Abstract :

Middle convolutions are quite effective to analyze linear differential equations on the Riemann sphere in cooperate with other operations such as gauge transformations, extensions to several variables and restrictions to curves, unfoldings and confluences etc. An analysis of the solutions to the equations using these operations will be explained. In particular, versal unfoldings, integral representations of the solutions, connection problems (Stokes coefficients) etc. will be discussed.

Abstract :

The lecture is based on works arXiv:2104.04818, arXiv:2008.11483, arXiv:2003.06997, arXiv:2002.05927 done with S. Dumitrescu, L. Heller, S. Heller, J. P. dos Santos and T. Mochizuki. We investigate the properties of the holomorphic connections on the trivial holomorphic bundle on a Riemann surface and more generally on a compact Kaehler manifold.

Abstract :

We will discuss two problems about transcendental features of Painlevé foliations in relationship with neighborhoods of compact curves in complex surfaces. One, about neighborhoods of rational curves, is a work in progress with Maycol Falla Luza. The second one, about neighborhoods of elliptic curves, is a work in progress with Gibran Espejo and Laura Ortiz, using recent classification obtained by Frederic Touzet, Sergei Voronin and the author.

Abstract :

Isomonodromic (that is, integrable) deformations of connections with irregular singularities in dimension one are well understood away from turning points of the parameter space. In general, at the turning points, the theorem of Kedlaya-Mochizuki is needed to understand the local behaviour of the Stokes structure, but it breaks the notion of deformation. Motivated by understanding boundaries of Frobenius manifolds, Cotti, Dubrovin and Guzzetti have analyzed some simple turning points and shown vanishing of certain entries of the Stokes matrices at the neighbourhood of these turning points. The talk will give a different point of view on these results.

Abstract :

We review series representations of tau functions of Painlevé equations and their relations to irregular conformal blocks, which are defined as expectation values of vertex operators for Virasoro algebra on irregular Verma modules. A conjectural combinatorial formula for a three point irregular conformal block is given. Toward proving that series representations of tau functions of Painlevé equations in terms of irregular conformal blocks satisfy bilinear equations, irregular vertex operators for a super Virasoro algebra(Neveu-Schwarz-Ramond algebra) are presented.

Abstract :

We present a method to construct certain families $\mathcal{M}$ of connections on the projective line. The fibres of the Riemann--Hilbert morphism $RH:\mathcal{M}\rightarrow \mathcal{R}$, where $\mathcal{R}$ denotes the family of analytic data, should be parametrized by a variable $t$. This produces a Lax pair and Painlevé type equations. The analytic classification of singularities of connections will be presented, because this is essential background for the construction of $\mathcal{M}$ and $\mathcal{R}$. The method produces besides the classical Painlevé equations, new families of Painlev\'e type equations.

Abstract :

We look at the moduli space of rank 2 logarithmic lambda-connections with quasi-parabolic structure on a curve. Up to the action of a groupoid of local gauge transformations, there is a Riemann-Hilbert correspondence. This in turn leads to the construction of the Deligne-Hitchin twistor space such that harmonic bundles give preferred sections. The relative tangent bundle along a preferred section has a mixed twistor structure where the weight two piece parametrizes the deformations of local monodromy transformations at the singularities.

Abstract :

We express discrete Painleve equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painleve equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.

Abstract :

The 4-dimensional Painlevé-type systems are tangible (in a sense) yet give nontrivial higher-dimensional analogs of the 2-dimensional Painlevé systems. In this talk, we study the genus two curves associated with the autonomous versions of the 4-dimensional Painlevé-type systems. As an application of the result, we can find a linear problem starting from a nonlinear problem. We will mention some connections to work on "Generalized Hitchin systems on rational surfaces" by Eric Rains. This talk is partially based on joint work with Eric Rains.

Abstract :

Complex analytic orbi-curves give rise to natural examples of twisted local systems, for which the fundamental group acts non-trivially on the coefficients. In this talk, we construct moduli varieties of twisted local systems, and prove that these are affine varieties over the complex numbers, whose (strong) topology can be studied through an appropriate version of the non-Abelian Hodge correspondence to the case of nonconstant coefficients. This partially answers a question of Carlos Simpson on the meaning of the Dolbeault moduli space in the nonconstant case.

Abstract : We study the birational structure of moduli spaces of rank 2 logarithmic connections on smooth projective curves. We generalize a previous result by F. Loray and M. -H. Saito in the projective line case. Our approach is to analyze the underlying parabolic bundles and apparent singularities of the parabolic connections.

Abstract :

In 2004, K Nishioka showed that if y_1, ... , y_n are solutions of the first Painlevé equation such that the transcendence degree of the extension of C(t) by y_1,y'_1, ... , y_n,y'_n is strictly less than 2n, then there exist i<j such that y_i = y_j. This result was extended to other Painlevé equations by J.Nagloo and A.Pillay in 2017 using Hsushovski-Skolovic trichotomy theorem in Model Theory of differential field of characteristic 0. In this talk, I will explain how the Galois pseudogroup defined independently by B. Malgrange and H. Umemura can be used as an alternative to the trichotomy theorem.

