## 離散数理モデリングセミナー

過去の記録 ～06/17｜次回の予定｜今後の予定 06/18～

担当者 | 時弘哲治, ウィロックス ラルフ |
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**過去の記録**

### 2024年05月01日(水)

13:00-15:00 数理科学研究科棟(駒場) 126号室

Semitoric systems and their symplectic invariants (English)

**Jaume Alonso 氏**(Technische Universität Berlin)Semitoric systems and their symplectic invariants (English)

[ 講演概要 ]

Semitoric systems are a special class of completely integrable systems defined on four-dimensional symplectic manifolds. One of the reasons that make these systems interesting is their classification in terms of five symplectic invariants proposed by Pelayo and Vũ Ngọc. In the last years, many efforts have been made in order to extend this classification to broader settings, to generate more examples and to compute their invariants. In this talk we will discuss some of the most important properties of semitoric systems and introduce some families of systems with one and more focus-focus singularities. We will also show how the symplectic invariants of these systems change as we move the parameters of the families and how they can be computed using mathematical software.

This is a joint work with H. Dullin, S. Hohloch and J. Palmer.

Semitoric systems are a special class of completely integrable systems defined on four-dimensional symplectic manifolds. One of the reasons that make these systems interesting is their classification in terms of five symplectic invariants proposed by Pelayo and Vũ Ngọc. In the last years, many efforts have been made in order to extend this classification to broader settings, to generate more examples and to compute their invariants. In this talk we will discuss some of the most important properties of semitoric systems and introduce some families of systems with one and more focus-focus singularities. We will also show how the symplectic invariants of these systems change as we move the parameters of the families and how they can be computed using mathematical software.

This is a joint work with H. Dullin, S. Hohloch and J. Palmer.

### 2024年04月24日(水)

13:30-15:00 数理科学研究科棟(駒場) 126号室

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups (English)

**Jaume Alonso 氏**(Technische Universität Berlin)Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups (English)

[ 講演概要 ]

In this talk we propose a new method for the exact computation of the degree $\deg (f^n)$ of the iterates of a birational map $f:\mathbb{P}^n \dashrightarrow \mathbb{P}^n$. The method is based on two main ingredients. Firstly, the factorisation of a polynomial under the pull-back by $f$, based on local indices of a polynomial associated to blow-ups used to resolve the singularity. Secondly, the propagation of these indices along the orbits of $f$. We will illustrate the method in different examples, showing its flexibility, since it does not require the construction of an algebraically stable lift of $f$, unlike other methods based on the Picard group.

This is a joint work with Yuri Suris and Kangning Wei.

In this talk we propose a new method for the exact computation of the degree $\deg (f^n)$ of the iterates of a birational map $f:\mathbb{P}^n \dashrightarrow \mathbb{P}^n$. The method is based on two main ingredients. Firstly, the factorisation of a polynomial under the pull-back by $f$, based on local indices of a polynomial associated to blow-ups used to resolve the singularity. Secondly, the propagation of these indices along the orbits of $f$. We will illustrate the method in different examples, showing its flexibility, since it does not require the construction of an algebraically stable lift of $f$, unlike other methods based on the Picard group.

This is a joint work with Yuri Suris and Kangning Wei.

### 2024年04月17日(水)

13:30-15:00 数理科学研究科棟(駒場) 126号室

Discrete Painlevé equations and pencils of quadrics in 3D (English)

**Jaume Alonso 氏**(Technische Universität Berlin)Discrete Painlevé equations and pencils of quadrics in 3D (English)

[ 講演概要 ]

In this talk we propose a new geometric interpretation of discrete Painlevé equations. From this point of view, the equations are birational transformations of $\mathbb{P}^3$ that preserve a pencil of quadrics and map each quadric of the pencil to a different one, according to a Möbius transformation of the pencil parameter. This allows for a classification of discrete Painlevé equations based on the classification of pencils of quadrics in $\mathbb{P}^3$. In this scheme, discrete Painlevé equations are obtained as deformations of the 3D QRT maps introduced in the previous talk, which consist of the composition of two involutions along the generators of the quadrics of a pencil of quadrics until they meet a second pencil. The deformation is then a birational (often linear) transformation in $\mathbb{P}^3$ under which the pencil remains invariant, but the individual quadrics do not.

This is a joint work with Yuri Suris and Kangning Wei.

In this talk we propose a new geometric interpretation of discrete Painlevé equations. From this point of view, the equations are birational transformations of $\mathbb{P}^3$ that preserve a pencil of quadrics and map each quadric of the pencil to a different one, according to a Möbius transformation of the pencil parameter. This allows for a classification of discrete Painlevé equations based on the classification of pencils of quadrics in $\mathbb{P}^3$. In this scheme, discrete Painlevé equations are obtained as deformations of the 3D QRT maps introduced in the previous talk, which consist of the composition of two involutions along the generators of the quadrics of a pencil of quadrics until they meet a second pencil. The deformation is then a birational (often linear) transformation in $\mathbb{P}^3$ under which the pencil remains invariant, but the individual quadrics do not.

This is a joint work with Yuri Suris and Kangning Wei.

