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過去の記録 ～04/24｜次回の予定｜今後の予定 04/25～
| 担当者 | 大島 利雄, 坂井 秀隆 |
|---|
過去の記録
2024年01月24日(水)
10:30-12:00 数理科学研究科棟(駒場) 122号室
Gergő Nemes 氏 (東京都立大学)
On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations
(English)
Gergő Nemes 氏 (東京都立大学)
On the Borel summability of formal solutions of certain higher-order linear ordinary differential equations
(English)
[ 講演概要 ]
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年10月31日(火)
10:30-14:30 数理科学研究科棟(駒場) 126号室
Benedetta Facciotti 氏 (University of Birmingham) 10:30-11:30
The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)
The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)
Benedetta Facciotti 氏 (University of Birmingham) 10:30-11:30
The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)
[ 講演概要 ]
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 氏 (University of Birmingham) 13:30-14:30In 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.
The Wild Riemann-Hilbert Correspondence via Groupoid Representations (ENGLISH)
[ 講演概要 ]
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年08月21日(月)
10:00-17:30 数理科学研究科棟(駒場) 123号室
Xiaomeng Xu 氏 (BICMR, China) 10:00-11:30
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 氏 (BICMR, China) 10:00-11:30
Stokes matrices of confluent hypergeometric systems and the isomonodromy deformation equations (ENGLISH)
[ 講演概要 ]
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 氏 (BICMR, China) 14:00-15:30This 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.
Stokes matrices of quantum confluent hypergeometric systems and the representation of quantum groups (ENGLISH)
[ 講演概要 ]
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 氏 (BICMR, China) 16:00-17:30This 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.
The WKB approximation of (quantum) confluent hypergeometric systems, Cauchy interlacing inequality and crystal basis (ENGLISH)
[ 講演概要 ]
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年09月29日(木)
11:30-17:00 数理科学研究科棟(駒場) 118号室
伊藤公毅 氏 (大阪電気通信大学) 11:30-12:00
差分加群とホモロジー 6 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学) 14:00-17:00
差分加群とホモロジー 7 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学) 11:30-12:00
差分加群とホモロジー 6 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学) 14:00-17:00
差分加群とホモロジー 7 (JAPANESE)
2022年09月28日(水)
11:30-17:00 数理科学研究科棟(駒場) 128号室
伊藤公毅 氏 (大阪電気通信大学) 11:30-12:00
差分加群とホモロジー 4 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学) 14:00-17:00
差分加群とホモロジー 5 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学) 11:30-12:00
差分加群とホモロジー 4 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学) 14:00-17:00
差分加群とホモロジー 5 (JAPANESE)
2022年09月27日(火)
11:30-17:00 数理科学研究科棟(駒場) 118号室
伊藤公毅 氏 (大阪電気通信大学) 11:30-12:00
差分加群とホモロジー 2 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学) 14:00-17:00
差分加群とホモロジー 3 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学) 11:30-12:00
差分加群とホモロジー 2 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学) 14:00-17:00
差分加群とホモロジー 3 (JAPANESE)
2022年09月26日(月)
14:00-17:00 数理科学研究科棟(駒場) 118号室
伊藤公毅 氏 (大阪電気通信大学)
差分加群とホモロジー 1 (JAPANESE)
伊藤公毅 氏 (大阪電気通信大学)
差分加群とホモロジー 1 (JAPANESE)
[ 講演概要 ]
（今回ほとんどはq差分）
以下2つのテーマについてお話ししたい：
