## Classical Analysis

Seminar information archive ～10/10｜Next seminar｜Future seminars 10/11～

**Seminar information archive**

### 2024/01/24

10:30-12:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations (English)

**Gergő Nemes**(Tokyo Metropolitan University)On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations (English)

[ Abstract ]

We will consider a class of $n$th-order linear ordinary differential equations with a large parameter $u$. Analytic solutions of these equations can be described by (divergent) formal series in

descending powers of $u$. We shall demonstrate that, given mild conditions on the potential functions of the equation, the formal solutions are Borel summable with respect to the parameter $u$ in large, unbounded domains of the independent variable. We will establish that the formal series expansions serve as asymptotic expansions, uniform with respect to the independent variable, for the Borel re-summed exact solutions. Additionally, the exact solutions can be expressed using factorial series in the parameter, and these expansions converge in half-planes, uniformly with respect to the independent variable. To illustrate our theory, we apply it to an $n$th-order Airy-type equation.

Related preprint: https://arxiv.org/abs/2312.14449

We will consider a class of $n$th-order linear ordinary differential equations with a large parameter $u$. Analytic solutions of these equations can be described by (divergent) formal series in

descending powers of $u$. We shall demonstrate that, given mild conditions on the potential functions of the equation, the formal solutions are Borel summable with respect to the parameter $u$ in large, unbounded domains of the independent variable. We will establish that the formal series expansions serve as asymptotic expansions, uniform with respect to the independent variable, for the Borel re-summed exact solutions. Additionally, the exact solutions can be expressed using factorial series in the parameter, and these expansions converge in half-planes, uniformly with respect to the independent variable. To illustrate our theory, we apply it to an $n$th-order Airy-type equation.

Related preprint: https://arxiv.org/abs/2312.14449

### 2023/10/31

10:30-14:30 Room #126 (Graduate School of Math. Sci. Bldg.)

The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)

The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)

**Benedetta Facciotti**(University of Birmingham) 10:30-11:30The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)

[ Abstract ]

In this talk, through simple examples, I will explain the basic idea behind the Riemann-Hilbert correspondence. It is a correspondence between two different moduli spaces: the de Rham moduli space parametrizing meromorphic differential equations, and the Betti moduli space describing local systems of solutions and the representations of the fundamental group defined by them. We will see why such a correspondence breaks down for higher order poles.

In this talk, through simple examples, I will explain the basic idea behind the Riemann-Hilbert correspondence. It is a correspondence between two different moduli spaces: the de Rham moduli space parametrizing meromorphic differential equations, and the Betti moduli space describing local systems of solutions and the representations of the fundamental group defined by them. We will see why such a correspondence breaks down for higher order poles.

**Nikita Nikolaev**(University of Birmingham) 13:30-14:30The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)

[ Abstract ]

I will explain an approach to extending the Riemann-Hilbert correspondence to the setting of equations with higher-order poles using the representation theory of holomorphic Lie groupoids. Each Riemann-Hilbert problem is associated with a suitable Lie algebroid that is integrable to a holomorphic Lie groupoid that can be explicitly constructed as a blowup of the fundamental groupoid. Then the Riemann-Hilbert correspondence can be formulated in rather familiar Lie theoretic terms as the correspondence between representations of algebroids and groupoids. An advantage of this approach is that groupoid representations can be investigated geometrically. Based on joint work with Benedetta Facciotti (Birmingham) and Marta Mazzocco (Birmingham), as well as joint work with Francis Bischoff (Regina) and Marco Gualtieri (Toronto).

I will explain an approach to extending the Riemann-Hilbert correspondence to the setting of equations with higher-order poles using the representation theory of holomorphic Lie groupoids. Each Riemann-Hilbert problem is associated with a suitable Lie algebroid that is integrable to a holomorphic Lie groupoid that can be explicitly constructed as a blowup of the fundamental groupoid. Then the Riemann-Hilbert correspondence can be formulated in rather familiar Lie theoretic terms as the correspondence between representations of algebroids and groupoids. An advantage of this approach is that groupoid representations can be investigated geometrically. Based on joint work with Benedetta Facciotti (Birmingham) and Marta Mazzocco (Birmingham), as well as joint work with Francis Bischoff (Regina) and Marco Gualtieri (Toronto).

