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2018RIMS 共同研究（公開型）

幾何構造と微分方程式― 対称性と特異点の視点から ―Symmetry and Singularityof Geometric Structures and Diﬀerential Equations

ProgramandAbstracts

18 – 21 December 2018
Ritsumeikan University
Biwako-Kusatsu Campus
Westwing 6F Colloquium room
Organizers:
Daisuke Tarama (多羅間 大輔, Ritsumeikan University)
Kenro Furutani (古谷 賢朗, Tokyo University of Science)
Hiroaki Yoshimura (吉村 浩明, Waseda University)
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Program Table
December 18 (Tuesday)
13:00 – 13:10
13:10 – 14:00
14:10 – 15:00
15:00 – 15:20
15:20 – 16:10
16:20 – 17:10

Speaker
Opening
01 Jean-Pierre Françoise
02 Kazuyuki Yagasaki
tea time
03 Wolfram Bauer
04 Tohru Morimoto

December 19 (Wednesday)
09:30 – 10:20
10:20 – 10:40
10:40 – 11:30
11:40 – 12:30
12:30 – 14:00
14:00 – 14:50
14:50 – 15:10
15:10 – 16:00
16:10 – 17:00
17:05 – 17:40

05 Elmar Schrohe
tea time
06 Sonja Hohloch
07 Wei Wanglunch time
08 Nguyen Tien Zung
tea time
09 Masanori Adachi
10 Le Bich Phuong
11 Abdellah Laaroussi

December 20 (Thursday)
09:30 – 10:20
10:20 – 10:40
10:40 – 11:30
11:40 – 12:30
12:30 – 14:00
14:00 – 14:50
14:50 – 15:10
15:10 – 15:50
15:55 – 16:35
18:00 –

12 Irina Markina
tea time
13 Chisato Iwasaki
14 Dmitry V. Zenkov
lunch time
15 Yoshio Uwano
tea time
16 Shhogo Yamanaka
17 Shouya Motonaga
Banquet @ Lake Biwa Otsu Prince Hotel

December 21 (Friday)
09:30 – 10:20
10:30 – 11:20
11:20 – 11:40
11:40 – 12:30
12:40 – 13:30
13:30 – 13:40

18 Tudor S. Ratiu
19 Toshihiro Iwaitea time
20 Linyu Peng
21 Hiroaki Yoshimura
Closing2(1) Program

Tuesday, 18/December/2018
13:00 – 13:10 Opening
13:10 – 14:00 Jean-Pierre Françoise
Integrable Systems: The Clebsch top and its associated Kummer Surface
14:10 – 15:00 Kazuyuki Yagasaki (矢ヶ崎 一幸)
Nonintegrability and chaos in Hamiltonian systems with saddle-centers
15:00 – 15:20 Tea Time
15:20 – 16:10 Wolfram Bauer
The fundamental solution of a class of ultra-hyperbolic operators on pseudo-H-typegroups
16:20 – 17:10 Tohru Morimoto (森本 徹)
Extrinsic geometries and diﬀerential equations

Wednesday, 19/December/2018
09:30 – 10:20 Elmar Schrohe
Elliptic operators associated with groups of quantized canonical transformations
10:20 – 10:40 Tea Time
10:40 – 11:30 Sonja Hohloch
Dynamics around focus-focus singularities in semitoric systems
11:40 – 12:30 Wei Wang
The tangential k-Cauchy-Fueter complexes and Hartogs’ phenomenon over the rightquaternionic Heisenberg group12:30 – 14:00 Lunch Time
14:00 – 14:50 Nguyen Tien Zung
On a universal conservation law in dynamics
14:50 – 15:10 Tea Time
15:10 – 16:00 Masanori Adachi (足立 真訓)
Sobolev estimates for the complex Green operator on Levi-flat manifolds
16:10 – 16:50 Le Bich Phuong
Convergence of stochastic gradient flows
15:55 – 16:35 Abdellah Laaroussi
Weyl asymptotics of a class of elliptic and hypoelliptic operators: methods and problems

