Program, Titles and Abstracts

Title : Deligne-Mumford semi-compactification of Painlevé VI equation

Abstract : In a work in progress with Gabriel Calsamiglia (UFF, Niteroi, Brasil) and Titouan Serandour (Univ Rennes/ENS Lyon) we investigate a partial compactification of the phase space of Painlevé VI equations by using Deligne-Mumford compactification of the moduli space of the 5-punctured sphere. Our motivation is to understand in a geometric way the asymptotics of Painlevé VI transcendents.

Title : Moduli of connections, Hilbert schemes and Equations of Painlevé type

Abstract : One can derive the differential equations of Pailevé type from the isomonodromic deformations of linear connections on curves. A family of moduli spaces of connections become the phase space of differential equations of Painlevé type. We will see that the canonical coordinates for these phase spaces are given by so-called apparent singularities and their duals and the moduli space of connections is birational to some Hilbert schemes of points on some algebraic surface. This is a part of joint work with Szilard Szabo.

Title : Long-Moody induction of braid representations and Katz' middle convolution

Abstract : In 1995, N. Katz introduced the theory of Rigid Local Systems in which the operation called Middle Convolution plays a central role. At the same time in 1994, D. Long introduced an inductive construction of braid group representations. In this talk, it will be pointed out that these two operations have many similarities, and I will define an operation, "middle convolution of braid representations", by mixing these operations up. And I will explain some properties of this middle convolution of braid representations. This talk is based on the joint work with Haru Negami.

Title : Dynamical system on KZ type equations and generalization of deformation

Abstract : A KZ type equation is a Pfaffian system with logarithmic singularities along the diagonal. The set of KZ type equations has a rich structure -- we may define several operations. KZ type equations are closely related to Fuchsian ordinary differential equations, and then the rich structure is brought to the set of Fuchsian equations.

In this talk we study a dynamical system on the moduli space of KZ type equations generated by Katz operations (middle convolution and addition) and restrictions. We look at the relation between Katz operations and restriction operations, and come to a notion of a generalized deformation. As an application, we find several non-rigid Fuchsian ordinary differential equations with integral representations of solutions.

There arise several fundamental problems on the dynamical system. The study of these problems will be interesting.

Title : Takasaki’s rational fourth Painleve-Calogero system and geometric regularisability of algebro-Painleve equations

Abstract : In the talk I shall mainly concentrate on a Hamiltonian system without the Painleve property and show that it admits a kind of regularisation on a bundle of rational surfaces with certain divisors removed, generalising Okamoto’s spaces of initial conditions for the Painleve differential equations. The system in question was obtained by Takasaki as part of the Painleve-Calogero correspondence and possesses the algebro-Painleve property, being related by an algebraic transformation to the fourth Painleve equation. An atlas for the bundle of surfaces in which the system has a global Hamiltonian structure, with all Hamiltonian functions being polynomial in coordinates just as in the case of Okamoto’s spaces, is discussed. I shall also briefly discuss similar results for the quasi-Painleve equations.

The talk will be mainly based on the joint work with A. Stokes, see https://arxiv.org/pdf/2209.10515.pdf

Title: Perturbative connection formulas for Heun equations

Abstract : Connection formulas relating Frobenius solutions of linear ODEs at different Fuchsian singular points can be expressed in terms of the large order asymptotics of the corresponding power series. I will show that for the Heun equation and some of its confluent versions, the series expansion of the relevant asymptotic amplitude in a suitable parameter can be systematically computed to arbitrary order. This allows to check a recent conjecture of Bonelli-Iossa-Panea-Tanzini expressing the Heun connection matrix in terms of quasiclassical Virasoro conformal blocks.

Title : Irregular conformal blocks and Painlevé equations

Abstract : Gamayun, Iorgov, Lisovyy discovered series representations of the tau functions of PVI, PV, PIII at regular singular points in terms of regular conformal blocks for the Virasoro algebra in 2012, 2013. It is natural to consider an extension of their works for irregular case. I review about irregular conformal blocks for the Virasoro algebra. They are defined as expectation values of irregular vertex operators. By using these irregular conformal blocks with the central charge c=1, conjectural formulas for the tau functions of PV, PIV, PIII, PII are given in terms of these irregular conformal blocks. It is known that we could take a limit of conformal blocks as c goes to infinity by Zamolodchikov (regular case), Lisovyy-Naidiuk (irregular case). I mention about quasiclassical limit of ramified irregular conformal blocks. Finally, I introduce irregular vertex operators for a super Virasoro algebra (Neveu-Schwarz-Ramond algebra).

Title: Lamé accessory parameters and the semiclassical limit of probabilistic conformal blocks

Abstract : Conformal blocks appear in several areas of mathematical physics from random geometry to black hole physics. The form of the semiclassical limit of (Liouville) conformal blocks was conjectured by Zamolodchikov in 1986. In this talk, I will outline the proof of the conjecture for the case of the one point torus starting from the recently formulated probabilistic conformal blocks by Promit Ghosal, Guillaume Remy, Xin Sun, Yi Sun (2003.03802). A key role is played by the information coming from isomonodromic equations on the one point torus and the accessory parameter of the Lamé equation. This talk is based on an upcoming paper with Promit Ghosal and Andrei Prokhorov.

Title : Convergence of Painlevé tau function series

Abstract : It is known that all tau functions of the Painlevé equations satisfy the fourth-order quadratic differential equation. Among them, for the III, V, and VI equations, it is possible to express the formal series solutions explicitly by using combinatorics. In this talk, we show the convergence of the formal series and their domain of convergence, including the solutions of more general equations. We also characterized the form of a quadratic homogeneous equation with a series solution similar to the tau functions of the Painlevé equations.

