早稲田可積分系セミナー
Waseda Integrable Systems Seminar
日時：２０２5年1月24日（金）午後5時 － 午後6時
場所：早稲田大学西早稲田キャンパス63号館04-22
講演者： Dr.
Amin Chabchoub （OIST）
概要: The
formation of wave localizations in nonlinear dispersive media can be described
by weakly nonlinear evolution equations such as the nonlinear Schrödinger
equation (NLSE). Within the class of exact NLSE breather solutions on a finite
background, the hierarchy of rational solutions, localized in both time and
space, are considered to provide appropriate prototypes to model extreme wave
dynamics on the water surface, plasma and electromagnetic Kerr media. The talk
will focus on the applicability and limitations of hydrodynamic NLSE and Mankov solitons and breathers in unidirectional and
directional wave systems. The time-reversal invariance and the effect of
dissipation on the evolution of such wave packets will be also elaborated upon.
Moreover, the critical role of breathers in wave engineering applications will
be highlighted.
日時：２０２４年７月５日（金）午後３時５分
－ 午後４時４５分
場所：早稲田大学西早稲田キャンパス53号館103教室
講演者： Prof.
Yuji Kodama （児玉裕治） （Ohio State University）
概要: I
will classify all the regular soliton solutions of the (good) Boussinesq equation. The classification is based on the
result on the 3-reduction of the KP hierarchy. In particular, we show that the
soliton solution can include at most one resonant solution in addition to two
sets of solitons propagating in opposite directions. I will then discuss a
possible application of the result to bidirectional model of soliton gas. I
will also give a classification of the KP solitons under the general
Gelfand-Dickey reductions, and provide a vertex
operator construction of the soliton solutions.
日時：２０２３年１２月１８日（月）午後３時３０分
− 午後５時２０分
場所：早稲田大学西早稲田キャンパス６２W号館 １階 大会議室B（西側）
講演１：午後３時３０分 −
午後４時２０分
題目：Connection between the symmetric discrete AKP system
and bilinear ABS lattice equations
講演者： Jing Wang (上海大学, 早稲田大学)
概要： In this talk,
we show that all the bilinear Adler-Bobenko-Suris (ABS) equations (except Q2 and Q4) can be
obtained from symmetric discrete AKP system by taking proper reductions
and continuum limits. Among the bilinear ABS equations, a simpler bilinear
form of the ABS H2 equation is given. As by-products, a
8-point 3-dimensional lattice equation and a 8-point 4-dimensional lattice
equation are given. Both of them can be considered as extensions of the
symmetric discrete AKP equations.
講演２：午後４時３０分 −
午後５時２０分
題目： Quantum dark solitons in a 1D Bose
gas
講演者： 金城佳世 (埼玉大学)
概要： In a weakly
interacting ultracold Bose gas, the mean-field approach leads to the the nonlinear Schroedinger (NLS)
equation. For repulsive interaction, the NLS equation has a dark soliton
solution, which is a density dip traveling with constant velocity. The Lieb-Liniger (LL) model describes a gas of one-dimensional Bose
particles interacting via a delta function potential under periodic boundary
conditions. In the LL model, the density profile of certain quantum states
agrees well with that of the NLS soliton solutions under periodic boundary
conditions; these are called quantum dark soliton states.
Using exact computational methods, we have successfully constructed quantum
dark soliton states which have several exotic features: a quantum dark soliton
state that is topologically wound and a quantum dark soliton which has two
notches.
日時：２０２３年１１月１３日（月）午後４時３０分
− 午後６時１０分
場所：早稲田大学西早稲田キャンパス６２W号館 １階 大会議室A（東側）
題目：Modulation instability and excitation of periodic waves
講演者：Prof. Nail Akhmediev (Australian National University)
概要： Modulation
instability is a fundamental physical effect that is in the roots of many
natural phenomena that include nonlinear water waves. Despite the relative
simplicity of its mathematical treatment, there are still some unresolved
issues that require further analysis. The confusing part of MI is the
exponential growth of modulation and the transformation of small periodic
modulation into large amplitude periodic waves.
When many unstable frequencies are involved, the MI is transformed into chaotic
waves. A few examples of such transitions will be considered in this talk.