### 2023/08/21

10:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Stokes matrices of confluent hypergeometric systems and the isomonodromy deformation equations (ENGLISH)

Stokes matrices of quantum confluent hypergeometric systems and the representation of quantum groups (ENGLISH)

The WKB approximation of (quantum) confluent hypergeometric systems, Cauchy interlacing inequality and crystal basis (ENGLISH)

**Xiaomeng Xu**(BICMR, China) 10:00-11:30Stokes matrices of confluent hypergeometric systems and the isomonodromy deformation equations (ENGLISH)

[ Abstract ]

This talk first gives an introduction to the Stokes matrices of a linear meromorphic system of Poncaré rank 1, and the associated nonlinear isomonodromy deformation equation. The nonlinear equation naturally arises from the theory of Frobenius manifolds, stability conditions, Poisson-Lie groups and so on, and can be seen as a higher rank generalizations of the sixth Painlevé equation. The talk then gives a parameterization of the asymptotics of the solutions of the isomonodromy equation at a critical point, the explicit formula of the monodromy/Stokes matrices of the linear problem via the parameterization, as well as a connection formula between two differential critical points. It can be seen as a generalization of Jimbo's work for the sixth Painlevé equation to a higher rank case. It is partially based on a joint work with Qian Tang.

This talk first gives an introduction to the Stokes matrices of a linear meromorphic system of Poncaré rank 1, and the associated nonlinear isomonodromy deformation equation. The nonlinear equation naturally arises from the theory of Frobenius manifolds, stability conditions, Poisson-Lie groups and so on, and can be seen as a higher rank generalizations of the sixth Painlevé equation. The talk then gives a parameterization of the asymptotics of the solutions of the isomonodromy equation at a critical point, the explicit formula of the monodromy/Stokes matrices of the linear problem via the parameterization, as well as a connection formula between two differential critical points. It can be seen as a generalization of Jimbo's work for the sixth Painlevé equation to a higher rank case. It is partially based on a joint work with Qian Tang.

**Xiaomeng Xu**(BICMR, China) 14:00-15:30Stokes matrices of quantum confluent hypergeometric systems and the representation of quantum groups (ENGLISH)

[ Abstract ]

This talk studies a quantum analog of Stokes matrices of confluent hypergeometric systems. It first gives an introduction to the Stokes phenomenon of an irregular Knizhnik–Zamolodchikov at a second order pole, associated to a regular semisimple element u and a representation $L(\lambda)$ of $gl_n$. It then shows that the Stokes matrices of the

irregular Knizhnik–Zamolodchikov equation define representation of $U_q(gl_n)$ on $L(\lambda)$. In then end, using the isomonodromy approach, it derives an explicit expression of the regularized limit of the Stokes matrices as the regular semisimple element u goes to the caterpillar point in the wonderful compactification.

This talk studies a quantum analog of Stokes matrices of confluent hypergeometric systems. It first gives an introduction to the Stokes phenomenon of an irregular Knizhnik–Zamolodchikov at a second order pole, associated to a regular semisimple element u and a representation $L(\lambda)$ of $gl_n$. It then shows that the Stokes matrices of the

irregular Knizhnik–Zamolodchikov equation define representation of $U_q(gl_n)$ on $L(\lambda)$. In then end, using the isomonodromy approach, it derives an explicit expression of the regularized limit of the Stokes matrices as the regular semisimple element u goes to the caterpillar point in the wonderful compactification.

**Xiaomeng Xu**(BICMR, China) 16:00-17:30The WKB approximation of (quantum) confluent hypergeometric systems, Cauchy interlacing inequality and crystal basis (ENGLISH)

[ Abstract ]

This talk studies the WKB approximation of the linear meromorphic systems of Poncaré rank 1 appearing in talk 1 and 2, via the isomonodromy approach. In the classical case, it unveils a relation between the WKB approximation, the Cauchy interlacing inequality and cluster algebras with the help of the spectral network; in the quantum case, motivated by the crystal limit of the quantum groups, it shows a relation between the WKB approximation and the gl_n-crystal structures. It is partially based on a joint work with

Anton Alekseev, Andrew Neitzke and Yan Zhou.