### 2024年04月10日(水)

13:30-15:00 数理科学研究科棟(駒場) 056号室

セミナー室変更：470セミナー室→056セミナー室

Integrable birational maps and a generalisation of QRT to 3D (English)

セミナー室変更：470セミナー室→056セミナー室

**Jaume Alonso 氏**(Technische Universität Berlin)Integrable birational maps and a generalisation of QRT to 3D (English)

[ 講演概要 ]

When completely integrable Hamiltonian systems are discretised, the resulting discrete-time systems are often no longer integrable themselves. This is the so-called problem of integrable discretisation. Two known exceptions to this situation in 3D are the Kahan-Hirota-Kimura discretisations of the Euler top and the Zhukovski-Volterra gyrostat with one non-zero linear parameter β, both birational maps of degree 3. The integrals of these systems define pencils of quadrics. By analysing the geometry of these pencils, we develop a framework that generalises QRT maps and QRT roots to 3D, which allows us to create new integrable maps as a composition of two involutions. We show that under certain geometric conditions, the new maps become of degree 3. We use these results to create new families of discrete integrable maps and we solve the problem of integrability of the Zhukovski-Volterra gyrostat with two β’s.

This is a joint work with Yuri Suris and Kangning Wei.

When completely integrable Hamiltonian systems are discretised, the resulting discrete-time systems are often no longer integrable themselves. This is the so-called problem of integrable discretisation. Two known exceptions to this situation in 3D are the Kahan-Hirota-Kimura discretisations of the Euler top and the Zhukovski-Volterra gyrostat with one non-zero linear parameter β, both birational maps of degree 3. The integrals of these systems define pencils of quadrics. By analysing the geometry of these pencils, we develop a framework that generalises QRT maps and QRT roots to 3D, which allows us to create new integrable maps as a composition of two involutions. We show that under certain geometric conditions, the new maps become of degree 3. We use these results to create new families of discrete integrable maps and we solve the problem of integrability of the Zhukovski-Volterra gyrostat with two β’s.

This is a joint work with Yuri Suris and Kangning Wei.

### 2023年08月30日(水)

16:30-17:30 数理科学研究科棟(駒場) 370号室

Classical structure theory for second-order semi-degenerate super-integrable systems (English)

**Joshua Capel 氏**(University of New South Wales)Classical structure theory for second-order semi-degenerate super-integrable systems (English)

[ 講演概要 ]

Second-order superintegrable systems have long been a subject of interest in the study of integral systems, primarily due to their close connection with separation of variables. Among these, the 'non-degenerate' second-order systems on constant curvature spaces have been particularly well studies, and are mostly understood.

A non-degenerate system in n-dimensions comprises an (n+1)-dimensional vector space of potentials, along with 2n-1 second-order constants that commute with the Hamiltonian of the system.

In contrast, a semi-degenerate system is characterised by having fewer than n+1 parameters. In this talk, we will discuss the structure theory of truly n-dimensional potentials (meaning potentials that are not just restrictions of (n+1)-dimensional counterparts). We will see that the classical structure theory appears just as rich as the non-degenerate case.

This talk is joint work with Jeremy Nugent and Jonathan Kress.

Second-order superintegrable systems have long been a subject of interest in the study of integral systems, primarily due to their close connection with separation of variables. Among these, the 'non-degenerate' second-order systems on constant curvature spaces have been particularly well studies, and are mostly understood.

A non-degenerate system in n-dimensions comprises an (n+1)-dimensional vector space of potentials, along with 2n-1 second-order constants that commute with the Hamiltonian of the system.

In contrast, a semi-degenerate system is characterised by having fewer than n+1 parameters. In this talk, we will discuss the structure theory of truly n-dimensional potentials (meaning potentials that are not just restrictions of (n+1)-dimensional counterparts). We will see that the classical structure theory appears just as rich as the non-degenerate case.

This talk is joint work with Jeremy Nugent and Jonathan Kress.

### 2023年07月13日(木)

17:30-18:30 オンライン開催

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On the Painlevé XXV - Ermakov equation (English)

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**Galina Filipuk 氏**(University of Warsaw)On the Painlevé XXV - Ermakov equation (English)

[ 講演概要 ]

We study a nonlinear second order ordinary differential equation which we call the Ermakov-Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced to the Ermakov or the Painlevé XXV equation. The Ermakov-Painlevé XXV equation also arises from a generalized Riccati equation and the related third order linear differential equation via the Schwarzian derivative. Starting from the Riccati equation and the second-order element of the Riccati chain as the simplest examples of linearizable equations, by introducing a suitable change of variables, it is shown how the Schwarzian derivative represents a key tool in the construction of solutions. Two families of Bäcklund transformations, which link the linear and nonlinear equations under investigation, are obtained. Some analytical examples will be given and discussed.

The talk will be mainly based on the paper

S. Carillo, A. Chichurin, G. Filipuk, F. Zullo, Schwarzian derivative, Painleve XXV--Ermakov equation and Backlund transformations, accepted in Mathematische Nachrichten, https://doi.org/10.1002/mana.202200180, available at arXiv:2201.02267 [nlin.SI].

We study a nonlinear second order ordinary differential equation which we call the Ermakov-Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced to the Ermakov or the Painlevé XXV equation. The Ermakov-Painlevé XXV equation also arises from a generalized Riccati equation and the related third order linear differential equation via the Schwarzian derivative. Starting from the Riccati equation and the second-order element of the Riccati chain as the simplest examples of linearizable equations, by introducing a suitable change of variables, it is shown how the Schwarzian derivative represents a key tool in the construction of solutions. Two families of Bäcklund transformations, which link the linear and nonlinear equations under investigation, are obtained. Some analytical examples will be given and discussed.