Abstract :

We study a quantum (non-commutative) representation of the affine Weyl group of type $E_8^{(1)}$.

The representation is given by birational actions on two variables $x, y$ with $q$-commutation relation $yx=qxy$.

We also construct a lift of the representation including the tau variables.

The Weyl group actions on tau variables are described by interesting quantum polynomials $F(x,y)$.

We give a characterization of the polynomials using their singularity structures as the $q$-difference operators.

As an application, the quantum mirror curve for 5d E-string is rederived by the Weyl group symmetry.

This talk is base on the joint work with S. Moriyama, arXiv.:2104.06661[math.QA].

Abstract :

The Fifth Painlevé equation (PV) is obtained from the Sixth one (PVI) by confluence. In principle, this allows to transfer knowledge from PVI to PV, however, to do this, one needs to be able to deal with divergence. For PVI a great deal of information can be obtained from its nonlinear monodromy group, the elements of which act on solutions by analytic continuation along loops, and which has a well known representation through the Riemann-Hilbert correspondence as an explicit action on the character variety of the (linear) monodromy data. For PV the analogical object is the ``nonlinear wild monodromy pseudogroup'' which expresses not only the nonlinear monodromy but also the nonlinear Stokes phenomenon at the singularity at infinity. The goal of the talk is to show how one can obtain the corresponding action of this pseudogroup on the "wild character variety" of the (linear) monodromy and Stokes data by studying the confluence. To do this I will try to explain how the confluence PVI -> PV works on both sides of the Riemann-Hilbert correspondence.

Abstract :

It is well known that irregular singular points of differential equations are obtained by the confluence of some regular singular points and then some analytic properties of these irregular singular points can be related to that of regular singular points via this confluence procedure. In this talk, I will explain a construction of flat families of moduli spaces of meromeprhic connections on the Riemann sphere in which generic fibers are moduli spaces of regular singular connections and specializations of deformation parameters correspond to the confluence of their regular singularities. Then I will show that every moduli space of connections with unramified irregular singularities has this kind of deformation.

Abstract :

We study meromorphic $\hbar$-connections on vector bundles over a Riemann surface. These are families of vector bundles with meromorphic connections, parameterised by a small complex parameter $\hbar$, which degenerate to a Higgs bundle in the limit as $\hbar \to 0$. I will show under rather general assumptions that (at least in rank two) the vector bundle has canonical flat filtrations whose limits as $\hbar \to 0$ in a halfplane converge to the eigendecomposition of the corresponding limiting Higgs bundle. These filtrations, which we call the WKB filtrations, are defined over certain open subsets given by real-flows of a certain holomorphic vector field obtained from the limiting Higgs field. The key to the construction of WKB filtrations is the ability to find exact solutions to a singularly perturbed Riccati equation. The name is derived from the fact that, in the special case where the $\hbar$-connection arises from a Schrödinger equation, the WKB filtrations are generated by local exact WKB solutions. We also show that near each pole of the connection, the WKB filtration (at every fixed nonzero value of $\hbar$) coincides with the local Levelt filtration determined by growth rates of flat sections as they are parallel transported into the singular point. As a result, the WKB filtration is a very special filtration on the vector bundle which converges to the Higgs bundle eigendecomposition as $\hbar \to 0$ and also to the eigendecomposition of the connection’s principal part as we parallel transport into the pole.

Abstract :

We propose a birational representation of an extended affine Weyl group of type $(A_{mn-1}+A_{m-1}+A_{m-1})^{(1)}$. It provides a generalization of Sakai's $q$-Garnier system as a group of translations. The affine Weyl group is formulated in two ways. One is a cluster mutation and the other is a Lax form with $mn\times mn$ matrices. If time permits, we discuss a particular solution in terms of a $q$-hypergeometric function.

Abstract :

First we explain how recent results on the asymptotic solutions of Hitchin's equations on a curve allow one to prove Simpson's Geometric P = W conjecture in the Painlevé 6 case. In the second part of the talk we outline the main ideas of our ongoing work on the extension of this result to the Garnier case with 5 parabolic points.

Abstract:

We study connection problems of q-Painlevé equations. This problem is divided Into three parts:

1. Study q-connection problems on hypergeometric equations

2. Study q-connection problems on linearized equations of the Painlevé equations

3. Consider q-connection problems on the Painlevé equations

Abstract : In this talk, I will discuss the quantisation of the Painlevé monodromy manifolds as a special class of quantum del Pezzo surfaces. In particular I will introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.

Abstract : The weight of a differential equation is defined through the Newton diagram of the equation. It gives the weighted projective space, that is the natural compactification of the phase space of a differential equation. In this talk, I show how to construct a weighted projective space, analysis of the Painlevé equation using the geometry of a weighted projective space, and related topics.