This talk studies the WKB approximation of the linear meromorphic systems of Poncaré rank 1 appearing in talk 1 and 2, via the isomonodromy approach. In the classical case, it unveils a relation between the WKB approximation, the Cauchy interlacing inequality and cluster algebras with the help of the spectral network; in the quantum case, motivated by the crystal limit of the quantum groups, it shows a relation between the WKB approximation and the gl_n-crystal structures. It is partially based on a joint work with

Anton Alekseev, Andrew Neitzke and Yan Zhou.

### 2022/09/29

11:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Difference module and Homology 6 (JAPANESE)

Difference module and Homology 7 (JAPANESE)

**Koki Ito**(Osaka Electro-Communication University) 11:30-12:00Difference module and Homology 6 (JAPANESE)

**Koki Ito**(Osaka Electro-Communication University) 14:00-17:00Difference module and Homology 7 (JAPANESE)

### 2022/09/28

11:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Difference module and Homology 4 (JAPANESE)

Difference module and Homology 5 (JAPANESE)

**Koki Ito**(Osaka Electro-Communication University) 11:30-12:00Difference module and Homology 4 (JAPANESE)

**Koki Ito**(Osaka Electro-Communication University) 14:00-17:00Difference module and Homology 5 (JAPANESE)

### 2022/09/27

11:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Difference module and Homology 2 (JAPANESE)

Difference module and Homology 3 (JAPANESE)

**Koki Ito**(Osaka Electro-Communication University) 11:30-12:00Difference module and Homology 2 (JAPANESE)

**Koki Ito**(Osaka Electro-Communication University) 14:00-17:00Difference module and Homology 3 (JAPANESE)

### 2022/09/26

14:00-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Difference module and Homology 1 (JAPANESE)

**Koki Ito**(Osaka Electro-Communication University)Difference module and Homology 1 (JAPANESE)

### 2018/11/02

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On the inverse problem of the discrete calculus of variations (ENGLISH)

**Giorgio Gubbiotti**(The University of Sydney)On the inverse problem of the discrete calculus of variations (ENGLISH)

[ Abstract ]

One of the most powerful tools in Mathematical Physics since Euler and Lagrange is the calculus of variations. The variational formulation of mechanics where the equations of motion arise as the minimum of an action functional (the so-called Hamilton's principle), is fundamental in the development of theoretical mechanics and its foundations are present in each textbook on this subject [1, 3, 6]. Beside this, the application of calculus of variations goes beyond mechanics as many important mathematical problems, e.g. the isoperimetrical problem and the catenary, can be formulated in terms of calculus of variations.

An important problem regarding the calculus of variations is to determine which system of differential equations are Euler-Lagrange equations for some variational problem. This problem has a long and interesting history, see e.g. [4]. The general case of this problem remains unsolved, whereas several important results for particular cases were presented during the 20th century.

In this talk we present some conditions on the existence of a Lagrangian in the discrete scalar setting. We will introduce a set of differential operators called annihilation operators. We will use these operators to

reduce the functional equation governing of existence of a Lagrangian for a scalar difference equation of arbitrary even order 2k, with k > 1 to the solution of a system of linear partial differential equations. Solving such equations one can either find the Lagrangian or conclude that it does not exist.

We comment the relationship of our solution of the inverse problem of the discrete calculus of variation with the one given in [2], where a result analogous to the homotopy formula [5] for the continuous case was proven.

References

[1] H. Goldstein, C. Poole, and J. Safko. Classical Mechanics. Pearson Education, 2002.

[2] P. E. Hydon and E. L. Mansfeld. A variational complex for difference equations. Found. Comp. Math., 4:187{217, 2004.

[3] L. D. Landau and E. M. Lifshitz. Mechanics. Course of Theoretical Physics. Elsevier Science, 1982.

[4] P. J. Olver. Applications of Lie Groups to Differential Equations. Springer-Verlag, Berlin, 1986.

[5] M. M. Vainberg. Variational methods for the study of nonlinear operators. Holden-Day, San Francisco, 1964.

[6] E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge, 1999.

One of the most powerful tools in Mathematical Physics since Euler and Lagrange is the calculus of variations. The variational formulation of mechanics where the equations of motion arise as the minimum of an action functional (the so-called Hamilton's principle), is fundamental in the development of theoretical mechanics and its foundations are present in each textbook on this subject [1, 3, 6]. Beside this, the application of calculus of variations goes beyond mechanics as many important mathematical problems, e.g. the isoperimetrical problem and the catenary, can be formulated in terms of calculus of variations.