Title : Voros coefficients of isomonodromy systems associated with the Painlevé equations and BPS invariants

Abstract : The series of works of Gaiotto-Moore-Neitzke showed that the wall-crossing phenomenon for BPS invariants in class S theory is closely related to Stokes phenomena for Voros coefficients (Fock-Goncharov cluster coordinates) in WKB analysis. In this talk, I'll briefly review the relationship and show an observation which implies that the BPS invariants (partially) appear in the Voros coefficients of the isomonodromy systems associated with the Painlevé equations.

Title : q-Painlevé systems, toric surfaces, and cluster Poisson varieties

Abstract : A space of initial conditions of q-Painlevé systems is constructed as a composition of blowups from a toric surface at smooth points on the toric boundary. We see that although the choice of toric surfaces and blowup points are not unique, they are related by a composition of simple operations called mutations. From this point of view, we see that a space of initial conditions has a structure of a cluster Poisson variety.

Title : Global analysis on the Painlevé equations

Abstract: I can study Painlevé equations a little, but I know very few about global behavior of the Painlevé equations. We can connect global data on linearlized equations (so called the Lax pair) to not only local asymptotics of Painlevé transcendents but also global behavior by means of the Riemann-Hilbert correspondence. I hope to talk some dreams to the future.

Title : Period of primitive forms, the space of Okubo-Saito potentials and the sixth Painleve equation

Abstract : K. Saito introduced the notion of ``flat structure'' (which is essentially equivalent to ``Frobenius manifold'' by B. Dubrovin) based on the study on the universal unfoldings of simple singularities. In particular, the notion of ``primitive form'' plays a crucial role in his construction.

In the talk, we will generalize the construction to the cases of flat structures (with or without metric). The key notion in our construction is ``the space of Okubo-Saito potentials''. Typical examples of the space of Okubo-Saito potentials include

• the space of period integrals of a primitive form,

• the dual space $V^*$ of the standard representation space $V$ of a well-generated unitary reflection group $G$.

Given the space of Okubo-Saito potentials, we can construct a unique semi-simple flat structure (with or without metric). In particular, if the space of Okubo-Saito potentials admits a monodromy-invariant non-degenerate symmetric bilinear form, then the corresponding flat structure can be equipped with a Frobenius metric.

In the talk, I will also refer to the correspondence between three dimensional semi-simple flat strucures (with or without metric) and generic solutions to the sixth Painlevé equation.

This correspondence enable us to introduce invariants of solutions to the sixth Painlevé equation.

Title: A space of monodromy data for the Jimbo-Sakai family of q-difference equations

Abstract: Jimbo and Sakai derived q-PVI from an ``isomonodromy'' condition, where the ``monodromy'' is understood to be encoded into Birkhoff's connection matrix. Using a variant of Birkhoff's matrix, we (= Ohyama, Ramis, Sauloy) defined and studied a relevant space of monodromy data and found it to be an algebraic surface. Instrumental was the use of a fascinating tool, the ``Mano decomposition''.

Title : Gq-summation and Fourier transforms

Abstract : The Borel-Laplace transforms, which play an important role in the study of the singular irrgular points of ODEs, may be viewed as a variant or elaboration of Fourier transforms. In our talk, we will talk about how to take a similar way for the q-Borel-Laplace transforms in matters related to the summation of q-Gevrey power series. Specially, a convolution product type formula will be given in the form of a Gaussian integral for the q-Borel transform of the product of two Gq-summable functions.

Title : Around Mahler equations

Abstract :In this talk, we will report on some recent results about linear Mahler equations. We will notably speak about automata, difference Galois theory, Hahn series.

Title : Rigid irreducible meromorphic connections in dimension one

Abstract : I will illustrate the Arinkin-Deligne-Katz algorithm for rigid irreducible meromorphic bundles with connection on the projective line by giving motivicity consequences similar to those given by Katz for rigid local systems.

Title : Canonical Dynamics on the Character Varieties of the Painlevé Equations

Abstract : The dynamic of the differential Painlevé VI equation is conjugated through the Riemann-Hilbert map to a rational symplectic dynamic on the character variety (the Fricke cubic surface). For the other Painlevé equations there are also (wild) character varieties: spaces of representations of wild fundamental groupoids. They are symplectic affine cubic surfaces, symplectically birationnally equivalent to the symplectic place (\C^2,\omega=du\wedge dv/(uv)). Using some natural subgroups of the Cremona Plane Symplectic Group, and pull-backs by some natural coordinates, we will define canonical dynamics on the character varieties of all the Painlevé equations, extending the case of Painlevé VI. These canonical dynamics are rational and symplectic and they contain special elements and subgroups: Stokes transformations, monodromies, exponential tori. In the Painlevé V case these special elements are conjugated to similar elements in the local wild dynamic of the Painlevé equation through the (wild) Riemann-Hilbert map. We will discuss also the other cases, in particular Painlevé II and Painlevé I

Title : Several dynamics related to the Painlevé V equation

Abstract : The dynamic of the differential Painlevé VI equation is conjugated through the Riemann-Hilbert map to a rational dynamic on a character variety. In order to extend this fact to the other Painlevé equations, the character variety related to the Painlevé V equation will be presented here from linear representations of a wild fundamental groupoid by the usual monodromy, Stokes operators and exponential torus. This point of view will allow us to consider a tame dynamic, and its extension by a confluent dynamic. We will also discuss a canonical dynamic on this character variety, using the pull-back of the Cremona symplectic group through canonical coordinates.

This presentation comes from a joint work with JP Ramis, and gives a focus in the case of the Painlevé V equation of his lecture.