## 古典解析セミナー

担当者 大島 利雄, 坂井 秀隆

### 2018年11月02日(金)

17:00-18:30   数理科学研究科棟(駒場) 122号室
Giorgio Gubbiotti 氏 (The University of Sydney)
On the inverse problem of the discrete calculus of variations (ENGLISH)
[ 講演概要 ]
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年06月01日(木)

16:30-18:00   数理科学研究科棟(駒場) 122号室

Homological and monodromy representations of framed braid groups
(JAPANESE)
[ 講演概要 ]
KZ方程式は配置空間上の可積分な微分方程式であり,そのモノドロミー表現を考えることで組みひも群の様々な表現が得られることはよく知られている. 2008年に神保-名古屋-Sunによって合流型のKZ方程式が導入された. この話では, 合流型のKZ方程式のモノドロミー表現を考えることで,枠付組みひも群(リボンの絡み方を表す群)の表現が得られることを説明する.
また, 枠付組みひも群の表現を, ある空間のホモロジー群を用いて構成し, 合流KZ方程式のモノドロミー表現との関係について説明する.

### 2016年12月06日(火)

16:45-18:15   数理科学研究科棟(駒場) 154号室
David Sauzin 氏 (CNRS)
Introduction to resurgence on the example of saddle-node singularities (ENGLISH)
[ 講演概要 ]
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年06月23日(木)

16:50-18:20   数理科学研究科棟(駒場) 118号室

Resurgence of formal series solutions of nonlinear differential and difference equations (JAPANESE)
[ 講演概要 ]
Resurgent analysis は1980年代に J. Ecalle により創始された. そこでは, alien derivatives 等の漸近解析における重要な概念が導入され, 近年数理物理学においても大きな注目を集めている. 本講演では Resurgent analysis の基本事項の概説から始め, 最近得られた非線形微(差)分方程式の形式解の resurgence に関する結果の紹介を行う.

### 2015年09月25日(金)

16:00-17:00   数理科学研究科棟(駒場) 126号室
Damiran Tseveennamjil 氏 (モンゴル生命科学大学)
Twistor理論からみた一般Schlesinger系の合流 (JAPANESE)
[ 講演概要 ]
Grassmann多様体上の一般超幾何関数に対する合流の操作の類似として、一般Schlesinger系を導く線形方程式系の合流を論じる。

### 2015年01月21日(水)

16:30-18:00   数理科学研究科棟(駒場) 128号室

Remarks on the number of accessory parameters (JAPANESE)
[ 講演概要 ]

このような作用素(の族)の大域的な構造を考察する上で,
アクセサリーパラメータは重要な役割を果たす.

このアクセサリーパラメータに関する考察を行う.

### 2014年11月10日(月)

16:00-17:00   数理科学研究科棟(駒場) 122号室
Jean-Pierre RAMIS 氏 (Toulouse)
DIFFERENTIAL GALOIS THEORY AND INTEGRABILITY OF DYNAMICAL SYSTEMS
[ 講演概要 ]
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年10月29日(水)

16:00-17:00   数理科学研究科棟(駒場) 117号室
Whittaker functions and Barnes-Type Lemmas (ENGLISH)
[ 講演概要 ]
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年07月08日(火)

16:00-17:30   数理科学研究科棟(駒場) 122号室

ABS equations arising from q-P((A2+A1)^{(1)}) (JAPANESE)
[ 講演概要 ]
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年06月24日(火)

16:00-17:30   数理科学研究科棟(駒場) 122号室

D7型離散パンルヴェ方程式の既約性
(JAPANESE)
[ 講演概要 ]

### 2014年03月19日(水)

16:00-17:00   数理科学研究科棟(駒場) 128号室
Anton Dzhamay 氏 (University of Northern Colorado)
Discrete Schlesinger Equations and Difference Painlevé Equations (ENGLISH)
[ 講演概要 ]
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年10月30日(水)

16:00-17:00   数理科学研究科棟(駒場) 122号室
Jacques Sauloy 氏 (Institute de Mathematiques de Toulouse, Universite Paul Sabatier)
The space of monodromy and Stokes data for q-difference equations (ENGLISH)
[ 講演概要 ]
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年12月05日(水)

16:00-17:30   数理科学研究科棟(駒場) 270号室
Andrei Kapaev 氏 (SISSA, Trieste, Italy)
On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case (ENGLISH)
[ 講演概要 ]
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年11月21日(水)