An important problem regarding the calculus of variations is to determine which system of differential equations are Euler-Lagrange equations for some variational problem. This problem has a long and interesting history, see e.g. [4]. The general case of this problem remains unsolved, whereas several important results for particular cases were presented during the 20th century.

In this talk we present some conditions on the existence of a Lagrangian in the discrete scalar setting. We will introduce a set of differential operators called annihilation operators. We will use these operators to

reduce the functional equation governing of existence of a Lagrangian for a scalar difference equation of arbitrary even order 2k, with k > 1 to the solution of a system of linear partial differential equations. Solving such equations one can either find the Lagrangian or conclude that it does not exist.

We comment the relationship of our solution of the inverse problem of the discrete calculus of variation with the one given in [2], where a result analogous to the homotopy formula [5] for the continuous case was proven.

References

[1] H. Goldstein, C. Poole, and J. Safko. Classical Mechanics. Pearson Education, 2002.

[2] P. E. Hydon and E. L. Mansfeld. A variational complex for difference equations. Found. Comp. Math., 4:187{217, 2004.

[3] L. D. Landau and E. M. Lifshitz. Mechanics. Course of Theoretical Physics. Elsevier Science, 1982.

[4] P. J. Olver. Applications of Lie Groups to Differential Equations. Springer-Verlag, Berlin, 1986.

[5] M. M. Vainberg. Variational methods for the study of nonlinear operators. Holden-Day, San Francisco, 1964.

[6] E. T. Whittaker. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge, 1999.

### 2017/06/01

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Homological and monodromy representations of framed braid groups

(JAPANESE)

**Akishi Ikeda**(IPMU, University of Tokyo)Homological and monodromy representations of framed braid groups

(JAPANESE)

### 2016/12/06

16:45-18:15 Room #154 (Graduate School of Math. Sci. Bldg.)

Introduction to resurgence on the example of saddle-node singularities (ENGLISH)

**David Sauzin**(CNRS)Introduction to resurgence on the example of saddle-node singularities (ENGLISH)

[ Abstract ]

Divergent power series naturally appear when solving such an elementary differential equation as x^2 dy = (x+y) dx, which is the simplest example of saddle-node singularity. I will discuss the formal classification of saddle-node singularities and illustrate on that case Ecalle's resurgence theory, which allows one to analyse the divergence of the formal solutions. One can also deal with resonant saddle-node singularities with one more dimension, a situation which covers the local study at infinity of some Painlevé equations.

Divergent power series naturally appear when solving such an elementary differential equation as x^2 dy = (x+y) dx, which is the simplest example of saddle-node singularity. I will discuss the formal classification of saddle-node singularities and illustrate on that case Ecalle's resurgence theory, which allows one to analyse the divergence of the formal solutions. One can also deal with resonant saddle-node singularities with one more dimension, a situation which covers the local study at infinity of some Painlevé equations.

### 2016/06/23

16:50-18:20 Room #118 (Graduate School of Math. Sci. Bldg.)

Resurgence of formal series solutions of nonlinear differential and difference equations (JAPANESE)

**Shingo Kamimoto**(Hiroshima University)Resurgence of formal series solutions of nonlinear differential and difference equations (JAPANESE)

### 2015/09/25

16:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Confluence of general Schlesinger systems from the viewpoint of Twistor theory (JAPANESE)

**Damiran Tseveennamjil**(Mongolian University of Life Sciences)Confluence of general Schlesinger systems from the viewpoint of Twistor theory (JAPANESE)

### 2015/01/21

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Remarks on the number of accessory parameters (JAPANESE)

**Shingo Kamimoto**(Kyoto University)Remarks on the number of accessory parameters (JAPANESE)

### 2014/11/10

16:00-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

DIFFERENTIAL GALOIS THEORY AND INTEGRABILITY OF DYNAMICAL SYSTEMS

**Jean-Pierre RAMIS**(Toulouse)DIFFERENTIAL GALOIS THEORY AND INTEGRABILITY OF DYNAMICAL SYSTEMS

[ Abstract ]

We will explain how to get obstructions to the integrability of analytic Hamiltonian Systems (in the classical Liouville sense) using Differential Galois Theory (introduced by Emile Picard at the end of XIX-th century). It is the so-called Morales-Ramis theory. Even if this sounds abstract, there exist efficient algorithms allowing to apply the theory and a lot of applications in various domains.

