie group valued moment maps, math.DG/9707021

and the extension of this formalism to complex groups that we use is in:

• Alekseev, Bursztyn, Meinrenken, Pure spinors on Lie groups, arXiv:0709.1452

The wild extension is in the sequence of papers:

1. Quasi-Hamiltonian Geometry of Meromorphic Connections, arXiv:math/0203161, published in Duke 2007 [[the generic case, for any reductive group $G$]].
2. Through the analytic halo: Fission via irregular singularities, arXiv:1305.6465, Ann. Inst. Fourier 59, 7 (2009) 2669-2684 [[the simplest non-generic case]
3. Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Math. 179 (2014) 301-365 (arXiv:1111.6228) [[the general untwisted case]]
4. P.Boalch and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091 [[the general case]]

An earlier analytic approach to these symplectic manifolds is in P.Boalch, Symplectic Manifolds and Isomonodromic Deformations, Adv. in Math. 163 (2001) 137-205, and the hyperkahler upgrade is in:

• P. Boalch and O. Biquard, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004) no. 1, 179-204

A “simple as possible” description of several intrinsic approaches to Stokes data for general linear groups is in

• arXiv:1903.12612 (Topology of the Stokes phenomenon).

The introduction of this paper aims to give a good guide to the Stokes data literature.

Several explicit descriptions of some of the simplest examples is in arXiv:1501.00930 (Wild Character Varieties, points on the Riemann sphere and Calabi's examples), and reviews of several different directions related to this story are in arXiv:1305.6593, arXiv:1703.10376.

Third Lecture

Firstly I will discuss the problem of the definition of the wild monodromy for an arbitrary irregular singularity in the linear case, in relation with Stokes phenomena, $k$-summability, multisummability, Laplace transform and resurgence. The source of this topic is, a century ago, a R. Garnier paper 1919.

I will detail the basic example: the monodromy and Stokes phenomena in the case of Hypergeometic Equations (classical and confluent). I will explain the confluence of monodromy towards wild monodromy.

In the second part of the lecture I will describe some non-linear Stokes phenomena and the corresponding unfoldings: saddle-nodes, symplectic saddle-nodes.

I will end with the application of all the tools to the case of the confluence of PVI towards PV. As a byproduct, it is possible to get a proof of the rationality of the wild dynamics of PV (M. Klimes). It is extremely technical and I will give only the (simple!) basic ideas and the main lines.

It is a first step towards a proof of the following conjecture (Ramis 2012):

The (wild) dynamics on the (wild) character variety of each Painlevé equation is rational and explicitely computable.

The moduli space of flat $C^\infty$ connections on rank $n$ vector bundles on a smooth complex algebraic curve $\Sigma$ has an explicit description as the space of representations of the fundamental group: $${\mathcal M}_B = {\rm Hom}(\pi_1(\Sigma),G)/G,\qquad G={\rm GL}_n({\mathbb C})$$ often called the character variety (or Betti moduli space). It has an algebraic Poisson structure with symplectic leaves ${\mathcal M}_B({\bf{\mathcal C}})\subset {\mathcal M}_B$ given by fixing the conjugacy classes of monodromy around each puncture. Choosing suitable generators of $\pi_1(\Sigma)$ this becomes the quotient by $G$ of a space of matrices satisfying a relation of the form $$[A_1,B_1]\cdots[A_g,B_g]M_1\cdots M_m=1$$ where $[a,b]=aba^{-1}b^{-1}$. Thus ${\mathcal M}_B$ “looks like” a multiplicative version of a symplectic (Marsden-Weinstein) quotient, with $G$-valued moment map $\mu$ given by the left-hand side of the relation, so that ${\mathcal M}_B=\mu^{-1}(1)/G$. This “quasi-Hamiltonian” theory was set-up by Alekseev-Malkin-Meinrenken, and leads to the construction of the symplectic manifolds ${\mathcal M}_B({\bf \mathcal C})$ as the multiplicative symplectic quotient of the fusion product of two basic types pieces (conjugacy classes ${\mathcal C}$ and doubles ${\rm \bf D}$).

However this story is just the tip of the iceberg: In algebraic geometry the space ${\mathcal M}_B$ just parametrises the special class of algebraic connections on rank $n$ algebraic vector bundles on $\Sigma$ with tame/regular singular behaviour at the punctures. In this course I’ll first review the topological description of the full category of algebraic connections in terms of Stokes local systems, and the resulting explicit presentations of the wild character varieties (many of which go back to Birkhoff 1913). In turn I’ll describe the extension of the above story, constructing the wild character varieties as algebraic symplectic/Poisson varieties, by fusing together and reducing some new basic pieces (the fission spaces ${\mathcal A}$ and ${\mathcal B}$).

If time permits I'll discuss other topics such as

1. the fact the Drinfeld-Jimbo quantum group quantises a quite simple wild character variety,
2. the notion of wild Riemann surface and the resulting wild mapping group action on the wild character varieties,
3. the upgrading of the symplectic structure to a hyperkahler structure, yielding the link to meromorphic Higgs bundles,
4. the quasi-Hamiltonian fusion approach to complex WKB.

Some references/sources:

Quasi-Hamiltonian geometry (for compact Lie groups) was defined in:

• Alekseev, Malkin, Meinrenken, Lie group valued moment maps, math.DG/9707021

and the extension of this formalism to complex groups that we use is in:

• Alekseev, Bursztyn, Meinrenken, Pure spinors on Lie groups, arXiv:0709.1452

The wild extension is in the sequence of papers:

1. Quasi-Hamiltonian Geometry of Meromorphic Connections, arXiv:math/0203161, published in Duke 2007 [[the generic case, for any reductive group $G$]].
2. Through the analytic halo: Fission via irregular singularities, arXiv:1305.6465, Ann. Inst. Fourier 59, 7 (2009) 2669-2684 [[the simplest non-generic case]
3. Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Math. 179 (2014) 301-365 (arXiv:1111.6228) [[the general untwisted case]]
4. P.Boalch and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091 [[the general case]]

An earlier analytic approach to these symplectic manifolds is in P.Boalch, Symplectic Manifolds and Isomonodromic Deformations, Adv. in Math. 163 (2001) 137-205, and the hyperkahler upgrade is in:

• P. Boalch and O. Biquard, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004) no. 1, 179-204

A “simple as possible” description of several intrinsic approaches to Stokes data for general linear groups is in

• arXiv:1903.12612 (Topology of the Stokes phenomenon).

The introduction of this paper aims to give a good guide to the Stokes data literature.

Several explicit descriptions of some of the simplest examples is in arXiv:1501.00930 (Wild Character Varieties, points on the Riemann sphere and Calabi's examples), and reviews of several different directions related to this story are in arXiv:1305.6593, arXiv:1703.10376.