The talk will be mainly based on the paper

S. Carillo, A. Chichurin, G. Filipuk, F. Zullo, Schwarzian derivative, Painleve XXV--Ermakov equation and Backlund transformations, accepted in Mathematische Nachrichten, https://doi.org/10.1002/mana.202200180, available at arXiv:2201.02267 [nlin.SI].

### 2023年06月08日(木)

18:15-19:15 オンライン開催

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Deformations of Zamolodchikov periodicity, discrete integrability and the Laurent property (English)

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**Andy Hone 氏**(University of Kent)Deformations of Zamolodchikov periodicity, discrete integrability and the Laurent property (English)

[ 講演概要 ]

Zamolodchikov periodicity was a conjectured property of Y-systems observed in exactly solvable models of quantum field theory associated with simple Lie algebras. The advent of Fomin & Zelevinsky's theory of cluster algebras provided an ideal mathematical framework for proving and formulating extensions of this property. Recently we have found a family of birational maps which deforms the periodic dynamics observed by Zamolodchikov, and destroys the Laurent property that is an inherent feature of cluster dynamics, but still preserves integrability. In this talk we present a variety of examples of deformed integrable maps in types A, B & D, and show how to restore the Laurent property in higher dimensions. This is the combined result of joint work with Grabowski, Kouloukas, Kim and Mase.

Zamolodchikov periodicity was a conjectured property of Y-systems observed in exactly solvable models of quantum field theory associated with simple Lie algebras. The advent of Fomin & Zelevinsky's theory of cluster algebras provided an ideal mathematical framework for proving and formulating extensions of this property. Recently we have found a family of birational maps which deforms the periodic dynamics observed by Zamolodchikov, and destroys the Laurent property that is an inherent feature of cluster dynamics, but still preserves integrability. In this talk we present a variety of examples of deformed integrable maps in types A, B & D, and show how to restore the Laurent property in higher dimensions. This is the combined result of joint work with Grabowski, Kouloukas, Kim and Mase.

### 2023年01月13日(金)

13:15-14:45 数理科学研究科棟(駒場) 126号室

An infinite sequence of Heron triangles with two rational medians (English)

**Andy Hone 氏**(University of Kent)An infinite sequence of Heron triangles with two rational medians (English)

[ 講演概要 ]

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, despite an assertion by Schubert that even two rational medians are impossible, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E(Q) with Mordell-Weil group Z x Z/2Z, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, despite an assertion by Schubert that even two rational medians are impossible, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E(Q) with Mordell-Weil group Z x Z/2Z, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.

### 2023年01月11日(水)

13:15-16:45 数理科学研究科棟(駒場) 056号室

Determinantal expressions for Ohyama polynomials (English)

Discrete dynamics, continued fractions and hyperelliptic curves (English)

**Joe Harrow 氏**(University of Kent) 13:15-14:45Determinantal expressions for Ohyama polynomials (English)

[ 講演概要 ]

The Ohyama polynomials provide algebraic solutions of the D7 case of the Painleve III equation at a particular sequence of parameter values. It is known that many special function solutions of Painleve equations are expressed in terms of tau functions that can be written in the form of determinants, but until now such a representation for the Ohyama polynomials was not known. Here we present two different determinantal formulae for these polynomials: the first, in terms of Wronskian determinants related to a Darboux transformation for a Lax pair of KdV type; and the second, in terms of Hankel determinants, which is related to the Toda lattice. If time permits, then connections with orthogonal polynomials, and with the recent Riemann-Hilbert approach of Buckingham & Miller, will briefly be mentioned.

The Ohyama polynomials provide algebraic solutions of the D7 case of the Painleve III equation at a particular sequence of parameter values. It is known that many special function solutions of Painleve equations are expressed in terms of tau functions that can be written in the form of determinants, but until now such a representation for the Ohyama polynomials was not known. Here we present two different determinantal formulae for these polynomials: the first, in terms of Wronskian determinants related to a Darboux transformation for a Lax pair of KdV type; and the second, in terms of Hankel determinants, which is related to the Toda lattice. If time permits, then connections with orthogonal polynomials, and with the recent Riemann-Hilbert approach of Buckingham & Miller, will briefly be mentioned.

**Andy Hone 氏**(University of Kent) 15:15-16:45Discrete dynamics, continued fractions and hyperelliptic curves (English)

[ 講演概要 ]

After reviewing some standard facts about continued fractions for quadratic irrationals, we switch from the real numbers to the field of Laurent series, and describe some classical and more recent results on continued fraction expansions for the square root of an even degree polynomial, and other functions defined on the associated hyperelliptic curve. In the latter case, we extend results of van der Poorten on continued fractions of Jacobi type (J-fractions), and explain the connection with a family of discrete integrable systems (including Quispel-Roberts-Thompson maps and Somos sequences), orthogonal polynomials, and the Toda lattice. If time permits, we will make some remarks on current work with John Roberts and Pol Vanhaecke, concerning expansions involving the square root of an odd degree polynomial, Stieltjes continued fractions, and the Volterra lattice.