Firstly I will present basics on Hamiltonian Systems and integrability on one side and on Differential Galois Theory on the other side. Then I will state the main theorems. Afterwards I will describe some applications.

We will explain how to get obstructions to the integrability of analytic Hamiltonian Systems (in the classical Liouville sense) using Differential Galois Theory (introduced by Emile Picard at the end of XIX-th century). It is the so-called Morales-Ramis theory. Even if this sounds abstract, there exist efficient algorithms allowing to apply the theory and a lot of applications in various domains.

Firstly I will present basics on Hamiltonian Systems and integrability on one side and on Differential Galois Theory on the other side. Then I will state the main theorems. Afterwards I will describe some applications.

### 2014/10/29

16:00-17:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Whittaker functions and Barnes-Type Lemmas (ENGLISH)

**Eric Stade**(University of Colorado Boulder)Whittaker functions and Barnes-Type Lemmas (ENGLISH)

[ Abstract ]

In the theory of automorphic forms on GL(n,R), which concerns harmonic analysis and representation theory of this group, certain special functions known as GL(n,R) Whittaker functions play an important role. These Whittaker functions are generalizations of classical Whittaker (or, more specifically, Bessel) functions.

Mellin transforms of products of GL(n,R) Whittaker functions may be expressed as certain Barnes type integrals, or equivalently, as hypergeometric series of unit argument. The general theory of automorphic forms predicts that these Mellin transforms reduce, in certain cases, to products of gamma functions. That this does in fact occur amounts to a whole family of generalizations of the so-called Barnes' Lemma and Barnes' Second Lemma, from the theory of hypergeometric series. We will explore these generalizations in this talk.

This talk will not require any specific knowledge of automorphic forms.

In the theory of automorphic forms on GL(n,R), which concerns harmonic analysis and representation theory of this group, certain special functions known as GL(n,R) Whittaker functions play an important role. These Whittaker functions are generalizations of classical Whittaker (or, more specifically, Bessel) functions.

Mellin transforms of products of GL(n,R) Whittaker functions may be expressed as certain Barnes type integrals, or equivalently, as hypergeometric series of unit argument. The general theory of automorphic forms predicts that these Mellin transforms reduce, in certain cases, to products of gamma functions. That this does in fact occur amounts to a whole family of generalizations of the so-called Barnes' Lemma and Barnes' Second Lemma, from the theory of hypergeometric series. We will explore these generalizations in this talk.

This talk will not require any specific knowledge of automorphic forms.

### 2014/07/08

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

ABS equations arising from q-P((A2+A1)^{(1)}) (JAPANESE)

**Nakazono Nobutaka**(University of Sydney)ABS equations arising from q-P((A2+A1)^{(1)}) (JAPANESE)

[ Abstract ]

The study of periodic reductions from ABS equations to discrete Painlevé equations have been investigated by many groups. However, there still remain open questions:

(i) How do we identify the discrete Painlevé equation that would result from applying a periodic reduction to an ABS equation?

(ii) Discrete Painlevé equations obtained by periodic reductions often have insufficient number of parameters. How do we obtain the general case with all essential parameters?

To solve these problems, we investigated the periodic reductions from the viewpoint of Painlevé systems.

In this talk, we show how to construct a lattice where ABS equations arise from relationships between $\\tau$ functions of Painlevé systems and explain how this lattice relates to a hyper cube associated with an ABS equation on each face.

In particular, we consider the $q$-Painlevé equations, which have the affine Weyl group symmetry of type $(A_2+A_1)^{(1)}$.

The study of periodic reductions from ABS equations to discrete Painlevé equations have been investigated by many groups. However, there still remain open questions:

(i) How do we identify the discrete Painlevé equation that would result from applying a periodic reduction to an ABS equation?

(ii) Discrete Painlevé equations obtained by periodic reductions often have insufficient number of parameters. How do we obtain the general case with all essential parameters?

To solve these problems, we investigated the periodic reductions from the viewpoint of Painlevé systems.

In this talk, we show how to construct a lattice where ABS equations arise from relationships between $\\tau$ functions of Painlevé systems and explain how this lattice relates to a hyper cube associated with an ABS equation on each face.

In particular, we consider the $q$-Painlevé equations, which have the affine Weyl group symmetry of type $(A_2+A_1)^{(1)}$.

