## 離散数理モデリングセミナー

過去の記録 ～01/28｜次回の予定｜今後の予定 01/29～

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**

### 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)

### 2019年04月22日(月)

17:15-18:30 数理科学研究科棟(駒場) 118号室

Geometry of the Kahan-Hirota-Kimura discretization

**Yuri Suris 氏**(Technische Universität Berlin)Geometry of the Kahan-Hirota-Kimura discretization

[ 講演概要 ]

We will report on some novel results concerning the bilinear discretization of quadratic vector fields.

We will report on some novel results concerning the bilinear discretization of quadratic vector fields.

### 2018年11月20日(火)

15:00-16:30 数理科学研究科棟(駒場) 002号室

情報幾何とその応用ー深層学習の解明に向けて

**甘利俊一 氏**(理化学研究所)情報幾何とその応用ー深層学習の解明に向けて

[ 講演概要 ]

情報幾何の基本的考えを説明した後で、深層学習にどう関係するのか、統計神経力学の立場から話をする。

情報幾何の基本的考えを説明した後で、深層学習にどう関係するのか、統計神経力学の立場から話をする。

### 2018年11月19日(月)

17:15-18:30 数理科学研究科棟(駒場) 056号室

Integrability for four-dimensional recurrence relations

**Dinh T. Tran 氏**(School of Mathematics and Statistics, The University of Sydney)Integrability for four-dimensional recurrence relations

[ 講演概要 ]

In this talk, we study some fourth-order recurrence relations (or mappings). These recurrence relations were obtained by assuming that they possess two polynomial integrals with certain degree patterns.

For mappings with cubic growth, we will reduce them to three-dimensional ones using a procedure called deflation. These three-dimensional maps have two integrals; therefore, they are integrable in the sense of Liouville-Arnold. Furthermore, we can reduce the obtained three-dimensional maps to two-dimensional maps of Quispel-Roberts-Thompsons (QRT) type. On the other hand, for recurrences with quadratic growth and two integrals, we will show that they are integrable in the sense of Liouville-Arnold by providing their Poisson brackets. Non-degenerate Poisson brackets can be found by using the existence of discrete Lagrangians and a discrete analogue of the Ostrogradsky transformation.

This is joint work with G. Gubbiotti, N. Joshi, and C-M. Viallet.

In this talk, we study some fourth-order recurrence relations (or mappings). These recurrence relations were obtained by assuming that they possess two polynomial integrals with certain degree patterns.

For mappings with cubic growth, we will reduce them to three-dimensional ones using a procedure called deflation. These three-dimensional maps have two integrals; therefore, they are integrable in the sense of Liouville-Arnold. Furthermore, we can reduce the obtained three-dimensional maps to two-dimensional maps of Quispel-Roberts-Thompsons (QRT) type. On the other hand, for recurrences with quadratic growth and two integrals, we will show that they are integrable in the sense of Liouville-Arnold by providing their Poisson brackets. Non-degenerate Poisson brackets can be found by using the existence of discrete Lagrangians and a discrete analogue of the Ostrogradsky transformation.

This is joint work with G. Gubbiotti, N. Joshi, and C-M. Viallet.

### 2018年06月25日(月)

17:30-18:30 数理科学研究科棟(駒場) 056号室

Gap Probabilities and discrete Painlevé equations

**Anton Dzhamay 氏**(University of Northern Colorado)Gap Probabilities and discrete Painlevé equations

[ 講演概要 ]

It is well-known that important statistical quantities, such as gap probabilities, in various discrete probabilistic models of random matrix type satisfy the so-called discrete Painlevé equations, which provides an effective way to computing them. In this talk we discuss this correspondence for a particular class of models, known as boxed plane partitions (equivalently, lozenge tilings of a hexagon). For uniform probability distribution, this is one of the most studied models of random surfaces. Borodin, Gorin, and Rains showed that it is possible to assign a very general elliptic weight to the distribution, with various degenerations of this weight corresponding to the degeneration cascade of discrete polynomial ensembles, such as Racah and Hahn ensembles and their q-analogues. This also correspond to the degeneration scheme of discrete Painlevé equations, due to Sakai. In this talk we consider the q-Hahn and q-Racah ensembles and corresponding discrete Painlevé equations of types q-P(A_{2}^{(1)}) and q-P(A_{1}^{(1)}).

This is joint work with Alisa Knizel (Columbia University)

It is well-known that important statistical quantities, such as gap probabilities, in various discrete probabilistic models of random matrix type satisfy the so-called discrete Painlevé equations, which provides an effective way to computing them. In this talk we discuss this correspondence for a particular class of models, known as boxed plane partitions (equivalently, lozenge tilings of a hexagon). For uniform probability distribution, this is one of the most studied models of random surfaces. Borodin, Gorin, and Rains showed that it is possible to assign a very general elliptic weight to the distribution, with various degenerations of this weight corresponding to the degeneration cascade of discrete polynomial ensembles, such as Racah and Hahn ensembles and their q-analogues. This also correspond to the degeneration scheme of discrete Painlevé equations, due to Sakai. In this talk we consider the q-Hahn and q-Racah ensembles and corresponding discrete Painlevé equations of types q-P(A_{2}^{(1)}) and q-P(A_{1}^{(1)}).