After reviewing some standard facts about continued fractions for quadratic irrationals, we switch from the real numbers to the field of Laurent series, and describe some classical and more recent results on continued fraction expansions for the square root of an even degree polynomial, and other functions defined on the associated hyperelliptic curve. In the latter case, we extend results of van der Poorten on continued fractions of Jacobi type (J-fractions), and explain the connection with a family of discrete integrable systems (including Quispel-Roberts-Thompson maps and Somos sequences), orthogonal polynomials, and the Toda lattice. If time permits, we will make some remarks on current work with John Roberts and Pol Vanhaecke, concerning expansions involving the square root of an odd degree polynomial, Stieltjes continued fractions, and the Volterra lattice.

### 2022年11月30日(水)

15:00-16:30 数理科学研究科棟(駒場) 128号室

Folding transformations for q-Painleve equations (English)

**Mikhail Bershtein 氏**(Skoltech・HSE / IPMU)Folding transformations for q-Painleve equations (English)

[ 講演概要 ]

Folding transformation of the Painleve equations is an algebraic (of degree greater than 1) transformation between solutions of different equations. In 2005 Tsuda, Okamoto and Sakai classified folding transformations of differential Painleve equations. These transformations are in correspondence with automorphisms of affine Dynkin diagrams. We give a complete classification of folding transformations of the q-difference Painleve equations, these transformations are in correspondence with certain subdiagrams of the affine Dynkin diagrams (possibly with automorphism). The method is based on Sakai's approach to Painleve equations through rational surfaces.

Based on joint work with A. Shchechkin [arXiv:2110.15320]

Folding transformation of the Painleve equations is an algebraic (of degree greater than 1) transformation between solutions of different equations. In 2005 Tsuda, Okamoto and Sakai classified folding transformations of differential Painleve equations. These transformations are in correspondence with automorphisms of affine Dynkin diagrams. We give a complete classification of folding transformations of the q-difference Painleve equations, these transformations are in correspondence with certain subdiagrams of the affine Dynkin diagrams (possibly with automorphism). The method is based on Sakai's approach to Painleve equations through rational surfaces.

Based on joint work with A. Shchechkin [arXiv:2110.15320]

### 2022年08月18日(木)

15:00-16:00 数理科学研究科棟(駒場) 117号室

Different Hamiltonians for Painlevé Equations and their identification using geometry of the space of

initial conditions (English)

**Anton Dzhamay 氏**(University of Northern Colorado)Different Hamiltonians for Painlevé Equations and their identification using geometry of the space of

initial conditions (English)

[ 講演概要 ]

It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique – there are many very different Hamiltonians that result in the same differential Painlevé equation. In this paper we describe a systematic procedure of finding

changes of coordinates transforming different Hamiltonian systems into some canonical form.

Our approach is based on the Okamoto-Sakai geometric approach to Painlevé equations. We explain this approach using the differential P-IV equation as an example, but the procedure is general and can be easily adapted to other Painlevé equations as well. (Joint work with Galina Filipuk, Adam Ligeza and Alexander Stokes.)

It is well-known that differential Painlevé equations can be written in a Hamiltonian form. However, a coordinate form of such representation is far from unique – there are many very different Hamiltonians that result in the same differential Painlevé equation. In this paper we describe a systematic procedure of finding

changes of coordinates transforming different Hamiltonian systems into some canonical form.

Our approach is based on the Okamoto-Sakai geometric approach to Painlevé equations. We explain this approach using the differential P-IV equation as an example, but the procedure is general and can be easily adapted to other Painlevé equations as well. (Joint work with Galina Filipuk, Adam Ligeza and Alexander Stokes.)

### 2021年10月28日(木)

19:00-20:00 オンライン開催

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Deformations of cluster mutations and invariant presymplectic forms

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**Andrew Hone 氏**(University of Kent)Deformations of cluster mutations and invariant presymplectic forms

[ 講演概要 ]

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4 we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions of the discrete sine-Gordon equation.

We consider deformations of sequences of cluster mutations in finite type cluster algebras, which destroy the Laurent property but preserve the presymplectic structure defined by the exchange matrix. The simplest example is the Lyness 5-cycle, arising from the cluster algebra of type A_2: this deforms to the Lyness family of integrable symplectic maps in the plane. For types A_3 and A_4 we find suitable conditions such that the deformation produces a two-parameter family of Liouville integrable maps (in dimensions two and four, respectively). We also perform Laurentification for these maps, by lifting them to a higher-dimensional space of tau functions with a cluster algebra structure, where the Laurent property is restored. More general types of deformed mutations associated with affine Dynkin quivers are shown to correspond to four-dimensional symplectic maps arising as reductions of the discrete sine-Gordon equation.