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Thursday, 20/December/2018
09:30 – 10:20 Irina Markina
Extremal curves on Stiefel and Grassmann manifolds
10:20 – 10:40 Tea Time
10:40 – 11:30 CHisato Iwasaki (岩崎 千里)
On the heat kernel for forms on the Heisenberg group
11:40 – 12:30 Dmitry V. Zenkov
Hamel’s Formalism for Infinite-Dimensional Mechanical Systems
12:30 – 14:00 Lunch Time
14:00 – 14:50 Uwano Yoshio (上野 嘉夫)
Symmetry and reduction for the α-geodesics on quantum statistical manifolds
14:50 – 15:10 Tea Time
15:10 – 15:50 Sogo Yamanaka (山中 祥五)
Nonintegrability of three-degree-of-freedom Birkhoﬀ normal forms of resonance degreetwo
15:55 –16:35 Shoya Motonaga (本永 翔也)
Nonpersistence of periodic orbits, homoclinic orbits, first integrals, and commutative
vector fields in perturbed systems
18:00 –

Banquet at Lake Biwa Prince Hotel Otsu

Friday, 21/December/2018
09:30–10:20 Tudor S. Ratiu
Diﬀerential character valued momentum maps
10:30 – 11:20 Toshihiro Iwai (岩井 敏洋)
The 2D Kramers-Dirac oscillator
11:20 – 11:40 Tea Time
11:40 – 12:30 Linyu Peng
Symmetries of semi-discrete variational problems and Noether’s theorems
12:40 – 13:30 Hiroaki Yoshimura (吉村 浩明)
A variational formulation, Dirac structures and dynamical systems for nonequilibrium
thermodynamics
13:30 – 13:40 Closing