日時：２０２０年１月１７日（金）午後４時３０分
− 午後６時
場所：早稲田大学西早稲田キャンパス６３号館０４−０８号室
題目：On Geometric Curve Flows and
Solitons
講演者：Dr. Hsiao-Fan Liu (Tamkang
University)
概要： Geometric curves
flows are curve evolutions whose invariants flow according to some soliton
equations. Such correspondences provide a systematic tool to study geometric
curve flows via soliton theory, and vice versa.
In this talk, we discuss certain geometric curve flows and explain
how we build relations between curves and soliton equations, how we use
the soliton theory to derive B\"acklund
transformations for these curve flows, and to study the existence of solutions
to the (periodic) Cauchy problems of curve flows. This also provides geometric
algorithms to solve periodic Cauchy problems numerically.
日時：２０１９年１２月１３日（金）午後４時
− 午後６時
場所：早稲田大学西早稲田キャンパス６３号館０４−０８号室
題目：Discrete Painlevé
Equations and Orthogonal Polynomials
講演者：Prof. Anton Dzhamay (University of Northern Colorado)
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).
日時：２０１９年１２月１１日（水）午後２時４５分
− 午後６時
場所：早稲田大学西早稲田キャンパス５１号館１７−０６号室
題目： Geometry of Discrete Painlevé Equations
講演者：Prof. Anton Dzhamay (University of Northern Colorado)
概要： In this talk we give
an introduction to some geometric ideas and tools used to study discrete
integrable systems. Our main goal is to give an introduction to Sakai’s
geometric theory of discrete Painlevé equations.
However, we first consider an autonomous example of a dynamical system known as
the QRT map. For this example we explain the geometry behind indeterminate
(or base)points of birational maps, as well as how to
fix such indeterminacies by changing the geometry of the configuration space
using the so-called blowup procedure. In the process we also introduce the
notions of the Picard lattice of the algebraic surface that is the
configuration space of the dynamics, the anti-canonical divisor class, and the
linearization of the mapping on the level of the Picard lattice. After
that we consider an idea of geometric deautonomzation.
Using this approach we introduce discrete Painlevé equations as deautonomizations
of QRT maps. We show how such deautonomization
results in the decomposition of the Picard lattice into complementary pairs of
the surface and symmetry sub-lattices and explain the construction of a
binational representation of affine Weyl symmetry groups that gives a complete
algebraic description of our non-linear dynamic. We show how to represent a
discrete Painlevé equation as a composition of
elementary birational transformations (Cremona isometries). We conclude the
tutorial by a brief introduction into Sakai’s classification scheme for
discrete Painlevé equations.This talk is based on joint work with
Stefan Carstea (Bucharest) and Tomoyuki Takenawa (Tokyo).
日時：２０１９年６月４日（火） 午後４時３０分 - 午後６時
場所：早稲田大学西早稲田キャンパス５１号館１７−０４号室
題目： Rational space curves and
solitons for the Gelfand-Dickey reductions of the KP hierarchy.
講演者：Prof. Yuji Kodama (Ohio State University)
概要： It is well known that the algebro-geometric
solutions of the KdV hierarchy are
constructed
from the Riemann theta functions associated with the hyperelliptic curves,
and
that the soliton solutions can be obtained by rational (singular) limits of the
hyperelliptic curves.
In
this talk, I will discuss certain class of KP solitons in the connections with
space curves,
which
are labeled by certain types of numerical semigroups. In particular, I will
show that
the
(singular and complex) KP solitons of the Gelfand-Dickey reduction
($l$-reduction)
are
associated with the rational space curves of $<l,lm+1,\ldots, lm+k>$ where $m\ge 1$ and
$1\le
k\le l-1$. This is a part of the PhD project of my student, Yuancheng
Xie.
日時：２０１９年３月７日（木） 午後４時
− 午後５時３０分
場所：早稲田大学西早稲田キャンパス５２号館１０３号室
題目： On the inverse spectral
transform for the conservative Camassa-Holm flow
講演者：Prof. Jonathan Eckhardt (Loughborough University)
概要： The Camassa-Holm
equation is a nonlinear partial differential equation that models
unidirectional wave propagation on shallow water. I will show how this equation
can be integrated by means of the inverse spectral transform method. The global
conservative solutions obtained in this way form into a train of solitons
(peakons) in the long-time limit.