### 2014/06/24

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Irreducibility of the discrete Painlev\\'e equation of type $D_7$ (JAPANESE)

**Nishioka Seiji**(Yamagata University)Irreducibility of the discrete Painlev\\'e equation of type $D_7$ (JAPANESE)

### 2014/03/19

16:00-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Discrete Schlesinger Equations and Difference Painlevé Equations (ENGLISH)

**Anton Dzhamay**(University of Northern Colorado)Discrete Schlesinger Equations and Difference Painlevé Equations (ENGLISH)

[ Abstract ]

The theory of Schlesinger equations describing isomonodromic

dynamic on the space of matrix coefficients of a Fuchsian system

w.r.t.~continuous deformations is well-know. In this talk we consider

a discrete version of this theory. Discrete analogues of Schlesinger

deformations are Schlesinger transformations that shift the eigenvalues

of the coefficient matrices by integers. By discrete Schlesinger equations

we mean the evolution equations on the matrix coefficients describing

such transformations. We derive these equations, show how they can be

split into the evolution equations on the space of eigenvectors of the

coefficient matrices, and explain how to write the latter equations in

the discrete Hamiltonian form. We also consider some reductions of those

equations to the difference Painlevé equations, again in complete parallel

to the differential case.

This is a joint work with H. Sakai (the University of Tokyo) and

T.Takenawa (Tokyo Institute of Marine Science and Technology).

The theory of Schlesinger equations describing isomonodromic

dynamic on the space of matrix coefficients of a Fuchsian system

w.r.t.~continuous deformations is well-know. In this talk we consider

a discrete version of this theory. Discrete analogues of Schlesinger

deformations are Schlesinger transformations that shift the eigenvalues

of the coefficient matrices by integers. By discrete Schlesinger equations

we mean the evolution equations on the matrix coefficients describing

such transformations. We derive these equations, show how they can be

split into the evolution equations on the space of eigenvectors of the

coefficient matrices, and explain how to write the latter equations in

the discrete Hamiltonian form. We also consider some reductions of those

equations to the difference Painlevé equations, again in complete parallel

to the differential case.

This is a joint work with H. Sakai (the University of Tokyo) and

T.Takenawa (Tokyo Institute of Marine Science and Technology).

### 2013/10/30

16:00-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

The space of monodromy and Stokes data for q-difference equations (ENGLISH)

**Jacques Sauloy**(Institute de Mathematiques de Toulouse, Universite Paul Sabatier)The space of monodromy and Stokes data for q-difference equations (ENGLISH)

[ Abstract ]

Riemann-Hilbert correspondance for fuchsian q-difference equations has been obtained by Sauloy along the lines of Birkhoff and then, for irregular equations, by Ramis, Sauloy and Zhang in terms of q-Stokes operators.

However, these correspondances are not formulated in geometric terms, which makes them little suitable for the study of isomonodromy or "iso-Stokes" deformations. Recently, under the impulse of Ohyama, we started to construct such a geometric description in order to apply it to the famous work of Jimbo-Sakai and then to more recent extensions. I shall describe this work.

Riemann-Hilbert correspondance for fuchsian q-difference equations has been obtained by Sauloy along the lines of Birkhoff and then, for irregular equations, by Ramis, Sauloy and Zhang in terms of q-Stokes operators.

However, these correspondances are not formulated in geometric terms, which makes them little suitable for the study of isomonodromy or "iso-Stokes" deformations. Recently, under the impulse of Ohyama, we started to construct such a geometric description in order to apply it to the famous work of Jimbo-Sakai and then to more recent extensions. I shall describe this work.

### 2012/12/05

16:00-17:30 Room #270 (Graduate School of Math. Sci. Bldg.)

On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case (ENGLISH)

**Andrei Kapaev**(SISSA, Trieste, Italy)On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case (ENGLISH)

[ Abstract ]

Equation $P_I^2$ is the second member in the hierarchy of ODEs associated with the classical Painlev\\’e first equation $P_I$ and can be solved via the Riemann-Hilbert (RH) problem approach. It is known also that solutions of equation $P_I^2$ as the functions of $x$ depending on the parameter $t$ can be used to construct a 4-parameter family of isomonodromic solutions to the KdV equation. Given the monodromy data, the set of points $(x,t)$, where the above mentioned RH problem is not solvable, is called the Malgrange divisor. The function $x=a(t)$, which parametrizes locally the Malgrange divisor, satisfies a nonlinear ODE which admits a Lax pair representation and can be also studied using an RH problem. We discuss the relations between these two kinds of the RH problems and the properties of their $t$-large genus 1 asymptotic solutions.