4(2) Abstracts

Title: Integrable Systems: The Clebsch top and its associated Kummer Surface
Jean-Pierre Françoise
Sorbonne Université, France
Abstract:
This is a joint article with Daisuke Tarama. It contains a comprehensive presentation ofa classical article of Weber (1878) about the Clebsch case of the motion of a rigid bodyin an ideal fluid. The article is revisited in the light of several more recent contributionsof Aomoto (1981), Enolski-Fedorov (2016) and Magri-Skrypyk (2016). We emphasize inparticular the existence of an invariant Kummer surface and the algebraic linearization on
the Jacobian of an hyperelliptic curve of genus two. We further discuss the existence ofinvariant subspaces on which the motion is equivalent to the free rigid body. Finally, we
introduce an explicit computation of the action-angle coordinates.
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Title: Nonintegrability and chaos in Hamiltonian systems with saddle-centers
Kazuyuki Yagasaki
Kyoto University, Japan
Abstract:
Nonintegrability and chaos are fundamental and central topics not only in Hamiltoniansystems but also in dynamical systems, as seen in the famous research of Poincare. It isclear that if it exhibits chaos, then a dynamical system is nonintegrable. However, theconverse of this statement may not necessarily hold. In this talk, we consider Hamiltoniansystems with saddle-centers and discuss this problem. Our main tools are Melnikov-type
methods for existence of chaos and the Morales-Ramis theory for nonintegrability. Joint
work with Shogo Yamanaka is included.
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Title: The fundamental solution of a class of ultra-hyperbolic operators on
pseudo-H-type groups
Wolfram Bauer
Leibniz Universität Hannover, Germany
Abstract:
Pseudo-H-type Lie groups Gr,s of signature (r, s) are defined via a module action of theCliﬀord algebra Cℓr,s on a vector space V ∼= R2n . They form a subclass of all 2-stepnilpotent Lie groups and based on their algebraic structure they can be equipped with aleft-invariant pseudo-Riemannian metric. Let Nr,s denote the Lie algebra corresponding toGr,s . A choice of left-invariant vector fields [X1 , · · · , X2n ] which generate a complement of
the center of Nr,s gives rise to a second order operator
)
) ( 2
(
2
+ · · · + X2n,∆r,s := X12 + · · · + Xn2 − Xn+1
which we call ultra-hyperbolic. In terms of classical special functions we present families
of fundamental solutions of ∆r,s in the case r = 0, s > 0 and study their properties. In
the case of r > 0 the operator ∆r,s admits no fundamental solution in the space tempered
distributions. This is a joint work with I. Markina (University of Bergen) and A. Froehly(formerly Leibniz Universität Hannover).
References[1] W. Bauer, K. Furutani, and C. Iwasaki, Spectral zeta function on pseudo H-typenilmanifolds, Indian J. Pure Appl. Math. 46 (4) (2015), 539-582.
[2] P. Ciatti, Scalar products on Cliﬀord modules and pseudo-H-type Lie algebras, Ann.
Mat. Pura Appl. 178 (4), 132 (2000).
[3] K. Furutani, I. Markina, Complete classification of pseudo- H-type Lie algebras: I,
Geom. Dedicata 190 (2017) 23-51.
[4] D. Müller, and F. Ricci, Analysis of second order diﬀerential operators on Heisenberg
groups I, Invent. Math. 101 (1990), 545-582.
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Title: Extrinsic geometries and diﬀerential equations
Tohru Morimoto
Seki Kowa Institute for Mathematics and Institut Mathématiques de Kiyoshi Oka, Japan
Abstract:
The distinction between intrinsic and extrinsic geometry is fundamental in geometry. The
former treats spaces and the latter figures. A space may be defined to be a set S equippedwith a geometric structure σ on S. A figure may be defined to be a morphism φ : X → A
of spaces, where the target space A is usually assumed to be a homogeneous space L/L0 of
a group L factored by a subgroup L0 .
Two spaces X and Y are said to be (intrinsically) equivalent if there exists an isomorphism
f : X → Y . Two figures φ : X → L/L0 and ψ : Y → L/L0 are said to be (extrinsically)equivalent if there exist an isomorphism f : X → Y and a ∈ L such that Λa ◦ φ = ψ ◦ f ,where Λa denotes the left translation by a.
Especially in smooth or analytic category one of the central problems is to find the complete
invariants of a space X or a figure φ : X → L/L0 in order to solve the so called equivalence
problems, that is, to decide whether two spaces or two figures are (locally) equivalent ornot.
For intrinsic geometry we have general theories originated by Cartan and developed mainly
by Kuranishi, Spencer, Singer- Sternberg, Tanaka, and Morimoto (cf [2]).
In this talk I will give a unified scheme to treat the equivalence problem for extrinsic
geometry, following [1]. In particular, I will show a general method to construct the complete
invariants of a morphem
φ : (M, f) → L/L0 ⊂ Flag(V, ϕ)from a filtered manifold (M, f) to a flag variety Flag(V, ϕ) of type ϕ, in which we prove thatthe invariants live in the cohomology spac H+1 (g− , l/ĝ) associated to the symbol of φ.
I will also discuss close relations among the geometries intrinsic and extrinsic, and the differential equations linear and non-linear, in particular, the functorial isomorphism between
the category of extrinsic geometries in flag varieties and the category of involutive systemsof linear diﬀerential equations, and various results and problems which arise from thesepoints of view.
References[1] B. Doubrov, Y. Machida, and T. Morimoto, Extrinsic geometries and linear diﬀerential equations, in preparation.
[2] T. Morimoto, Lie algebras, geometric structures and diﬀerential equations on filteredmanifolds, Advanced Studies in Pure Mathematics 37, 2002, 205–252.
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Title: Elliptic operators associated with groups of quantized canonical transformations
Elmar Schrohe
Leivniz Universität Hannover, Germany
Abstract:
Given a closed manifold M and a discrete group G of quantized canonical transformations
Φg , g ∈ G, we are developing an index theory for the algebra of operators of the form
∑
A=
Dg Φg
g