日時：２０１８年１１月１９日（月） 午後４時３０分
− 午後６時
場所：早稲田大学西早稲田キャンパス６３号館０４―２２
題目： Detecting and determining
preserved measures and integrals of rational maps
講演者：Prof. Reinout Quispel (La Trobe University)
概要： The search for preserved measures and integrals of ordinary
differential equations has been at the forefront of mathematical physics since
the time of Galileo and Newton. In this talk our aim will be to develop an
analogous theory for the (arguably more general) discrete-time case. This will
lead to linear algorithms for detecting and determining preserved measures and
integrals of rational maps.
日時：２０１８年７月１４日（土） 午後３時３０分
− 午後４時３０分
場所：早稲田大学西早稲田キャンパス６２W号館一階大会議室
題目：GBDT version of Backlund-Darboux transformation and evolution of Weyl functions
講演者：Prof. Alexander Sakhnovich，Universität Wien
概要： We
consider applications of GBDT to dynamical systems and integrable nonlinear
equations including nonlocal NLS equation and second harmonic generation
equation. The initial-boundary problem for second harmonic generation equation
will be discussed as well.
（本セミナーは講演会を兼ねています）
日時：２０１８年６月２３日（土） 午後２時 − 午後３時３０分
場所：早稲田大学西早稲田キャンパス５２号館１０２号室
題目：On the rational solutions and the solitons of the KP hierarchy
講演者：Prof. Yuji Kodama （児玉裕治），Ohio State University
概要： It is well known that the Schur polynomials satisfy the Hirota bilinear equations of the KP hierarchy, and that each Schur polynomial can be parametrized by a unique Young diagram. We also know that the KP solitons (exponential solutions) can be parametrized by certain decomposition of the Grassmannians. In the talk, I will explain the connection between the rational solutions and the KP solitons in terms of the Young diagrams. More explicitly, I will show how one gets a rational solution from a KP soliton. I will also discuss a connection between quasi-periodic solutions (theta or sigma functions) and the KP solitons.
（本セミナーは講演会を兼ねています）
日時：２０１８年５月２８日月曜日 午後４時３０分 − 午後６時
場所：早稲田大学西早稲田キャンパス６２W号館１階大会議室
題目：The nonlinear Schroedinger equation and variety of it's nontrivial extensions
講演者：Prof. Nail Akhmediev，Australian National University
概要： The NLSE represents a dynamical system with an infinite number of degrees of freedom and as such it has an infinite number of solutions that includes solitons, breathers, rogue waves, radiation waves and their combinations. Every extension of the NLSE expands dramatically variety of it's solutions. Both conservative and dissipative extensions will be considered in this talk.
（本セミナーは講演会を兼ねています）
日時：２０１８年５月２１日月曜日 午後４時３０分 − 午後６時
場所：早稲田大学西早稲田キャンパス６２W号館１階大会議室
題目: Mach Reflection of a Solitary Wave: Experiments
講演者：Prof. Harry Yeh，Oregon State University
概要： Laboratory and numerical experiments are presented for Mach reflection of an obliquely incident solitary wave at a vertical wall. The numerical model is based on the pseudo-spectral method for the full Euler formulation. With the aid of a laser sheet in the laboratory, the wave profiles are measured optically in sub-millimeter precision. Discrepancies reported in previous works are now substantially improved, partly because of the higher-order KP theory and in part because of the advancement in computational power and laboratory instrumentation. While the theory predicts the maximum of four-fold (4.0) amplification of the Mach stem, the maximum observed in the laboratory was 2.922 (the previous laboratory study had achieved the amplification of 2.4), while our numerical simulation reached the maximum of 3.91 (previously reported amplification was 2.897). Also presented are other laboratory realizations of soliton-soliton.