### 2021年07月07日(水)

17:15-19:00 オンライン開催

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K-理論版特殊多項式の組み合わせ論、自由フェルミオン表示と可積分系 (Japanese)

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**岩尾慎介 氏**(東海大学)K-理論版特殊多項式の組み合わせ論、自由フェルミオン表示と可積分系 (Japanese)

[ 講演概要 ]

旗多様体にまつわる代数的組み合わせ論は、表現論・幾何学・可積分系などの分野と絡み合い、様々な解釈を持つ豊かな分野として今も研究が進んでいる。

旗多様体のコホモロジー環はある多項式環と自然に同一視され、そのなかでシューベルト多様体はシューベルト多項式という特殊多項式に対応する。コホモロジー環の部分をK理論に置き換えて同じ議論を行うと、今度はグロタンディーク多項式が得られる。以上の事実やその一般化は多くの研究者によって調べられており、シューベルト多項式・グロタンディーク多項式は多くの組み合わせ論的題材を提供している。本講演では、ソリトン方程式の理論と関係の深い自由フェルミオンを用いて、対称グロタンディーク多項式（とその双対）の新しい表示法を与える。この自由フェルミオン表示が有用であることは、組み合わせ論的観点からは説明しやすい。しかし、この結果がソリトン方程式の理論とどうつながるのかは未知であり、興味深い問題である。この講演ではこの問題点を提示し、ソリトン方程式の理論とK-理論版特殊多項式の関係性について考察したい。

旗多様体にまつわる代数的組み合わせ論は、表現論・幾何学・可積分系などの分野と絡み合い、様々な解釈を持つ豊かな分野として今も研究が進んでいる。

旗多様体のコホモロジー環はある多項式環と自然に同一視され、そのなかでシューベルト多様体はシューベルト多項式という特殊多項式に対応する。コホモロジー環の部分をK理論に置き換えて同じ議論を行うと、今度はグロタンディーク多項式が得られる。以上の事実やその一般化は多くの研究者によって調べられており、シューベルト多項式・グロタンディーク多項式は多くの組み合わせ論的題材を提供している。本講演では、ソリトン方程式の理論と関係の深い自由フェルミオンを用いて、対称グロタンディーク多項式（とその双対）の新しい表示法を与える。この自由フェルミオン表示が有用であることは、組み合わせ論的観点からは説明しやすい。しかし、この結果がソリトン方程式の理論とどうつながるのかは未知であり、興味深い問題である。この講演ではこの問題点を提示し、ソリトン方程式の理論とK-理論版特殊多項式の関係性について考察したい。

### 2021年06月30日(水)

17:15-18:45 オンライン開催

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Affine A and D cluster algebras: Dynamical systems, triangulated surfaces and friezes (English)

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**Joe PALLISTER 氏**(千葉大学)Affine A and D cluster algebras: Dynamical systems, triangulated surfaces and friezes (English)

[ 講演概要 ]

We first review the dynamical systems previously obtained for affine A and D type cluster algebras, given by the "cluster map", and the periodic quantities found for these systems. Then, by viewing the clusters as triangulations of appropriate surfaces, we show that all cluster variables either:

(i) Appear after applying the cluster map

(ii) Can be written as a determinant function of the periodic quantities.

Finally we show that the sets of cluster variables (i) and (ii) both form friezes.

We first review the dynamical systems previously obtained for affine A and D type cluster algebras, given by the "cluster map", and the periodic quantities found for these systems. Then, by viewing the clusters as triangulations of appropriate surfaces, we show that all cluster variables either:

(i) Appear after applying the cluster map

(ii) Can be written as a determinant function of the periodic quantities.

Finally we show that the sets of cluster variables (i) and (ii) both form friezes.

### 2021年06月23日(水)

18:00-19:30 オンライン開催

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Singularity confinement in delay-differential Painlevé equations (English)

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**Alexander STOKES 氏**(University College London)Singularity confinement in delay-differential Painlevé equations (English)

[ 講演概要 ]

Singularity confinement is a phenomenon first proposed as an integrability criterion for discrete systems, and has been used to great effect to obtain discrete analogues of the Painlevé differential equations. Its geometric interpretation has played a role in novel connections between discrete integrable systems and birational algebraic geometry, including Sakai's geometric framework and classification scheme for discrete Painlevé equations.

Examples of delay-differential equations, which involve shifts and derivatives with respect to a single independent variable, have been proposed as analogues of the Painlevé equations according to a number of viewpoints. Among these are observations of a kind of singularity confinement and it is natural to ask whether this could lead to the development of a geometric theory of delay-differential Painlevé equations.

In this talk we review previously proposed examples of delay-differential Painlevé equations and what is known about their singularity confinement behaviour, including some recent results establishing the existence of infinite families of confined singularities. We also propose a geometric interpretation of these results in terms of mappings between jet spaces, defining certain singularities analogous to those of interest in the singularity analysis of discrete systems, and what it means for them to be confined.

Singularity confinement is a phenomenon first proposed as an integrability criterion for discrete systems, and has been used to great effect to obtain discrete analogues of the Painlevé differential equations. Its geometric interpretation has played a role in novel connections between discrete integrable systems and birational algebraic geometry, including Sakai's geometric framework and classification scheme for discrete Painlevé equations.

Examples of delay-differential equations, which involve shifts and derivatives with respect to a single independent variable, have been proposed as analogues of the Painlevé equations according to a number of viewpoints. Among these are observations of a kind of singularity confinement and it is natural to ask whether this could lead to the development of a geometric theory of delay-differential Painlevé equations.

In this talk we review previously proposed examples of delay-differential Painlevé equations and what is known about their singularity confinement behaviour, including some recent results establishing the existence of infinite families of confined singularities. We also propose a geometric interpretation of these results in terms of mappings between jet spaces, defining certain singularities analogous to those of interest in the singularity analysis of discrete systems, and what it means for them to be confined.