on L2 (M ), where the sum is finite and the Dg are zero order pseudodiﬀerential operators.
This framework encompasses a variety of situations studied earlier, such as the AtiyahWeinstein index problem or the index theory for operators with shifts.
As a first step we introduced in [1] suitable notions of symbols and ellipticity and showed
the Fredholm propertyof elliptic elements. Here, also the more general case of operators∫of the form I + G Dg Φg dµG (g) for finite-dimensional Lie groups was considered.
As a first step towards an index formula, we focus on the case of finite extensions of abeliangroups. We introduce the localized algebraic index of the complete symbol of an elliptic
operator. With the help of a calculus of semiclassical quantized canonical transformations,a version of Egorov’s theorem and a theorem on trace asymptotics for semiclassical Fourier
integral operators we show that the localized analytic index and the localized algebraic
index coincide. As a corollary, we express the Fredholm index in terms of the algebraic
index. In forthcoming work we will try to apply methods of algebraic index theory in orderto obtain an index formula.
Joint work with Anton Savin and Boris Sternin.
References[1] A. Savin, E. Schrohe, B. Sternin, Elliptic operators associated with groups of quantized canonical transformations, Bull. Sci. Math., to appear.
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Title: Dynamics around focus-focus singularities in semitoric systems
Sonja Hohloch
University of Antwerp, Belgium
Abstract:
A semitoric integrable Hamiltonian system, briefly a semitoric system, is given by two autonomous Hamiltonian systems on a 4-dimensional manifold whose flows Poisson-commuteand induce an (S1 × R)-action that has only nondegenerate, nonhyperbolic singularities.
Semitoric systems have been symplectically classified a few years ago by Pelayo & Vu Ngoc
by means of five invariants.
Three of these five invariants are the so-called Taylor series invariant, the height invariant,and the twisting index. Roughly, the first one describes the behaviour of the system near
the focus-focus singular fibre, the second one the position of the focus-focus value, and thethird one compares the ‘rotation’ within regular fibres close to one focus-focus singularity
to that close to another focus-focus singularity.
Recently there has been made considerable progress in understanding and computing theseinvariants and, in this talk, we present the (results of the) finished and ongoing projects
with J. Alonso (Antwerp), H. Dullin (Sydney), and J. Palmer (Rutgers):• Taylor series and twisting index for coupled spin oscillators.
• Taylor series, height invariant, and twisting index for coupled angular momenta.
• Putting the Taylor series and twisting index in relation with wellknown notions fromclassical dynamical systems like rotation number and rotation vector etc.
• Change of the Taylor series and twisting index when varying the parameters of thesesystems.
References:• Symplectic classification of coupled angular momenta (with J. Alonso and H. Dullin), 59p.,
arXiv:1808.05849.
• Taylor series and twisting-index invariants of coupled spin-oscillators (with J. Alonso and
H. Dullin), to appear in Journal of Geometry and Physics.
• A family of compact semitoric systems with two focus-focus singularities (with J. Palmer),
Journal of Geometric Mechanics 2018, 10(3): 331–357.
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Title: The tangential k-Cauchy-Fueter complexes and Hartogs’ phenomenonover the right quaternionic Heisenberg group
Wei Wang
Zhejiang University, China
Abstract:
I will discuss the construction of the tangential k-Cauchy-Fueter complexes on the rightquaternionic Heisenberg group, as the quaternionic counterpart of ∂¯b complex on the Heisenberg group in the theory of several complex variables. We can use the L2 estimate to solve
the nonhomogeneous tangential k-Cauchy-Fueter equation under the compatibility condition over this group modulo a lattice. This solution has an important vanishing property
when the group is higher dimensional. It allows us to prove the Hartogs’ extension phenomenon for k-CF functions, which are the quaternionic counterpart of CR functions.
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Title: On a universal conservation law in dynamics
Nguyen Tien Zung
University of Toulouse, France
Abstract:
In this talk I want to explain the following conservation law, its manifestations and applications, especially in the local case near a singular point of a dynamical system: anythingwhich is conserved by a dynamical system is also conserved by its associated torus actions.
Part of the talk is based on a joint work with Kai Jiang and Tudor Ratiu on simultaneous
normalization of dynamical systems with a singular underlying geometric structure.
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Title: Sobolev estimates for the complex Green operator on Levi-flat manifolds
Masanori Adachi
Shizuoka University, Japan
Abstract:
Levi-flat manifolds are odd-dimensional real manifolds that has a foliation of real codimension one whose leaves admit smoothly varying complex structures. Although the leafwise
Cauchy-Riemann operator is not hypoelliptic, it enjoys a Sobolev estimate of finite orderdue to Ohsawa-Sibony (2000). In this talk, I would like to explain this Sobolev estimate
in detail for a special but interesting case, the unit circle bundle of hyperbolic Riemannsurface endowed with unstable foliation of the geodesic flow.
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Title: Convergence of stochastic gradient flows
Le Bich Phuong
Hanoi University of Mining and Geology, Vietnam
Abstract:
In machine learning one often uses a stochastic gradient flow with respect to a loss function
on the space of parameters of a neural net to find an optimal prediction model. In this talk,which is based on a work in progress under the supervision of Prof. Nguyen Tien Zung, I
want to discuss, via simple toy models, two mathematical problems aﬀecting this generaldiﬀerential learning method, namely:- In general, the stochastic gradient flow does not converge to the global minimal point of
the loss in general (due to the noise), and only to a stochastic equilibrium near a global
minimal point. The sharper the loss function, the higher the noise, and the further away
the stochastic equilibrium will be from the minimal point.
- Also, in general, the minimal point of the loss function is not the optimal solution for theartificial intelligence problem, due to the bias created by ”data imbalance”. If the data is
too much imbalanced then the minimal point of the loss function is actually a very badsolution.