（本セミナーは講演会を兼ねています）
日時：２０１７年１２月２７日水曜日 午後１時３０分から
場所：早稲田大学西早稲田キャンパス６２W号館大会議室
題目：直交多項式と古典・量子ランダムウォーク
講演者：三木 啓司 （気象大学校)
概要：直交多項式は可積分系・可解な数理モデルとの対応が非常によく知られている。本セミナーでは，1次元の出生死滅過程やスピン鎖モデルにおいて直交多項式がどういう役割を果たしているかについて述べ，これらの高次元拡張の結果を紹介する。
日時：２０１７年１１月６日月曜日 午後２時４５分 − 午後４時１５分
場所：早稲田大学西早稲田キャンパス６２W号館１階大会議室
題目： Interaction solutions between lumps and solitons via symbolic computations
講演者：Prof. Wen-Xiu Ma，University of South Florida
概要： We will talk about interaction solutions between lump solutions and soliton solutions to integrable equations. A computational algorithm will be discussed, based on the bilinear formulation; and illustrative examples in the cases of the (2+1)-dimensional KP and Ito equations will be presented through Maple symbolic computations.
（本セミナーは講演会を兼ねています）
日時：２０１７年７月１１日火曜日 午後５時１５分 − 午後６時４５分
場所：早稲田大学西早稲田キャンパス６２W号館大会議室
題目：On nonlocal nonlinear Schrodinger equation and its discrete version
講演者：Prof. Zuo-Nong Zhu, Shanghai Jiao Tong University, P. R. China
概要： Very recently, Ablowitz and Musslimani introduced reverse space, reverse time, and reverse space-time nonlocal nonlinear integrable equations including the reverse space nonlocal NLS equation, the real and complex reverse space-time nonlocal mKdV, sine-Gordon, Davey-Stewartson equations, et.al. In this talk, we will show that, under the gauge transformations, the nonlocal focusing NLS (and it discrete version) and the nonlocal defocusing NLS (and it discrete version) are, respectively, gauge equivalent to the coupled Heisenberg equation (and it discrete version) and the coupled modified Heisenberg equation (and it discrete version). We will discuss the construction of discrete soliton solutions for the discrete nonlocal focusing NLS. We will demonstrate that the discrete soliton yields soliton of nonlocal focusing NLS under the continuous limit. The relations of these solutions between nonlocal NLS and classical NLS will be given. This is a joint work with Dr. Li-yuan Ma.
（本セミナーは講演会を兼ねています）
日時：２０１７年６月１３日火曜日 午後４時３０分 − 午後６時
場所：早稲田大学西早稲田キャンパス５１号館１７−０８
題目：半古典的波束のハミルトン力学
講演者：Prof. Tomoki Ohsawa （大澤知己），(University of Texas at Dallas, USA)
概要： 古典力学と量子力学の基礎方程式はどちらも、適切なシンプレクティック構造を用いて、
ハミルトン力学系として記述されることはよく知られている。この講演では、そういった幾何学的構造を用いて、
量子力学と古典力学の境界である半古典領域の力学をハミルトン力学系として記述する。
特にHagedornによる半古典的波束の力学を対象とする。
日時：２０１７年６月１０日土曜日 午後１時 − 午後４時
場所：早稲田大学西早稲田キャンパス５４号館１０１
題目：Hypergeometric functions and integrable hydrodynamic systems
講演者：Prof. Yuji Kodama （児玉裕治）, Ohio State University, USA
概要：I will show an interesting
connection of (generalized) hypergeometric
functions with integrable hydrodynamic-type systems. The lecture contains the
following subjects.
(a) Integrable hydrodynamic systems generated by Lauricella functions.
(b) Confluence of the Lauricella functions and non-diagonalizable
hydrodynamic-type systems.
（本セミナーは講演会を兼ねています）
日時：２０１７年６月６日火曜日 午後３時 − 午後６時
場所：早稲田大学西早稲田キャンパス５１号館１７-０８
題目：KP solitons
講演者：Prof. Yuji Kodama （児玉裕治）, Ohio State University, USA
概要： I will discuss some
combinatorial aspects of the KP solitons. This lecture is to explain the
following subjects.
(a) Mathematical background of the regular soliton solutions (the totally
non-negative Grassmannians and their parametrization).