### 2021年01月13日(水)

17:00-18:00 オンライン開催

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Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subalgebra of quantum superalgebras (in Japanese)

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**米山 瑛仁 氏**(東京大学大学院・総合文化研究科)Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subalgebra of quantum superalgebras (in Japanese)

[ 講演概要 ]

We study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter equation was proposed by Isaev and Kulish as a 3D analog of the reflection equation of Cherednik. As a by-product of our approach, the Bazhanov-Sergeev solution to the Zamolodchikov tetrahedron equation is characterized as the transition matrix for a particular case of type A, which clarifies an algebraic origin of it. Our work is inspired by the recent developments connecting transition matrices for quantum non-super algebras with intertwiners of irreducible representations of quantum coordinate rings. We also discuss the crystal limit of transition matrices, which gives a super analog of transition maps of Lusztig's parametrizations of the canonical basis.

https://arxiv.org/abs/2012.13385

We study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter equation was proposed by Isaev and Kulish as a 3D analog of the reflection equation of Cherednik. As a by-product of our approach, the Bazhanov-Sergeev solution to the Zamolodchikov tetrahedron equation is characterized as the transition matrix for a particular case of type A, which clarifies an algebraic origin of it. Our work is inspired by the recent developments connecting transition matrices for quantum non-super algebras with intertwiners of irreducible representations of quantum coordinate rings. We also discuss the crystal limit of transition matrices, which gives a super analog of transition maps of Lusztig's parametrizations of the canonical basis.

https://arxiv.org/abs/2012.13385

### 2020年12月09日(水)

17:00-18:30 オンライン開催

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Gap probabilities in the Laguerre unitary ensemble and discrete Painlevé equations (English)

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**Anton DZHAMAY 氏**(University of Northern Colorado)Gap probabilities in the Laguerre unitary ensemble and discrete Painlevé equations (English)

[ 講演概要 ]

We use Sakai’s geometric theory of discrete Painlevé equations to study a recurrence relation that can be used to generate ladder operators for the Laguerre unitary ensemble. Using a recently proposed identification procedure for discrete Painlevé equations we show how this recurrence can be transformed into one of the standard equations on the affine D5-algebraic surface. This is a joint work with Yang Chen and Jie Hu.

We use Sakai’s geometric theory of discrete Painlevé equations to study a recurrence relation that can be used to generate ladder operators for the Laguerre unitary ensemble. Using a recently proposed identification procedure for discrete Painlevé equations we show how this recurrence can be transformed into one of the standard equations on the affine D5-algebraic surface. This is a joint work with Yang Chen and Jie Hu.

### 2020年11月18日(水)

17:00-18:00 オンライン開催

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Recent developments on variational difference equations and their classification (English)

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**Giorgio GUBBIOTTI 氏**(The University of Sydney, School of Mathematics and Statistics)Recent developments on variational difference equations and their classification (English)

[ 講演概要 ]

We review some recent development in the theory of variational difference equations of order higher than two. In particular we present our recent solution of the inverse problem of calculus variations. Then, we present the application of such solution in the classification of variational fourth-order difference equations. To be more specific, we will present the most general form of variational additive and multiplicative fourth-order difference equations.

We review some recent development in the theory of variational difference equations of order higher than two. In particular we present our recent solution of the inverse problem of calculus variations. Then, we present the application of such solution in the classification of variational fourth-order difference equations. To be more specific, we will present the most general form of variational additive and multiplicative fourth-order difference equations.

### 2020年11月11日(水)

17:00-18:00 オンライン開催

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離散ソボレフ不等式とその応用ー周期格子からC60フラーレンまで (Japanese)

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**永井 敦 氏**(津田塾・情報科学科)離散ソボレフ不等式とその応用ー周期格子からC60フラーレンまで (Japanese)

[ 講演概要 ]

境界値問題のグリーン関数は適切なヒルベルト空間の再生核であり、その応用としてソボレフ不等式の最良定数（等号成立条件）を具体的に計算できる。次にグラフ上の離散ソボレフ不等式を導出し、その応用としてC60フラーレンとその異性体上の離散ソボレフ不等式の最良定数を計算する。

Y. Kametaka, K. Watanabe, A. Nagai, K. Takemura, H. Yamagishi, H. Sekido, The best constant of discrete Sobolev inequality on 1812 C60 fullerene isomers, JSIAM Letters 2020 Volume 12 pp. 49-52.

境界値問題のグリーン関数は適切なヒルベルト空間の再生核であり、その応用としてソボレフ不等式の最良定数（等号成立条件）を具体的に計算できる。次にグラフ上の離散ソボレフ不等式を導出し、その応用としてC60フラーレンとその異性体上の離散ソボレフ不等式の最良定数を計算する。

Y. Kametaka, K. Watanabe, A. Nagai, K. Takemura, H. Yamagishi, H. Sekido, The best constant of discrete Sobolev inequality on 1812 C60 fullerene isomers, JSIAM Letters 2020 Volume 12 pp. 49-52.

