## 離散数理モデリングセミナー

担当者 時弘哲治, ウィロックス ラルフ

### 2021年07月07日(水)

17:15-19:00   オンライン開催
Zoomを用いてオンラインで行います．参加希望の方はウィロックスまでZoomのリンクをお尋ねください．

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 ]

### 2021年06月30日(水)

17:15-18:45   オンライン開催
Zoomを用いてオンラインで行います．参加希望の方はウィロックスまでZoomのリンクをお尋ねください．
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.

### 2021年06月23日(水)

18:00-19:30   オンライン開催
Zoomを用いてオンラインで行います．参加希望の方はウィロックスまでZoomのリンクをお尋ねください．
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.

### 2021年01月13日(水)

17:00-18:00   オンライン開催
Zoomを用いてオンラインで行います．参加希望の方はウィロックスまでZoomのリンクをお尋ねください．

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

### 2020年12月09日(水)

17:00-18:30   オンライン開催
Zoomを用いてオンラインで行います．参加希望の方はウィロックスまでZoomのリンクをお尋ねください．
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.

### 2020年11月18日(水)

17:00-18:00   オンライン開催
Zoomを用いてオンラインで行います．参加希望の方はウィロックスまでZoomのリンクをお尋ねください．
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.

### 2020年11月11日(水)

17:00-18:00   オンライン開催
Zoomを用いてオンラインで行います．参加希望の方はウィロックスまでZoomのリンクをお尋ねください．

[ 講演概要 ]

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
Zoom を用いてオンラインで行います．参加希望の方はウィロックスまでZoomのリンクをお尋ねください．
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.

### 2020年02月17日(月)

16:30-18:30   数理科学研究科棟(駒場) 126号室
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).

### 2020年02月13日(木)

16:30-18:30   数理科学研究科棟(駒場) 126号室
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).

### 2019年11月22日(金)

17:00-18:00   数理科学研究科棟(駒場) 056号室
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.

### 2019年10月10日(木)

17:00-18:00   数理科学研究科棟(駒場) 056号室
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.

### 2019年10月04日(金)

17:30-18:30   数理科学研究科棟(駒場) 118号室
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)

### 2019年04月22日(月)

17:15-18:30   数理科学研究科棟(駒場) 118号室
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.

### 2018年11月20日(火)

15:00-16:30   数理科学研究科棟(駒場) 002号室

[ 講演概要 ]

### 2018年11月19日(月)

17:15-18:30   数理科学研究科棟(駒場) 056号室
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.

### 2018年06月25日(月)

17:30-18:30   数理科学研究科棟(駒場) 056号室
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)

### 2018年01月17日(水)

17:00-18:45   数理科学研究科棟(駒場) 056号室
Samuel Colin 氏 (CBPF, Rio de Janeiro, Brasil) 17:00-17:50
Quantum 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.
Thomas Durt 氏 (Aix Marseille Université, Centrale Marseille, Institut Fresnel) 17:50-18:40
Mass 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.

### 2017年11月01日(水)

17:00-18:00   数理科学研究科棟(駒場) 056号室
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.

### 2017年10月31日(火)

16:30-17:30   数理科学研究科棟(駒場) 126号室
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.

### 2017年10月31日(火)

15:30-16:30   数理科学研究科棟(駒場) 126号室
Fon-Che Liu 氏 (National Taiwan University)
A hierarchy of approximate regularity of functions (ENGLISH)
[ 講演概要 ]
A hierarchy of a certain weakly sensed regularity of functions defined on subsets of Euclidean n-space which originated from the well-known Lusin theorem that characterizes measurable functions in terms of approximate continuity will be introduced. Its intimate relations with the ordinary hierarchy of regularity in terms of order of continuous differentiability will be exposed and explained.

### 2017年04月26日(水)

15:30-17:00   数理科学研究科棟(駒場) 056号室

nonlocalな古典可積分系に関する最近の話題 (JAPANESE)
[ 講演概要 ]
ここでいうnonlocalな可積分系とは、離散系のことではなく、特異積分変換項を持った微分方程式の可積分系をさします。そのような方程式に関しては、1.nonlocalな量子可積分系との関連について、3.テータ関数解について、3.佐藤理論から見た対称性についてといった話があります。最近の話題ということで、一応2.を中心にお話しようと思っています。技術的にはリーマン面上に特殊な第三種アーベル積分を構成する話で、地味に見えるかもしれませんが、きっと応用がある話だと思っています。

### 2017年02月09日(木)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Dinh Tran 氏 (University of New South Wales, Sydney, Australia)
Growth of degrees of lattice equations and its signatures over finite fields (ENGLISH)
[ 講演概要 ]
We study growth of degrees of autonomous and non-autonomous lattice equations, some of which are known to be integrable. We present a conjecture that helps us to prove polynomial growth of a certain class of equations including $Q_V$ and its non-autonomous generalization. In addition, we also study growth of degrees of several non-integrable equations. Exponential growth of degrees of these equations is also proved subject to a conjecture. Our technique is to determine the ambient degree growth of the equations and a conjectured growth of their common factors at each vertex, allowing the true degree growth to be found. Moreover, our results can also be used for mappings obtained as periodic reductions of integrable lattice equations. We also study signatures of growth of degrees of lattice equations over finite fields.