16:00-17:30   数理科学研究科棟(駒場) 128号室
Philip Boalch 氏 (ENS-DMA & CNRS Paris)
Beyond the fundamental group (ENGLISH)
[ 講演概要 ]
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年11月07日(水)

16:00-17:30   数理科学研究科棟(駒場) 128号室

パンルヴェ函数のWKB解析とパラメトリックStokes現象 (JAPANESE)
[ 講演概要 ]
パンルヴェ函数の漸近挙動の研究、および接続問題は、他分野への応用という観点からも非常に重要なテーマである。今回は(2階の古典的な)パンルヴェ方程式を完全WKB解析の立場から考察する。特に、方程式に含まれる「独立変数以外のパラメータ」の値を変化させた際にもある種のStokes現象(パラメトリックStokes現象)が起こることを紹介し、漸近挙動の不連続な変化を記述する接続公式を導く。

### 2012年07月18日(水)

16:00-17:30   数理科学研究科棟(駒場) 128号室

Free divisors, holonomic systems and algebraic Painlev\\'{e} sixth solutions (ENGLISH)
[ 講演概要 ]
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年07月11日(水)

14:00-17:30   数理科学研究科棟(駒場) 128号室

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)

### 2012年07月04日(水)

15:30-17:00   数理科学研究科棟(駒場) 128号室

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

### 2012年06月27日(水)

16:00-17:30   数理科学研究科棟(駒場) 128号室

### 2011年07月08日(金)

14:30-16:00   数理科学研究科棟(駒場) 128号室

$q$離散ドリンフェルト・ソコロフ階層と$q$パンルヴェ方程式、$q$超幾何函数 (JAPANESE)
[ 講演概要 ]
ドリンフェルト・ソコロフ階層はKP階層のアフィン・リー代数への一般化であり、相似簡約と呼ばれる簡約操作によって様々なパンルヴェ型微分方程式を導くことが知られている。

すなわち、DS階層のq離散化を定式化し、そこから神保・坂井によるqパンルヴェVI方程式の高階化となるものを導き、その特殊解と
してq超幾何函数が現れることを示す。

### 2011年06月24日(金)

15:00-16:30   数理科学研究科棟(駒場) 128号室

A Schwarz map of Appell's $F_2$ whose monodromy group is
related to the reflection group of type $D_4$ (JAPANESE)
[ 講演概要 ]
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.

### 2011年02月18日(金)

11:00-15:45   数理科学研究科棟(駒場) 126号室
T. Morita 氏 (Osaka University) 11:00-12:00
Connection problem on the Hahn-Exton $q$-Bessel functions (ENGLISH)
M. Yamaguchi 氏 (University of Tokyo) 13:30-14:30
Rigidity index and middle convolution of $q$-difference equations (Joint work with H. Sakai)
(ENGLISH)
L. Di Vizio 氏 (Universite Paris 7) 14:45-15:45
Arithmetic theory of $q$-difference equations and applications (Joint work with C. Hardouin)
(ENGLISH)

### 2011年02月18日(金)

10:15-10:45   数理科学研究科棟(駒場) 126号室
Y. Ohyama 氏 (Osaka University)
Degeneration shceme of basic hypergeometric equations and $q$-Painlev¥'e equations (ENGLISH)

### 2011年02月17日(木)

11:00-17:00   数理科学研究科棟(駒場) 126号室
L. Di Vizio 氏 (Universite Paris 7) 11:00-12:00
Overview of local theory of $q$-difference equations and summation, 1
(ENGLISH)
Y. Katsushima 氏 (University of Tokyo) 13:30-14:30
Bounded operators on Gevrey spaces and additive difference operators (in a view of differential operators of infinite order) (ENGLISH)
K. Matsuya 氏 (University of Tokyo) 14:45-15:45
Blow-up of solutions for a nonlinear difference equation (ENGLISH)
L. Di Vizio 氏 (Universite Paris 7) 16:00-17:00
Overview of local theory of $q$-difference equations and summation, 2 (ENGLISH)

### 2011年02月16日(水)

13:30-17:00   数理科学研究科棟(駒場) 126号室
H. Sakai 氏 (University of Tokyo) 13:30-14:30
Isomonodromic deformation and 4-dimensional Painlev\\'e type equations (ENGLISH)
H. Kawakami 氏 (University of Tokyo) 14:45-15:45
Degeneration scheme of 4-dimensional Painlev¥'e type equations
(Joint work with H. Sakai and A. Nakamura)

(ENGLISH)
S. Nishioka 氏 (University of Tokyo) 16:00-17:00
Solvability of difference Riccati equations (ENGLISH)