## Discrete mathematical modelling seminar

Organizer(s) Tetsuji Tokihiro, Ralph Willox

Seminar information archive

### 2021/10/28

19:00-20:00   Online
This seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Andrew Hone (University of Kent)
Deformations of cluster mutations and invariant presymplectic forms
[ Abstract ]
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.

### 2021/07/07

17:15-19:00   Online
This seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Iwao Shinsuke (Tokai University)
Combinatorics of K-theoretic special polynomials -- free fermion representation and integrable systems (Japanese)

### 2021/06/30

17:15-18:45   Online
This seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Joe PALLISTER (Chiba University)
Affine A and D cluster algebras: Dynamical systems, triangulated surfaces and friezes (English)
[ Abstract ]
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.

### 2021/06/23

18:00-19:30   Online
This seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Alexander STOKES (University College London)
Singularity confinement in delay-differential Painlevé equations (English)
[ Abstract ]
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.

### 2021/01/13

17:00-18:00   Online
This seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Akihito Yoneyama (Institute of Physics, Graduate School of Arts and Sciences, the University of Tokyo)
Tetrahedron and 3D reflection equation from PBW bases of the nilpotent subalgebra of quantum superalgebras (in Japanese)
[ Abstract ]
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

### 2020/12/09

17:00-18:30   Online
This seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Anton DZHAMAY (University of Northern Colorado)
Gap probabilities in the Laguerre unitary ensemble and discrete Painlevé equations (English)
[ Abstract ]
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.

### 2020/11/18

17:00-18:00   Online
This seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Giorgio GUBBIOTTI (The University of Sydney, School of Mathematics and Statistics)
Recent developments on variational difference equations and their classification (English)
[ Abstract ]
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.

### 2020/11/11

17:00-18:00   Online
The seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Atsushi NAGAI (Tsuda University, Department of Computer Science)
Discrete Sobolev inequalities and their applications -- from periodic lattices to 1812 C60 fullerene isomers (Japanese)

### 2020/06/24

15:00-16:30   Online
The seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Martin Skrodzki (RIKEN iTHEMS)
Combinatorial and Asymptotical Results on the Neighborhood Grid Data Structure (English)
[ Abstract ]
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.

### 2020/02/17

16:30-18:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Sanjay Ramassamy (IPhT, CEA Saclay)
Cluster algebras, dimer models and geometric dynamics
[ Abstract ]
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).

### 2020/02/13

16:30-18:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Sanjay Ramassamy (IPhT, CEA Saclay)
Embeddings adapted to two-dimensional models of statistical mechanics (English)
[ Abstract ]
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).

### 2019/11/22

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Adam Doliwa (University of Warmia and Mazury)
The Hopf algebra structure of coloured non-commutative symmetric functions
[ Abstract ]
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.

### 2019/10/10

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Boris Konopelchenko (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)
[ Abstract ]
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.

### 2019/10/04

17:30-18:30   Room #118 (Graduate School of Math. Sci. Bldg.)
Anton Dzhamay (University of Northern Colorado)
Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations (English)
[ Abstract ]
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)

### 2019/04/22

17:15-18:30   Room #118 (Graduate School of Math. Sci. Bldg.)
Yuri Suris (Technische Universität Berlin)
Geometry of the Kahan-Hirota-Kimura discretization
[ Abstract ]
We will report on some novel results concerning the bilinear discretization of quadratic vector fields.

### 2018/11/20

15:00-16:30   Room #002 (Graduate School of Math. Sci. Bldg.)

### 2018/11/19

17:15-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Dinh T. Tran (School of Mathematics and Statistics, The University of Sydney)
Integrability for four-dimensional recurrence relations
[ Abstract ]
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.

### 2018/06/25

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Anton Dzhamay (University of Northern Colorado)
Gap Probabilities and discrete Painlevé equations
[ Abstract ]
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)

### 2018/01/17

17:00-18:45   Room #056 (Graduate School of Math. Sci. Bldg.)
Samuel Colin (CBPF, Rio de Janeiro, Brasil) 17:00-17:50
Quantum matter bounce with a dark energy expanding phase (ENGLISH)
[ Abstract ]
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 (Aix Marseille Université, Centrale Marseille, Institut Fresnel) 17:50-18:40
Mass of the vacuum: a Newtonian perspective (ENGLISH)
[ Abstract ]
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.

### 2017/11/01

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Basile Grammaticos (Université de Paris VII・XI)
Discrete Painlevé equations associated with the E8 group (ENGLISH)
[ Abstract ]
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.

### 2017/10/31

16:30-17:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Basile Grammaticos (Université de Paris VII・XI)
The end of the World (ENGLISH)
[ Abstract ]
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.

### 2017/10/31

15:30-16:30   Room #126 (Graduate School of Math. Sci. Bldg.)
Fon-Che Liu (National Taiwan University)
A hierarchy of approximate regularity of functions (ENGLISH)
[ Abstract ]
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.

### 2017/04/26

15:30-17:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yohei Tutiya (Kanagawa Institute of Technology)
[Recent topics in nonlocal classical integrable systems] (JAPANESE)

### 2017/02/09

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Dinh Tran (University of New South Wales, Sydney, Australia)
Growth of degrees of lattice equations and its signatures over finite fields (ENGLISH)
[ Abstract ]
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.

### 2016/12/19

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Anton Dzhamay (University of Northern Colorado)
Discrete Painlevé equations on the affine A3 surface (ENGLISH)
[ Abstract ]
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.