This is joint work with Alisa Knizel (Columbia University)

### 2018年01月17日(水)

17:00-18:45 数理科学研究科棟(駒場) 056号室

Quantum matter bounce with a dark energy expanding phase (ENGLISH)

Mass of the vacuum: a Newtonian perspective (ENGLISH)

**Samuel Colin 氏**(CBPF, Rio de Janeiro, Brasil) 17:00-17:50Quantum matter bounce with a dark energy expanding phase (ENGLISH)

[ 講演概要 ]

The ``matter bounce'' is an alternative scenario to inflationary cosmology, according to which the universe undergoes a contraction, followed by an expansion, the bounce occurring when the quantum effects become important. In my talk, I will show that such a scenario can be unambiguously analyzed in the de Broglie-Bohm pilot-wave interpretation of quantum mechanics. More specifically, I will apply the pilot-wave theory to a Wheeler-DeWitt equation obtained from the quantization of a simple classical mini-superspace model, and show that there are numerical solutions describing bouncing universes with many desirable physical features. For example, one solution contains a dark energy phase during the expansion, without the need to postulate the existence of a cosmological constant in the classical action.

This work was done in collaboration with Nelson Pinto-Neto (CBPF, Rio de Janeiro, Brasil). Further details available at https://arxiv.org/abs/1706.03037.

The ``matter bounce'' is an alternative scenario to inflationary cosmology, according to which the universe undergoes a contraction, followed by an expansion, the bounce occurring when the quantum effects become important. In my talk, I will show that such a scenario can be unambiguously analyzed in the de Broglie-Bohm pilot-wave interpretation of quantum mechanics. More specifically, I will apply the pilot-wave theory to a Wheeler-DeWitt equation obtained from the quantization of a simple classical mini-superspace model, and show that there are numerical solutions describing bouncing universes with many desirable physical features. For example, one solution contains a dark energy phase during the expansion, without the need to postulate the existence of a cosmological constant in the classical action.

This work was done in collaboration with Nelson Pinto-Neto (CBPF, Rio de Janeiro, Brasil). Further details available at https://arxiv.org/abs/1706.03037.

**Thomas Durt 氏**(Aix Marseille Université, Centrale Marseille, Institut Fresnel) 17:50-18:40Mass of the vacuum: a Newtonian perspective (ENGLISH)

[ 講演概要 ]

One could believe that special relativity forces us to totally renounce to the idea of an aether, but the aether reappears in general relativity which teaches us that space-time is structured by the local metrics. It also reappears in quantum field theory which teaches us that even at zero temperature space is filled by the quantum vacuum energy. Finally, aether reappears in modern astronomy where it was necessary to introduce ill-defined concepts such as dark matter and dark energy in order to explain apparent deviations from Newtonian dynamics (at the level of galactic rotation curves).

Newton dynamics being the unique limit of general relativistic dynamics in the classical regime, dark matter and dark energy can be seen as an ultimate, last chance strategy, aimed at reconciling the predictions of general relativity with astronomical data.

In our talk we shall describe a simple model, derived in the framework of Newtonian dynamics, aimed at explaining puzzling astronomical observations realized at the level of the solar system (Pioneer anomaly) and at the galactic scale (rotation curves), without adopting ad hoc hypotheses about the existence of dark matter and/or dark energy.

The basic idea is that Newtonian gravity is modified due to the presence of a (negative) density, everywhere in space, of mass-energy.

One could believe that special relativity forces us to totally renounce to the idea of an aether, but the aether reappears in general relativity which teaches us that space-time is structured by the local metrics. It also reappears in quantum field theory which teaches us that even at zero temperature space is filled by the quantum vacuum energy. Finally, aether reappears in modern astronomy where it was necessary to introduce ill-defined concepts such as dark matter and dark energy in order to explain apparent deviations from Newtonian dynamics (at the level of galactic rotation curves).

Newton dynamics being the unique limit of general relativistic dynamics in the classical regime, dark matter and dark energy can be seen as an ultimate, last chance strategy, aimed at reconciling the predictions of general relativity with astronomical data.

In our talk we shall describe a simple model, derived in the framework of Newtonian dynamics, aimed at explaining puzzling astronomical observations realized at the level of the solar system (Pioneer anomaly) and at the galactic scale (rotation curves), without adopting ad hoc hypotheses about the existence of dark matter and/or dark energy.

The basic idea is that Newtonian gravity is modified due to the presence of a (negative) density, everywhere in space, of mass-energy.

### 2017年11月01日(水)

17:00-18:00 数理科学研究科棟(駒場) 056号室

Discrete Painlevé equations associated with the E8 group (ENGLISH)

**Basile Grammaticos 氏**(Université de Paris VII・XI)Discrete Painlevé equations associated with the E8 group (ENGLISH)

[ 講演概要 ]

I'll present a summary of the results of the Paris-Tokyo-Pondicherry group on equations associated with the affine Weyl group E8. I shall review the various parametrisations of the E8-related equations, introducing the trihomographic representation and the ancillary variable. Several examples of E8-associated equations will be given including what we believe is the simplest form for the generic elliptic discrete Painlevé equation.

I'll present a summary of the results of the Paris-Tokyo-Pondicherry group on equations associated with the affine Weyl group E8. I shall review the various parametrisations of the E8-related equations, introducing the trihomographic representation and the ancillary variable. Several examples of E8-associated equations will be given including what we believe is the simplest form for the generic elliptic discrete Painlevé equation.

### 2017年10月31日(火)

16:30-17:30 数理科学研究科棟(駒場) 126号室

The end of the World (ENGLISH)

**Basile Grammaticos 氏**(Université de Paris VII・XI)The end of the World (ENGLISH)

[ 講演概要 ]

This is not a seminar on astrophysics or cosmology. I am not going to talk about something that will happen in billions of years. I will rather explain the menace to our civilisation and to the human species. Inspired from the works of several authors I will explain the existing risks. I will also present mathematical models which show that a general collapse is possible in the decades that follow.

This is not a seminar on astrophysics or cosmology. I am not going to talk about something that will happen in billions of years. I will rather explain the menace to our civilisation and to the human species. Inspired from the works of several authors I will explain the existing risks. I will also present mathematical models which show that a general collapse is possible in the decades that follow.