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Title:Weyl asymptotics of a class of elliptic and hypoelliptic operators: methods
and problems
Abdellah Laaroussi
Leibniz University Hannover, Germany
Abstract:
We discuss the asymptotics of the eigenvalue counting function of elliptic and hypoelliptic
operators. In the elliptic case, there are diﬀerent methods to obtain such asymptotics, suchas method of the hyperbolic operator, asymptotics of the heat trace. In the hypoelliptic
case, we focus on the sub-Laplacian on 2-step Nilmanifolds and give some results and open
problems.
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Title: Extremal curves on Stiefel and Grassmann manifolds
Irina Markina
University of Bergen, Norway
Abstract:
In the talk we describe various left-invariant sub-Riemannian systems on Lie groups that
admit explicit solutions with certain properties, and provides geometric origins for a classof important curves on Stiefel manifolds, called quasi-geodesics, that project on Grassmann
manifolds as Riemannian geodesics. We show that quasi-geodesics are the projections of
sub-Riemannian geodesics generated by certain left-invariant distributions on Lie groups
that act transitively on each Stiefel manifold. This is join work with V. Jurdjevic, Universityof Toronto, Canada, and F. Silva Leite, University of Coimbra, Portugal.
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Title: On the heat kernel for forms on the Heisenberg group
Chisato Iwasaki
University of Hyogo, Japan
Abstract:
TBA

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Title: Hamel’s Formalism for Infinite-Dimensional Mechanical Systems
Dmitry V. Zenkov
North Carolina State University, USA
Abstract:
Separation of the position and velocity measurements in mechanics originated in Euler’s
work on the rigid body and fluid dynamics. Hamel extended this formalism from therigid body setting to arbitrary finite-dimensional Lagrangian mechanical systems. This
formalism proved to be useful in the mechanics and numerical simulations of multibody
systems. This talk will introduce Hamel’s formalism for infinite-dimensional mechanicalsystems, with an emphasis on the dynamics of constrained systems with symmetry.
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Title: Symmetry and reduction for the α-geodesics on quantum statistical manifolds
Yoshio Uwano
Kyoto Pharmaceutical University, Japan
Abstract:
This talk aims to report on the ‘matrix averaged Hebbian learning equations’ (MAHLE)
whose trajectories realize the exponential-type (e-) geodesics on the quantum statisticalmanifolds. The Hamiltonian system describing the MAHLE is shown to be in quadrature.
The symplectic reduction technique works well to show the quadrature. This may be thefirst attempt to develop Hamiltonian mechanical study on the cotangent-bundle QSM a .
The quantum statistical manifold, denoted by Qn , is the space of n × n positive-definite
Hermitean matrices with unit trace. On Qn , the MAHLE takes the form
(0.1)