### 2020年06月24日(水)

15:00-16:30 オンライン開催

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Combinatorial and Asymptotical Results on the Neighborhood Grid Data Structure (English)

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**Martin Skrodzki 氏**(RIKEN iTHEMS)Combinatorial and Asymptotical Results on the Neighborhood Grid Data Structure (English)

[ 講演概要 ]

In 2009, Joselli et al. introduced the Neighborhood Grid data structure for fast computation of neighborhood estimates in point clouds. Even though the data structure has been used in several applications and shown to be practically relevant, it is theoretically not yet well understood. The purpose of this talk is to present a polynomial-time algorithm to build the data structure. Furthermore, we establish the presented algorithm to be time-optimal. This investigations leads to several combinatorial questions for which partial results are given.

In 2009, Joselli et al. introduced the Neighborhood Grid data structure for fast computation of neighborhood estimates in point clouds. Even though the data structure has been used in several applications and shown to be practically relevant, it is theoretically not yet well understood. The purpose of this talk is to present a polynomial-time algorithm to build the data structure. Furthermore, we establish the presented algorithm to be time-optimal. This investigations leads to several combinatorial questions for which partial results are given.

### 2020年02月17日(月)

16:30-18:30 数理科学研究科棟(駒場) 126号室

Cluster algebras, dimer models and geometric dynamics

**Sanjay Ramassamy 氏**(IPhT, CEA Saclay)Cluster algebras, dimer models and geometric dynamics

[ 講演概要 ]

Cluster algebras were introduced by Fomin and Zelevinsky at the beginning of the 21st century and have since then been related to several areas of mathematics. In this talk I will describe cluster algebras coming from quivers and give two concrete situations were they arise. The first is the bipartite dimer model coming from statistical mechanics. The second is in several dynamics on configurations of points/lines/circles/planes.

This is based on joint work with Niklas Affolter (TU Berlin), Max Glick (Google) and Pavlo Pylyavskyy (University of Minnesota).

Cluster algebras were introduced by Fomin and Zelevinsky at the beginning of the 21st century and have since then been related to several areas of mathematics. In this talk I will describe cluster algebras coming from quivers and give two concrete situations were they arise. The first is the bipartite dimer model coming from statistical mechanics. The second is in several dynamics on configurations of points/lines/circles/planes.

This is based on joint work with Niklas Affolter (TU Berlin), Max Glick (Google) and Pavlo Pylyavskyy (University of Minnesota).

### 2020年02月13日(木)

16:30-18:30 数理科学研究科棟(駒場) 126号室

Embeddings adapted to two-dimensional models of statistical mechanics (English)

**Sanjay Ramassamy 氏**(IPhT, CEA Saclay)Embeddings adapted to two-dimensional models of statistical mechanics (English)

[ 講演概要 ]

A discrete model of statistical mechanics in 2D (for example simple random walk on the infinite square grid) can be defined on a graph without specifying a particular embedding of this graph. However, when stating that such a model converges to a conformally invariant object in the scaling limit, one needs to specify an embedding of the graph. For models which possess a local move, such as a star-triangle transformation, one would like the choice of the embedding to be compatible with that local move.

In this talk I will present a candidate for an embedding adapted to the 2D dimer model (a.k.a. random perfect matchings) on bipartite graphs, that is, graphs whose faces all have an even degree. This embedding is obtained by considering centers of circle patterns with the combinatorics of the graph on which the dimer model lives.

This is based on joint works with Dmitry Chelkak (École normale supérieure), Richard Kenyon (Yale University), Wai Yeung Lam (Université du Luxembourg) and Marianna Russkikh (MIT).

A discrete model of statistical mechanics in 2D (for example simple random walk on the infinite square grid) can be defined on a graph without specifying a particular embedding of this graph. However, when stating that such a model converges to a conformally invariant object in the scaling limit, one needs to specify an embedding of the graph. For models which possess a local move, such as a star-triangle transformation, one would like the choice of the embedding to be compatible with that local move.

In this talk I will present a candidate for an embedding adapted to the 2D dimer model (a.k.a. random perfect matchings) on bipartite graphs, that is, graphs whose faces all have an even degree. This embedding is obtained by considering centers of circle patterns with the combinatorics of the graph on which the dimer model lives.

This is based on joint works with Dmitry Chelkak (École normale supérieure), Richard Kenyon (Yale University), Wai Yeung Lam (Université du Luxembourg) and Marianna Russkikh (MIT).

### 2019年11月22日(金)

17:00-18:00 数理科学研究科棟(駒場) 056号室

The Hopf algebra structure of coloured non-commutative symmetric functions

**Adam Doliwa 氏**(University of Warmia and Mazury)The Hopf algebra structure of coloured non-commutative symmetric functions

[ 講演概要 ]

The Hopf algebra of symmetric functions (Sym), especially its Schur function basis, plays an important role in the theory of KP hierarchy. The Hopf algebra of non-commutative symmetric functions (NSym) was introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon. In my talk I would like to present its "A-coloured" version NSym_A and its graded dual - the Hopf algebra QSym_A of coloured quasi-symmetric functions. It turns out that these two algebras are both non-commutative and non-cocommutative (for |A|>1), and their product and coproduct operations allow for simple combinatorial meaning. I will also show how the structure of the poset of sentences over alphabet A (A-coloured compositions) gives rise to a description of the corresponding coloured version of the ribbon Schur basis of NSym_A.

