## Classical Analysis

Seminar information archive ～12/02｜Next seminar｜Future seminars 12/03～

**Seminar information archive**

### 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)

### 2012/06/27

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

Moduli space of meromorphic connections with ramified irregular singularities on principal bundles (JAPANESE)

**Daisuke Yamakawa**(Tokyo Institute of Technology)Moduli space of meromorphic connections with ramified irregular singularities on principal bundles (JAPANESE)

### 2011/07/08

14:30-16:00 Room #128 (Graduate School of Math. Sci. Bldg.)

$q$-Drinfeld-Sokolov hierarchy, $q$-Painlev¥'e equations, and $q$-hypergeometric functions (JAPANESE)

**T. Suzuki**(Osaka Prefecture University)$q$-Drinfeld-Sokolov hierarchy, $q$-Painlev¥'e equations, and $q$-hypergeometric functions (JAPANESE)

### 2011/06/24

15:00-16:30 Room #128 (Graduate School of Math. Sci. Bldg.)

A Schwarz map of Appell's $F_2$ whose monodromy group is

related to the reflection group of type $D_4$ (JAPANESE)

**J. Sekiguchi**(Tokyo University of Agriculture and Technology)A Schwarz map of Appell's $F_2$ whose monodromy group is

related to the reflection group of type $D_4$ (JAPANESE)

[ Abstract ]

The system of differential equations for Appell's hypergeometric function $F_2(a,b,b',c,c';x,y)$ has four fundamental solutions.

Let $u_1,u_2,u_3,u_4$ be such solutions. If the monodromy group of the system is finite, the closure of the image of the Schwarz map $U(x,y)=(u_1(x,y),u_2(x,y),u_3(x,y),u_4(x,y))$

is a hypersurface $S$ of the 3-dimensional projective space ${\\bf P}^3$. Then $S$ is defined by $P(u_1,u_2,u_3,u_4)=0$ for a polynomial $P(t_1,t_2,t_3,t_4)$.

It is M. Kato (Univ. Ryukyus) who determined the parameter

$a,b,b',c,c'$ such that the monodromy group of the system for $F_2(a,b,b',c,c';x,y)$ is finite. It follows from his result that such a group is the semidirect product of an irreducible finite reflection group $G$ of rank four by an abelian group.

In this talk, we treat the system for $F_2(a,b,b',c,c';x,y)$ with

$(a,b,b',c,c')=(1/2,1/6,-1/6,1/3,2/3$. In this case, the monodromy group is the semidirect group of $G$ by $Z/3Z$, where $G$ is the reflection group of type $D_4$. The polynomial $P(t_1,t_2,t_3,t_4)$ in this case is of degree four. There are 16 ordinary singular points in the hypersurface $S$.

In the rest of my talk, I explain the background of the study.

The system of differential equations for Appell's hypergeometric function $F_2(a,b,b',c,c';x,y)$ has four fundamental solutions.

Let $u_1,u_2,u_3,u_4$ be such solutions. If the monodromy group of the system is finite, the closure of the image of the Schwarz map $U(x,y)=(u_1(x,y),u_2(x,y),u_3(x,y),u_4(x,y))$

is a hypersurface $S$ of the 3-dimensional projective space ${\\bf P}^3$. Then $S$ is defined by $P(u_1,u_2,u_3,u_4)=0$ for a polynomial $P(t_1,t_2,t_3,t_4)$.

It is M. Kato (Univ. Ryukyus) who determined the parameter

$a,b,b',c,c'$ such that the monodromy group of the system for $F_2(a,b,b',c,c';x,y)$ is finite. It follows from his result that such a group is the semidirect product of an irreducible finite reflection group $G$ of rank four by an abelian group.

In this talk, we treat the system for $F_2(a,b,b',c,c';x,y)$ with

$(a,b,b',c,c')=(1/2,1/6,-1/6,1/3,2/3$. In this case, the monodromy group is the semidirect group of $G$ by $Z/3Z$, where $G$ is the reflection group of type $D_4$. The polynomial $P(t_1,t_2,t_3,t_4)$ in this case is of degree four. There are 16 ordinary singular points in the hypersurface $S$.

In the rest of my talk, I explain the background of the study.