ρ̇ = (ρC + Cρ) − 2Tr (Cρ)ρ (ρ ∈ Qn ),where C is a constant Hermitean matrix b . The e-geodesics on Qn are the α-geodesics with
α = 1 which are the autoparallel curves with respect to the parallel translation,
1(0.2)
X0 ∈ Tρ0 Qn 7→ (ρ1 Lρ0 (X0 ) + Lρ0 (X0 )ρ1 ) − Tr (ρ1 Lρ0 (X0 ))ρ1 ∈ Tρ1 Qn ,2where both of the tangent spaces, Tρj Qn (j = 1, 2), are identified with the space of traceless
Hermitean matrices and Lρ0 (X0 ) stands for the SLD (symmetric logarithmic derivertive)of X0 determined uniquely by X0 = (ρ0 Lρ0 (X0 ) + Lρ0 (X0 )ρ0 )/2.
Proposition 1 (U). The e-geodesic ρ(t) subject to ρ(0) = ρ0 and ρ̇(0) = X0 is the trajectoryof the MAHLE with C = Lρ0 (X0 )/2.
Consider the tangent bundle T ∗ Hn+ of Hn , the space of positive-definite Hermitean matrices,which can be identified with Hn+ × Hn where Hn denotes the space of Hermitean matrices.
On endowing the canonical symplectic form, say Ω, with T ∗ Hn+ , we can complete thesymplectic reduction of (T ∗ Hn+ , Ω, K # ) with K # (r, P ) = Tr (CrP + P Cr) to (T ∗ Qn , ω, K)
by the R-action, (r, P ) ∈ T ∗ Hn+ 7→ (es r, e−s P ) ∈ T ∗ Hn+ (s ∈ R), where T ∗ Qn is the
cotangent bundle of Qn , ω the canonical symplectic form on that.
Proposition 2 (U). The Hamiltonian system (T ∗ Hn+ , Ω, K # ) is in quadrature.
Proposition 3 (U). The Hamiltonian system (T ∗ Qn , ω, K) desscribes the MAHLE.
Referrences• Y.Nakamura, Japan J. Indust. Appl. Math., 11, 11-20 (1994).
• Y.Uwano, Mathematical Modelling and Geometry, 4, 19-33 (2016).
• Y.Uwano, in preparationaOn concerning classical statistical manifolds (CSM), a pioneering work by Nakamura (1994) is made on

the tangent bundle of a CSM.
bThe condition to C has been made weaker than that in Uwano (2016). Equation (0.1) also describes avariant of Moser’s form of the finite nonperiodic Toda lattice extended on Qn .
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Title: Nonintegrability of three-degree-of-freedom Birkhoﬀ normal forms ofresonance degree two
Shogo Yamanaka
Kyoto University, Japan
Abstract:
Birkhoﬀ normal forms are special normal forms of Hamiltonian systems near equilibria andthe problem of determining whether they are non-integrable has attracted much attention
in the field of dynamical systems. We show that three-degree-of-freedom Birkhoﬀ normal
forms are non-integrable when they are in 1 : 2 : ω resonance for ω = 1, 2, or 4. Recently,
O. Christov [1] also studied non-integrability of the same normal forms although his proofs
contained some errors.
References[1] O. Christov, Non-integrability of first order resonances in Hamiltonian systems inthree degrees of freedom, Celest. Mech. Dyn. Astr., 112 (2012), 149-167.
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Title: Nonpersistence of periodic orbits, homoclinic orbits, first integrals, andcom- mutative vector fields in perturbed systems
Shoya Motonaga
Kyoto University, Japan
Abstract:
Determination of whether periodic orbits, homoclinic orbits, first integrals or commutative
vector fields may persist under perturbations is one of the most important problems in thefield of dynamical systems. In this talk, we give several theorems on necessary conditions
for their persistence in general perturbed systems. Moreover, we consider periodic perturbations of one-degree-of-freedom Hamiltonian systems and describe some relationships
between our results and the standard Melnikov method for periodic orbits and homoclinic
orbits.