Equation $P_I^2$ is the second member in the hierarchy of ODEs associated with the classical Painlev\\’e first equation $P_I$ and can be solved via the Riemann-Hilbert (RH) problem approach. It is known also that solutions of equation $P_I^2$ as the functions of $x$ depending on the parameter $t$ can be used to construct a 4-parameter family of isomonodromic solutions to the KdV equation. Given the monodromy data, the set of points $(x,t)$, where the above mentioned RH problem is not solvable, is called the Malgrange divisor. The function $x=a(t)$, which parametrizes locally the Malgrange divisor, satisfies a nonlinear ODE which admits a Lax pair representation and can be also studied using an RH problem. We discuss the relations between these two kinds of the RH problems and the properties of their $t$-large genus 1 asymptotic solutions.

### 2012/11/21

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Beyond the fundamental group (ENGLISH)

**Philip Boalch**(ENS-DMA & CNRS Paris)Beyond the fundamental group (ENGLISH)

[ Abstract ]

Moduli spaces of representations of the fundamental group of a Riemann surface have been studied from numerous points of view and appear in many parts of mathematics and theoretical physics. They form an interesting class of symplectic manifolds, they often have Kahler or hyperkahler metrics (in which case they are diffeomorphic to spaces of Higgs bundles, i.e. Hitchin integrable systems), and they admit nonlinear actions of braid groups and mapping class groups with fascinating dynamical properties. The aim of this talk is to describe some aspects of this story and sketch their extension to the context of the "wild fundamental group", which naturally appears when one considers {\\em meromorphic} connections on Riemann surfaces. In particular some new examples of hyperkahler manifolds appear in this way, some of which are familiar from classical work on the Painleve equations.

Moduli spaces of representations of the fundamental group of a Riemann surface have been studied from numerous points of view and appear in many parts of mathematics and theoretical physics. They form an interesting class of symplectic manifolds, they often have Kahler or hyperkahler metrics (in which case they are diffeomorphic to spaces of Higgs bundles, i.e. Hitchin integrable systems), and they admit nonlinear actions of braid groups and mapping class groups with fascinating dynamical properties. The aim of this talk is to describe some aspects of this story and sketch their extension to the context of the "wild fundamental group", which naturally appears when one considers {\\em meromorphic} connections on Riemann surfaces. In particular some new examples of hyperkahler manifolds appear in this way, some of which are familiar from classical work on the Painleve equations.

### 2012/11/07

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

WKB analysis of the Painlev\\'e functions and parameteric Stokes phenomena (JAPANESE)

**Kohei IWAKI**(Kyoto University)WKB analysis of the Painlev\\'e functions and parameteric Stokes phenomena (JAPANESE)

### 2012/07/18

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Free divisors, holonomic systems and algebraic Painlev\\'{e} sixth solutions (ENGLISH)

**Jiro Sekiguchi**(Tokyo University of Agriculture and Technology)Free divisors, holonomic systems and algebraic Painlev\\'{e} sixth solutions (ENGLISH)

[ Abstract ]

In this talk, I will report an attempt to treat algebraic solutions of Painlev\\'{e} VI equation in a unified manner.

A classification of algebraic solutions of Painlev\\'{e} VI equation was accomplished by O. Lisovyy and Y. Tykhyy after efforts on the construction of such solutions by many authors, K. Iwasaki N. J. Hitchin, P. Boalch, B. Dubrovin, M. Mazzocco, A. V. Kitaev, R. Vidunas and others.

The outline of my approach is as follows.

Let $t$ be a variable and let $w$ be its algebraic function such that $w$ is a solution of Painlev\\'{e} sixth equation. Suppose that both $t$ and $w$ are rational functions of a parameter. Namely $(t,w)$ defines a rational curve.

(1) Find a polynomial $P(u)$ such that $t=\\frac{P(-u)}{P(u)}$.