テーマI---q差分ド・ラームコホモロジーとqサイクルのホモロジー
(i) q差分ド・ラーム・コホモロジーの定義
(ii) そのdualとしてあるべきqサイクルのホモロジーの定義
(ii)' それらのpairingであるべきq積分の定義（というか、(ii)'をみながら(ii)をつくる、というのが思考の流れではある）
テーマII---差分加群の理論整備にむけて
(i) コーシー問題とは何か
(ii) 非特性的とは
(iii) holonomicとは
特殊函数（とりわけ超幾何系）の研究で、積分表示が強力な武器となることは皆さんよくご承知と思う。積分表示的手法は、位相幾何や代数幾何も活用する「複素積分の理論」（ツイスト・ド・ラーム理論）として現代化・整備されるに至った。さて、q特殊函数の研究にもジャクソン積分表示が有効な武器になることが明らかになってきており、q差分ド・ラーム理論が提案されるに至った。この事情について、
https://www.jstage.jst.go.jp/pub/pdfpreview/sugaku1947/49/4_49_4_350.jpg
（雑誌「数学」49(1997)-4, 350-364）
を予めみておくと把握できよう。この論説を理解したい、あるいは、もう少し明快に再定式化したいというのが今回お話しすることを考え始めた動機である。たとえば、qの世界では、接空間や余接空間の概念として十分といえるものは未だみつかっていない（と思う）。従って、q差分形式といっても、自然な定式化がみえにくい。また、ライプニッツ則のズレや座標変換への強い制約などの困難がある。ここについて、１つの有望な打開策が「q差分加群」による定式化である。q差分加群とはD加群のq差分版である。q差分ド・ラーム・コホモロジー（複体）も「q差分加群のド・ラーム・コホモロジー」として自然に定義される。（論理的には、q差分形式を飛び越えて直接定義できる（本地）。但し、「手で扱える」ようにするためにq差分形式（垂迹）をとることになる。）ここで、q差分加群を導入するために、新たな位相（あるグロタンディーク位相）を考える必要が出てくる。今回は、グロタンディーく位相の定義の復習からお話しする。また、コホモロジー類の積分についても、これまで、ジャクソン積分を用いたり複素周回積分を用いたりと込み入っている。今回の講演では、これらを含む「q積分」を導入する。現在、ほぼ出来上がっている１次元の場合について上記のことを述べることにする。高次元について、できているところに限りお話ししたい。ここまでが第一のテーマである。
q差分加群なるものを登場させた以上、その基礎理論の整備は必須であろう。これが第２のテーマである。D加群の理論が(主として線形)偏微分方程式の一般論を与えるものであるものなら、q差分加群の理論は偏q差分方程式の一般論を与えねばなるまい。しかしながら、偏(q)差分方程式の一般論はおろか、各論だって多くは知られていないのではないだろうか？（どなたかご存じの方は、この機会にお教えいただけると有難いです。）手始めに、「コーシー問題とは何か」「非特性的とは」「表象とは」についての考察を述べる。その延長上にホロノミックを位置づけることを試みる。ただし、この部分については、現在進行形で完成形ではない。（前述の通り、余接空間にあたるものが不在であるため、聊かアドホックな感が否めない。）
今回の講演では、現在進行形の部分もあり、お聞き苦しいところが出てくるかもしれませんが、どうかよろしくお願いいたします。
（今回ほとんどはq差分）
以下2つのテーマについてお話ししたい：
テーマI---q差分ド・ラームコホモロジーとqサイクルのホモロジー
(i) q差分ド・ラーム・コホモロジーの定義
(ii) そのdualとしてあるべきqサイクルのホモロジーの定義
(ii)' それらのpairingであるべきq積分の定義（というか、(ii)'をみながら(ii)をつくる、というのが思考の流れではある）
テーマII---差分加群の理論整備にむけて
(i) コーシー問題とは何か
(ii) 非特性的とは
(iii) holonomicとは
特殊函数（とりわけ超幾何系）の研究で、積分表示が強力な武器となることは皆さんよくご承知と思う。積分表示的手法は、位相幾何や代数幾何も活用する「複素積分の理論」（ツイスト・ド・ラーム理論）として現代化・整備されるに至った。さて、q特殊函数の研究にもジャクソン積分表示が有効な武器になることが明らかになってきており、q差分ド・ラーム理論が提案されるに至った。この事情について、
https://www.jstage.jst.go.jp/pub/pdfpreview/sugaku1947/49/4_49_4_350.jpg
（雑誌「数学」49(1997)-4, 350-364）
を予めみておくと把握できよう。この論説を理解したい、あるいは、もう少し明快に再定式化したいというのが今回お話しすることを考え始めた動機である。たとえば、qの世界では、接空間や余接空間の概念として十分といえるものは未だみつかっていない（と思う）。従って、q差分形式といっても、自然な定式化がみえにくい。また、ライプニッツ則のズレや座標変換への強い制約などの困難がある。ここについて、１つの有望な打開策が「q差分加群」による定式化である。q差分加群とはD加群のq差分版である。q差分ド・ラーム・コホモロジー（複体）も「q差分加群のド・ラーム・コホモロジー」として自然に定義される。（論理的には、q差分形式を飛び越えて直接定義できる（本地）。但し、「手で扱える」ようにするためにq差分形式（垂迹）をとることになる。）ここで、q差分加群を導入するために、新たな位相（あるグロタンディーク位相）を考える必要が出てくる。今回は、グロタンディーく位相の定義の復習からお話しする。また、コホモロジー類の積分についても、これまで、ジャクソン積分を用いたり複素周回積分を用いたりと込み入っている。今回の講演では、これらを含む「q積分」を導入する。現在、ほぼ出来上がっている１次元の場合について上記のことを述べることにする。高次元について、できているところに限りお話ししたい。ここまでが第一のテーマである。
q差分加群なるものを登場させた以上、その基礎理論の整備は必須であろう。これが第２のテーマである。D加群の理論が(主として線形)偏微分方程式の一般論を与えるものであるものなら、q差分加群の理論は偏q差分方程式の一般論を与えねばなるまい。しかしながら、偏(q)差分方程式の一般論はおろか、各論だって多くは知られていないのではないだろうか？（どなたかご存じの方は、この機会にお教えいただけると有難いです。）手始めに、「コーシー問題とは何か」「非特性的とは」「表象とは」についての考察を述べる。その延長上にホロノミックを位置づけることを試みる。ただし、この部分については、現在進行形で完成形ではない。（前述の通り、余接空間にあたるものが不在であるため、聊かアドホックな感が否めない。）
今回の講演では、現在進行形の部分もあり、お聞き苦しいところが出てくるかもしれませんが、どうかよろしくお願いいたします。
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)
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.