The Hopf algebra of symmetric functions (Sym), especially its Schur function basis, plays an important role in the theory of KP hierarchy. The Hopf algebra of non-commutative symmetric functions (NSym) was introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon. In my talk I would like to present its "A-coloured" version NSym_A and its graded dual - the Hopf algebra QSym_A of coloured quasi-symmetric functions. It turns out that these two algebras are both non-commutative and non-cocommutative (for |A|>1), and their product and coproduct operations allow for simple combinatorial meaning. I will also show how the structure of the poset of sentences over alphabet A (A-coloured compositions) gives rise to a description of the corresponding coloured version of the ribbon Schur basis of NSym_A.

### 2019年10月10日(木)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Universal parabolic regularization of gradient catastrophes for the Burgers-Hopf equation and singularities of the plane into plane mappings of parabolic type (English)

**Boris Konopelchenko 氏**(INFN, sezione di Lecce, Lecce, Italy)Universal parabolic regularization of gradient catastrophes for the Burgers-Hopf equation and singularities of the plane into plane mappings of parabolic type (English)

[ 講演概要 ]

Two intimately connected topics, namely, regularization of gradient catastrophes of all orders for the Burgers-Hopf equation via the Jordan chain and the singularities of the plane into plane mappings

associated with two-component hydrodynamic type systems of parabolic type are discussed.

It is shown that the regularization of all gradient catastrophes (generic and higher orders) for the Burgers-Hopf equation is achieved by the step by step embedding of the Burgers-Hopf equation into multi-component parabolic systems of quasilinear PDEs with the most degenerate Jordan blocks. Infinite parabolic Jordan chain provides us with the complete regularization. This chain contains Burgers and KdV equations as particular reductions.

It is demonstrated that the singularities of the plane into planes mappings associated with the two-component system of quasilinear PDEs of parabolic type are quite different from those in hyperbolic and elliptic cases. Impediments arising in the application of the original Whitney's approach to such case are discussed. It is shown that flex is the lowest singularity while higher singularities are given by ( k+1,k+2) curves which are of cusp type for k=2n+1, n=1,2,...,. Regularization of these singularities is discussed.

Presentation is based on two publications:

1. B. Konopelchenko and G. Ortenzi, Parabolic regularization of the gradient catastrophes for the Burgers-Hopf equation and Jordan chain, J. Phys. A: Math. Theor., 51 (2018) 275201.

2. B.G. Konopelchenko and G. Ortenzi, On the plane into plane mappings of hydrodynamic type. Parabolic case. Rev. Math. Phys.,32 (2020) 2020006. Online access. arXiv:1904.00901.

Two intimately connected topics, namely, regularization of gradient catastrophes of all orders for the Burgers-Hopf equation via the Jordan chain and the singularities of the plane into plane mappings

associated with two-component hydrodynamic type systems of parabolic type are discussed.

It is shown that the regularization of all gradient catastrophes (generic and higher orders) for the Burgers-Hopf equation is achieved by the step by step embedding of the Burgers-Hopf equation into multi-component parabolic systems of quasilinear PDEs with the most degenerate Jordan blocks. Infinite parabolic Jordan chain provides us with the complete regularization. This chain contains Burgers and KdV equations as particular reductions.

It is demonstrated that the singularities of the plane into planes mappings associated with the two-component system of quasilinear PDEs of parabolic type are quite different from those in hyperbolic and elliptic cases. Impediments arising in the application of the original Whitney's approach to such case are discussed. It is shown that flex is the lowest singularity while higher singularities are given by ( k+1,k+2) curves which are of cusp type for k=2n+1, n=1,2,...,. Regularization of these singularities is discussed.

Presentation is based on two publications:

1. B. Konopelchenko and G. Ortenzi, Parabolic regularization of the gradient catastrophes for the Burgers-Hopf equation and Jordan chain, J. Phys. A: Math. Theor., 51 (2018) 275201.

2. B.G. Konopelchenko and G. Ortenzi, On the plane into plane mappings of hydrodynamic type. Parabolic case. Rev. Math. Phys.,32 (2020) 2020006. Online access. arXiv:1904.00901.

### 2019年10月04日(金)

17:30-18:30 数理科学研究科棟(駒場) 118号室

Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations (English)

**Anton Dzhamay 氏**(University of Northern Colorado)Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations (English)

[ 講演概要 ]

Over the last decade it became clear that the role of discrete Painlevé equations in applications has been steadily growing. Thus, the question of recognizing a certain non-autonomous recurrence as a discrete Painlevé equation and understanding its position in Sakai’s classification scheme, recognizing whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such example, becomes one of the central ones. Fortunately, Sakai’s geometric theory provides an almost algorithmic procedure of answering this question.

In this work we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for orthogonal polynomials can be expressed in terms of solutions of some discrete Painlevé equation. In this work we study orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form.

This is joint work with Galina Filipuk (University of Warsaw, Poland) and Alexander Stokes (University College, London, UK)

Over the last decade it became clear that the role of discrete Painlevé equations in applications has been steadily growing. Thus, the question of recognizing a certain non-autonomous recurrence as a discrete Painlevé equation and understanding its position in Sakai’s classification scheme, recognizing whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such example, becomes one of the central ones. Fortunately, Sakai’s geometric theory provides an almost algorithmic procedure of answering this question.

In this work we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for orthogonal polynomials can be expressed in terms of solutions of some discrete Painlevé equation. In this work we study orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form.

This is joint work with Galina Filipuk (University of Warsaw, Poland) and Alexander Stokes (University College, London, UK)