(2) From $P(u)$, define a weighted homogeneous polynomial $f(x_1,x_2,x_3)=x_3f_1(x_1,x_2,x_3)$ of three variables $x_1,x_2,x_3$, where $(1,2,n)$ is the weight system of $(x_1,x_2,x_3)$ with $n=\\deg P(u)$. The hypersurface $D:f(x)=0$ is a free divisor in ${\\bf C}^3$. Note that $\\deg_{x_3}f_1=2$.

(3) Construct a holonomic system ${\\sl M}$ on ${\\bf C}^3$ of rank two with singularities along $D$.

(4) Construct an ordinary differential equation from the holonomic system ${\\sl M}$ with respect to $x_3$. This differential equation has three singular points $z_0,z_1,a_s$ in $x_3$-line.

(5) Putting $t=\\frac{z_1}{z_0},\\lambda=\\frac{a_s}{z_0}$, we conclude that $(t,\\lambda)$ is equivalent to the pair $(t,w)$.

Our study starts with showing the existence of $P(u)$ in (1). From the classification by Losovyy and Tykhyy, I find that the existence of $P(u)$ is guaranteed for Solutions III, IV, Solutions $k$ ($1\\le k\\le 21$, $k\\not= 4,13,14,20$) and Solution 30. We checked whether (1)-(5) are true or not in these cases separately and as a consequence (1)-(5) hold for the all these cases except Solutions 19, 21.

In this talk, I will report an attempt to treat algebraic solutions of Painlev\\'{e} VI equation in a unified manner.

A classification of algebraic solutions of Painlev\\'{e} VI equation was accomplished by O. Lisovyy and Y. Tykhyy after efforts on the construction of such solutions by many authors, K. Iwasaki N. J. Hitchin, P. Boalch, B. Dubrovin, M. Mazzocco, A. V. Kitaev, R. Vidunas and others.

The outline of my approach is as follows.

Let $t$ be a variable and let $w$ be its algebraic function such that $w$ is a solution of Painlev\\'{e} sixth equation. Suppose that both $t$ and $w$ are rational functions of a parameter. Namely $(t,w)$ defines a rational curve.

(1) Find a polynomial $P(u)$ such that $t=\\frac{P(-u)}{P(u)}$.

(2) From $P(u)$, define a weighted homogeneous polynomial $f(x_1,x_2,x_3)=x_3f_1(x_1,x_2,x_3)$ of three variables $x_1,x_2,x_3$, where $(1,2,n)$ is the weight system of $(x_1,x_2,x_3)$ with $n=\\deg P(u)$. The hypersurface $D:f(x)=0$ is a free divisor in ${\\bf C}^3$. Note that $\\deg_{x_3}f_1=2$.

(3) Construct a holonomic system ${\\sl M}$ on ${\\bf C}^3$ of rank two with singularities along $D$.

(4) Construct an ordinary differential equation from the holonomic system ${\\sl M}$ with respect to $x_3$. This differential equation has three singular points $z_0,z_1,a_s$ in $x_3$-line.

(5) Putting $t=\\frac{z_1}{z_0},\\lambda=\\frac{a_s}{z_0}$, we conclude that $(t,\\lambda)$ is equivalent to the pair $(t,w)$.

Our study starts with showing the existence of $P(u)$ in (1). From the classification by Losovyy and Tykhyy, I find that the existence of $P(u)$ is guaranteed for Solutions III, IV, Solutions $k$ ($1\\le k\\le 21$, $k\\not= 4,13,14,20$) and Solution 30. We checked whether (1)-(5) are true or not in these cases separately and as a consequence (1)-(5) hold for the all these cases except Solutions 19, 21.

### 2012/07/11

14:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

On a q-analog of Painlevé III (D_7^{(1)}) and its algebraic function solutions (Joint work with N. Nakazono) (JAPANESE)

First order systems of linear ordinary differential equations and

representations of quivers (ENGLISH)

**Seiji Nishioka**(Yamagata University) 14:00-15:30On a q-analog of Painlevé III (D_7^{(1)}) and its algebraic function solutions (Joint work with N. Nakazono) (JAPANESE)

**Kazuki Hiroe**(Kyoto University) 16:00-17:30First order systems of linear ordinary differential equations and

representations of quivers (ENGLISH)

### 2012/07/04

15:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Symmetries of quantum Lax equations for the Painlev\\'e equations (JAPANESE)

**Hajime Nagoya**(Kobe University)Symmetries of quantum Lax equations for the Painlev\\'e equations (JAPANESE)