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
池田 曉志 氏 (東京大学 IPMU)
Homological and monodromy representations of framed braid groups
(JAPANESE)
池田 曉志 氏 (東京大学 IPMU)
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.
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)
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.
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
神本晋吾 氏 (広島大学)
Resurgence of formal series solutions of nonlinear differential and difference equations (JAPANESE)
神本晋吾 氏 (広島大学)
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.
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)
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。
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
神本 晋吾 氏 (京都大学)
Remarks on the number of accessory parameters (JAPANESE)
神本 晋吾 氏 (京都大学)
Remarks on the number of accessory parameters (JAPANESE)
[ 講演概要 ]
代数的な常微分作用素を考える.
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.
本講演では特異摂動論的な立場から,
このアクセサリーパラメータに関する考察を行う.
代数的な常微分作用素を考える.
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.
本講演では特異摂動論的な立場から,
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
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.
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
Eric Stade 氏 (University of Colorado Boulder)
Whittaker functions and Barnes-Type Lemmas (ENGLISH)
Eric Stade 氏 (University of Colorado Boulder)
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.
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
中園信孝 氏 (シドニー大学)
ABS equations arising from q-P((A2+A1)^{(1)}) (JAPANESE)
中園信孝 氏 (シドニー大学)
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 $(A_2+A_1)^{(1)}$.
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
西岡斉治 氏 (山形大学)
D7型離散パンルヴェ方程式の既約性
(JAPANESE)
西岡斉治 氏 (山形大学)
D7型離散パンルヴェ方程式の既約性
(JAPANESE)
[ 講演概要 ]
離散パンルヴェ方程式は2階代数的差分方程式で、パンルヴェ方程式と呼ばれる2階代数的微分方程式の差分方程式における対応物である。ここでは特にD7型を扱う。登場当初からパンルヴェ方程式が線形微分方程式に帰着されるか、という問題が議論された。結論は否定的であり、さらに楕円関数・アーベル関数を用いても解を表示できないとされる。この性質は既約性や還元不能性と呼ばれている。一方、離散パンルヴェ方程式に対しても同様の議論ができる。今回はD7型離散パンルヴェ方程式の既約性の証明を紹介する。なお、D7型はq差分方程式ではない。
離散パンルヴェ方程式は2階代数的差分方程式で、パンルヴェ方程式と呼ばれる2階代数的微分方程式の差分方程式における対応物である。ここでは特にD7型を扱う。登場当初からパンルヴェ方程式が線形微分方程式に帰着されるか、という問題が議論された。結論は否定的であり、さらに楕円関数・アーベル関数を用いても解を表示できないとされる。この性質は既約性や還元不能性と呼ばれている。一方、離散パンルヴェ方程式に対しても同様の議論ができる。今回はD7型離散パンルヴェ方程式の既約性の証明を紹介する。なお、D7型はq差分方程式ではない。
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)
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).
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)
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.
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)
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.
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)
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.
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
岩木耕平 氏 (京都大学)
パンルヴェ函数のWKB解析とパラメトリックStokes現象 (JAPANESE)
岩木耕平 氏 (京都大学)
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。
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
関口 次郎 氏 (東京農工大学)
Free divisors, holonomic systems and algebraic Painlev\\'{e} sixth solutions (ENGLISH)
関口 次郎 氏 (東京農工大学)
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.
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
西岡斉治 氏 (山形大学) 14:00-15:30
On a q-analog of Painlevé III (D_7^{(1)}) and its algebraic function solutions (Joint work with N. Nakazono) (JAPANESE)
廣惠一希 氏 (京都大学) 16:00-17:30
First order systems of linear ordinary differential equations and
representations of quivers (ENGLISH)
西岡斉治 氏 (山形大学) 14:00-15:30
On a q-analog of Painlevé III (D_7^{(1)}) and its algebraic function solutions (Joint work with N. Nakazono) (JAPANESE)
廣惠一希 氏 (京都大学) 16:00-17:30
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
名古屋 創 氏 (神戸大学)
Symmetries of quantum Lax equations for the Painlev\\'e equations (JAPANESE)
名古屋 創 氏 (神戸大学)
Symmetries of quantum Lax equations for the Painlev\\'e equations (JAPANESE)