(b) Applications of the KP solitons to shallow water waves (Mach reflection and
rogue waves).
（本セミナーは講演会を兼ねています）
日時：２０１７年５月１７日水曜日 午後３時 − 午後６時
場所：早稲田大学西早稲田キャンパス６２W号館１階大会議室
題目：Riemann-Hilbert Methods in Integrable Systems II
Lecture 3: Asymptotic Analysis of Riemann-Hilbert Problems, Part I
Lecture 4: Asymptotic Analysis of Riemann-Hilbert Problems, Part II
講演者：Prof. Peter Miller, University of Michigan, USA
Lecture 3, Asymptotic Analysis of Riemann-Hilbert Problems, Part I:
The Deift-Zhou steepest descent method is a powerful set of techniques applicable to Riemann-Hilbert problems that are generalizations of the classical steepest descent method for the asymptotic expansion of certain contour integrals. This lecture will focus on the Fokas-Its'-Kitaev Riemann-Hilbert problem characterizing orthogonal polynomials with exponentially varying weights and the asymptotic limit of large degree as an example of the steepest descent method. The goal of this lecture is to deform the Riemann-Hilbert problem to the point where it appears at a formal level to be asymptotically simple.
Lecture 4, Asymptotic Analysis of Riemann-Hilbert Problems, Part II:
This lecture picks up where Lecture 3 left off. The deformed Riemann-Hilbert problem suggests an approximate solution, known as a parametrix. The parametrix will be constructed explicitly with the help of elementary and special functions. Then by comparing the parametrix to the exact solution we will arrive at a Riemann-Hilbert problem of small-norm type (cf., Lecture 2). Estimates on the solution of the latter problem yield explicit leading-order asymptotic formulae for the orthogonal polynomials and related quantities of interest in applications such as random matrix theory.
（本セミナーは講演会を兼ねています）
日時：２０１７年５月１６日火曜日 午後３時 − 午後６時
場所：早稲田大学西早稲田キャンパス６２W号館１階大会議室
題目：Riemann-Hilbert Methods in Integrable Systems II
Lecture 3: Asymptotic Analysis of Riemann-Hilbert Problems, Part I
Lecture 4: Asymptotic Analysis of Riemann-Hilbert Problems, Part II
講演者：Prof. Peter Miller, University of Michigan, USA
Lecture 3, Asymptotic Analysis of Riemann-Hilbert Problems, Part I:
The Deift-Zhou steepest descent method is a powerful set of techniques applicable to Riemann-Hilbert problems that are generalizations of the classical steepest descent method for the asymptotic expansion of certain contour integrals. This lecture will focus on the Fokas-Its'-Kitaev Riemann-Hilbert problem characterizing orthogonal polynomials with exponentially varying weights and the asymptotic limit of large degree as an example of the steepest descent method. The goal of this lecture is to deform the Riemann-Hilbert problem to the point where it appears at a formal level to be asymptotically simple.
Lecture 4, Asymptotic Analysis of Riemann-Hilbert Problems, Part II:
This lecture picks up where Lecture 3 left off. The deformed Riemann-Hilbert problem suggests an approximate solution, known as a parametrix. The parametrix will be constructed explicitly with the help of elementary and special functions. Then by comparing the parametrix to the exact solution we will arrive at a Riemann-Hilbert problem of small-norm type (cf., Lecture 2). Estimates on the solution of the latter problem yield explicit leading-order asymptotic formulae for the orthogonal polynomials and related quantities of interest in applications such as random matrix theory.