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Title :Diﬀerential character valued momentum maps
Tudor S. Ratiu
Shanghai Jiao Tong University, China
Abstract:
In this talk I will introduce an extended version of the momentum map with values in
diﬀerential character groups. This momentum map is able to capture topological invariants
that are conserved on the flow of symmetric Hamiltonians. These momentum maps arebuilt on the model of Poisson Lie groups and have important applications when working in
infinite dimensions, namely when the group is that of diﬀeomorphisms preserving certain
geometric structures or the automorphism group and the gauge group of a bundle. Noneof the groups for which this momentum map exists, have a Poisson Lie structure to myknowledge.
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The 2D Kramers-Dirac oscillator
Toshihiro Iwai
Kyoto University, Japan
Abstract:
The Dirac oscillator was initially introduced as a Dirac operator which is linear in momentum and position variables, and has been developed with interest in integrable Diracequations. This talk starts with a review of free Dirac equations with boundary condition
and then proceeds to a review of the Dirac oscillators with interest in band rearrangement
against a parameter. After that, the 2D Kramers-Dirac oscillator is introduced. In contrast
to the usual 2D Dirac oscillator, the 2D Kramers-Dirac oscillator admits the time-reversal
symmetry, which is a reason for the present nomenclature. It is shown that there exists aneigenstate associated with an eigenvalue linear in the control parameter, which goes downfrom positive values to negative values as the parameter varies in the positive direction.
The other eigenvalues are broken up into two bands, positive and negative. In conclusion,a relativistic mechanical view of the 2D Kramers-Dirac oscillator is touched upon.
This talk is based on joint works with B. Zhilinskii at Université du Littoral Côte d Opale.
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Title: Symmetries of semi-discrete variational problems and Noether’s theorems
Linyu Peng
Waseda University, Japan
Abstract:
Symmetries, local groups of transformations, were introduced by Sophus Lie in the 1880s
in order to study solutions of ordinary diﬀerential equations. It is evidential that the symmetry method is essential in solving diﬀerential equations, e.g. [3]; for instance, almost
all solvable diﬀerential equations have symmetries behind the mystery. In 1918, exactly
one century ago, Emmy Noether published the paper ‘Invariante Variationsprobleme’ withthe celebrated Noether Theorems [2]. Her first theorem tells that every symmetry of anaction, the integral of a Lagrangian function, amounts to a conservation law of the Euler–
Lagrange equations, while her second theorem establishes a one-to-one correspondence between infinite-dimensional Lie algebra of variational symmetries and diﬀerential relations
among the Euler–Lagrange equations.
Noether’s theorems have been extended to finite diﬀerence and finite element variational
problems since 1980s, leaving the semi-discrete case a blank page even though symmetry
analysis of semi-discrete equations were greatly investigated for almost three decades, e.g.
[1]. The main diﬃculty may lie on the lack of a proper symmetry prolongation, in particular
the lack of an evolutionary representative making the integration and summation by parts
accessible [4]. In this talk, I will show how we can possibly overcome such a diﬃculty andthe extension of Noether’s theorems to semi-discrete equations.
References[1] D. Levi and P. Winternitz, Symmetries and conditional symmetries of diﬀerentialdiﬀerence equations, J. Math. Phys. 34 (1993), 3713–3730.
[2] E. Noether, Invariante Variationsprobleme, Nachr. v. d. Ges. d. Wiss. zu
Göttingen, Math-phys. Klasse 2 (1918), 235–257. (English transl.: Transport Theory Statist. Phys. 1 (1971), 186–207.)[3] P. J. Olver, Applications of Lie Groups to Diﬀerential Equations, (2nd ed.),Springer-Verlag, New York, 1993.
[4] L. Peng, Symmetries, conservation laws, and Noether’s theorem for diﬀerentialdiﬀerence equations, Stud. Appl. Math. 139 (2017), 457–502.
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Title: A variational formulation, Dirac structures and dynamical systems fornonequilibrium thermodynamics
Hiroaki Yoshimura
Waseda University, Japan
Abstract:
We present a Lagrangian variational formulation for nonequilibrium thermodynamic, which
is a natural extension of Hamilton’s principle in mechanics to thermodynamics including
irreversible processes. The irreversibility can be encoded into a nonlinear phenomenologicalconstraint, which is represented by nonlinear nonholonomic constraints due to the entropy
production associated with all the irreversible processes. We show that the variational formulation enables to treat various (finite dimensional) discrete systems such as mechanicalsystems with friction, matter transfer, electric circuits, chemical reactions, and diﬀusion
across membranes as well as (infinite dimensional) continuum systems such as multicomponent reacting viscous fluids. In addition to the variational formulation, we clarify theunderlying structures of the nonequilibrium thermodynamics, where the nonlinear nonholonomic constraint of thermodynamic type can be incorporated into Dirac structures and withthe associated Dirac dynamical systems.
(3) Organizers

Daisuke Tarama
Ritsumeikan University, Japandtarama[at]fc.ritsumei.ac.jp
Kenro Furutani
Tokyo University of Science, Japan
Hiroaki Yoshimura
Waseda University, Japan