### 2016年12月19日(月)

17:30-18:30   数理科学研究科棟(駒場) 056号室
Anton Dzhamay 氏 (University of Northern Colorado)
Discrete Painlevé equations on the affine A3 surface (ENGLISH)
[ 講演概要 ]
We explain how to construct the birational representation of the extended affine Weyl symmetry group D5 and consider examples of discrete Painlevé equations that correspond to certain translation elements in this group. One of the examples is the famous q-PV equation of Jimbo-Sakai. Some other examples are conjugated to it via explicit change of variables and we explain how representing translation elements as words in the group allows us to see the corresponding change of coordinates explicitly. We also show a new example of a discrete Painlevé equation that is elementary (short translation), but at the same time is different from the q-PVI equation.

### 2016年12月17日(土)

10:00-18:00   数理科学研究科棟(駒場) 056号室
Anton Dzhamay 氏 (University of Northern Colorado) 10:00-10:50
Factorization of Rational Mappings and Geometric Deautonomization (ENGLISH)
[ 講演概要 ]
This talk is the first of two talks describing the joint project with Tomoyuki Takenawa and Stefan Carstea on geometric deautonomization.
The goal of this project is to develop a systematic approach for deautonomizing discrete integrable mappings, such as the QRT mappings, to non-automonous mappings in the discrete Painlevé family, based on the action of the mapping on the Picard lattice of the surface and a choice of an elliptic fiber. In this talk we will explain the main ideas behind this approach and describe the technique that allows us to recover explicit formulas defining the mapping from the known action on the divisor group (the factorization technique). We illustrate our approach by reconstructing the famous example of the q-PVI equation of Jimbo-Sakai from a simple QRT mapping.
Tomoyuki Takenawa 氏 (Tokyo University of Marine Science and Technology) 11:00-11:50
From the QRT maps to elliptic difference Painlevé equations (ENGLISH)
[ 講演概要 ]
This talk is the second part of the joint project with Anton Dzhamay and Stefan Carstea on geometric deautonomization and focuses on the elliptic case and the special symmetry groups. It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated.
In this talk we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. Especially, in the case where the fiber is smooth elliptic, imposing certain restrictions on such non autonomous mappings, we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.
Hiroshi Kawakami 氏 (Aoyama Gakuin University) 13:30-14:20
The complete degeneration scheme of four-dimensional Painlevé-type equations (ENGLISH)
[ 講演概要 ]
In the joint work with H. Sakai and A. Nakamura, we constructed the degeneration scheme of four-dimensional Painlevé-type equations associated with unramified linear equations. In this talk I present the "complete" degeneration scheme of the four-dimensional Painlevé-type equations, which is constructed by means of the degeneration of HTL forms of associated linear equations.
Akane Nakamura 氏 (Josai University) 14:30-15:20
Degeneration of the Painlevé divisors (ENGLISH)
[ 講演概要 ]
There are three types of curves associated with 4-dimensional algebraically completely integrable systems, namely the spectral curve, the Painlevé divisors, and the separation curve. I am going to explain these three curves of genus two taking examples derived from the isospectral limit of the 4-dimensional Painlevé-type equations and study the Namikawa-Ueno type degeneration.
Teruhisa Tsuda 氏 (Hitotsubashi University) 16:00-16:50
Rational approximation and Schlesinger transformation (ENGLISH)
[ 講演概要 ]
We show how rational approximation problems for functions are related to the construction of Schlesinger transformations. Also we discuss their applications to the theory of isomonodromic deformations or Painlevé equations. This talk is based on a joint work with Toshiyuki Mano.
Takafumi Mase 氏 (the University of Tokyo) 17:00-17:50
Spaces of initial conditions for nonautonomous mappings of the plane (ENGLISH)
[ 講演概要 ]
Spaces of initial conditions are one of the most important and powerful tools to analyze mappings of the plane. In this talk, we study the basic properties of general nonautonomous equations that have spaces of initial conditions. We will consider the minimization of spaces of initial conditions for nonautonomous systems and we shall discuss a classification of nonautonomous integrable mappings of the plane with a space of initial conditions.