（本セミナーは講演会を兼ねています）
日時：２０１６年１２月２７日火曜日 午後１時００分 −
場所：早稲田大学西早稲田キャンパス６２W号館１階大会議室
題目：可積分なコマの方程式の差分化の現状
講演者：木村欣司（京都大学）
概要：オイラー、ラグランジュ、コワレフスカヤの場合については、コマの方程式は可積分であることが分かっており、そのうち、オイラー、ラグランジュの場合については、その離散アナログとして、保存量ならびに楕円関数解を持つものが知られている。しかし、それらの離散アナログに対するLax対は、不明であった。今回、連続時間のオイラーのコマを拡張したものからPainleve-6方程式を導出できるという事実を、差分方程式の世界で実現することを試みた。その結果、楕円差分Painleve方程式のパラメータが少ない場合の方程式を得られることがわかった。そのような計算結果を考察することで、離散オイラーのコマのLax対を構成できた。離散オイラーのコマのLax対からの類推として、より一般化された高階の離散オイラーのコマを手に入れることができる。しかし、その可積分性は、わかっていない。一方で、連続時間では、すでに可積分と証明されている高階のオイラーのコマが存在する。その離散アナログについての予想も存在する。こちらの方程式についても、その可積分性は、わかっていない。以上のように、この分野は、未整理な状態にある。講演では、この分野の研究の現状について、詳細に報告する。
日時：２０１５年３月１６日月曜日 午後３時００分 − 午後５時００分
場所：早稲田大学西早稲田キャンパス６２W号館１階大会議室
題目：From Schlesinger Transformations to Difference Painlevé Equations
講演者：Prof. Anton Dzhamay, School of Mathematical Sciences, University of Northern Colorado, USA
概要：The goal of the talk is to explain some recent joint work with T.Takenawa (Tokyo University of Marine Science and Technology) on reduction
from elementary Schlesinger transformations, including higher-rank transformations, to difference Painlevé equations. This is a continuation of an earlier project with
T.Takenawa and H. Sakai (the University of Tokyo). In the first part of the talk we will consider the general formalism of discrete evolution equations that describe
elementary Schlesinger transformations and see how this equations can be written in a discrete Hamiltonian form. In the second part we consider some examples of
reductions from these discrete evolutionary Schlesinger equations to difference Painlevé equations. We will particularly emphasize the role played by the geometry
of the Okamoto space of initial conditions in understanding the structure of these reductions.
（本セミナーは早稲田大学理工学術院講演会を兼ねています）
日時：２０１４年7月25日金曜日午後２時４５分 – 午後６時
場所：早稲田大学西早稲田キャンパス６３号館１階数学応数会議室
講演１：午後２時４５分 – 午後４時１５分
題目： Beach waves and line-solitons of the KP equation
講演者： Sarbarish Chakravarty, Department of Mathematics, The University of Colorado at Colorado Springs, USA
Nonlinear interactions among small amplitude, long wavelength, obliquely propagating waves on the surface of shallow water often generate web-like patterns.
In this talk, we discuss how line-soliton solutions of the Kadomtsev-Petviashvili (KP) equation can approximate such web-pattern
in shallow water wave. We describe an ``inverse problem" which maps a certain set of measurable data from the solitary waves in the given
pattern to the parameters required to construct an exact KP soliton that describes the non-stationary dynamics of the pattern.
We illustrate the inverse problem using explicit examples of shallow water wave pattern.
講演２：午後４時３０分 – 午後６時
題目： Triangulations of convex polygon and solitons in 2-dimension
講演者： Yuji Kodama, Department of Mathematics, Ohio State University, USA
We give an explicit connection between the patterns (soliton graphs) generated by soliton solutions (also obtained by hyperplane arrangements) and triangulations (subdivisions)
of polygons inscribed in conic curves. Two dimensional integrable systems admitting those soliton solutions include the KP equation (for parabola),
two-dimensional Toda lattice (for hyperbola) and the Davey-Stewartson systems (for ellipse).
（本セミナーは講演会を兼ねています）
日時：２０１４年６月２日月曜日午後２時４５分 – 午後４時１５分
場所：早稲田大学西早稲田キャンパス５１号館１７−０６
題目： q-Painleve 方程式の特殊解とリーマン・ヒルベルト問題
講演者： 大山陽介（大阪大学大学院情報科学研究科情報基礎数学専攻）
アブストラクト
q-Painleve 方程式は、q-線型方程式の接続係数保存変形として得られる。 q-Painleve 方程式の特殊解の中で、接続係数を決定可能な例を探るとともに、接続係数に関するバーコフのリーマン・ヒルベルト問題について解説する。
セミナー世話人：丸野健一
（可積分系、数理物理の周辺の話題についてのセミナーを不定期に開催します。話題を提供して下さる方は大歓迎です。）