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RIMS 共同研究（公開型）

幾何構造と微分方程式― 対称性・特異点及び量子化の視点から ―Geometric Structures and Diﬀerential Equations– Symmetry, Singularity, and Quantization –ProgramandAbstracts

14 – 18 March 2022
Zoom meeting/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, Osaka City University / Tokyo University of Science)
Hiroaki Yoshimura (吉村 浩明, Waseda University)
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Time Schedule
Remark: The following schedule is written in
Japanese Standard Time (JST = UTC + 0900) / Central European Time (CET = UTC + 0100).
March 14 (Monday)
12:50 – 13:00 (JST) 4:50 – 5:00 (CET)
13:00 – 14:00 (JST) 5:00 – 6:00 (CET)
14:20 – 15:20 (JST) 6:20 – 7:20 (CET)
16:00 – 17:00 (JST) 8:00 – 9:00 (CET)
17:20 – 18:20 (JST) 9:20 – 10:20 (CET)March 15 (Tuesday)

Opening
01 Takuma Tomihisa
02 Hidekazu Ito
03 Jean-Pierre Françoise
04 Sonja Hohloch

13:00 – 14:00 (JST) 5:00 – 6:00 (CET)
14:20 – 15:20 (JST) 6:20 – 7:20 (CET)
16:00 – 17:00 (JST) 8:00 – 9:00 (CET)
17:20 – 18:20 (JST) 9:20 – 10:20 (CET)March 16 (Wednesday)

05 Akira Kitaoka
06 Kazuyuki Yagasaki
07 Tudor S. Ratiu
08 Wolfram Bauer

13:00 – 14:00 (JST) 5:00 – 6:00 (CET)
14:20 – 15:20 (JST) 6:20 – 7:20 (CET)
16:00 – 17:00 (JST) 8:00 – 9:00 (CET)
17:20 – 18:20 (JST) 9:20 – 10:20 (CET)March 17 (Thursday)

09 Yuji Hirota
10 Tomoyuki Kakehi
11 Elmar Schrohe
12 Abdellah Laaroussi

13:00 – 14:00 (JST) 5:00 – 6:00 (CET)
14:20 – 15:20 (JST) 6:20 – 7:20 (CET)
16:00 – 17:00 (JST) 8:00 – 9:00 (CET)
17:20 – 18:20 (JST) 9:20 – 10:20 (CET)March 18 (Friday)

13 Shoya Motonaga
14 Hiroyuki Chihara
15 Ryszard Nest
16 Irina Markina

13:00 – 14:00 (JST)
14:20 – 15:20 (JST)
15:20 – 15:30 (JST)17 Genki Ishikawa
18 Setsuro FujiiéClosing

5:00 – 6:00 (CET)
6:20 – 7:20 (CET)
7:20 – 7:30 (CET)2(0) Basic Information

Dates: From 14/March/2022 till 18/March/2022
Participation: Through Zoom meeting.
Please register through the following URL:
https://ritsumei-ac-jp.zoom.us/meeting/register/tJYpfuyoqDsoGd0OoIvjRCGAW494F6ES2TJT

Web-site:
http://www.math.ritsumei.ac.jp/~dtarama/GSDE2022/index.html

3(1) Program
Remark: The following schedule is written in
Japanese Standard Time (JST = UTC + 0900) / Central European Time (CET = UTC + 0100).

Monday, 14/March/2022
12:50 – 13:00 (JST) / 4:50 – 5:00 (CET) Opening
13:00 – 14:00 (JST) / 5:00 – 6:00 (CET) Takuma Tomihisa (Waseda University)
Higher spin Dirac operators
14:20 – 15:20 (JST) / 6:20 – 7:20 (CET) Hidekazu Ito (Kanagawa University)
Birkhoﬀ normalization for a family of superintegrable symplectic maps and its application
16:00 – 17:00 (JST) / 8:00 – 9:00 (CET) Jean-Pierre Françoise (Sorbonne University)
Integrable Systems and their discretization
17:20 – 18:20 (JST) / 9:20 – 10:20 (CET) Sonja Hohloch (University of Antwerp)
S 1 -actions, semitoric systems, and hyperbolic singularities

Tuesday, 15/March/2022
13:00 – 14:00 (JST) / 5:00 – 6:00 (CET) Akira Kitaoka (University of Tokyo)
Ray-Singer torsion and the Rumin Laplacian on lens spaces
14:20 – 15:20 (JST) / 6:20 – 7:20 (CET) Kazuyuki Yagasaki (Kyoto University)
Some recent results on nonintegrability of dynamical systems
16:00 – 17:00 (JST) / 8:00 – 9:00 (CET) Tudor S. Ratiu (Shanghai Jiao Tong University)
Teichmüller space and diﬀerential character valued momentum maps
17:20 – 18:20 (JST) / 9:20 – 10:20 (CET) Wolfram Bauer (Leibniz University Hannover)
Trivializable subriemannian structures on spheres and subelliptic heat equation

Wednesday, 16/March/2022
13:00 – 14:00 (JST) / 5:00 – 6:00 (CET) Yuji Hirota (Azabu University)
Dirac structures over the space of connections on 3 and 4 dimensional Riemannian manifolds
14:20 – 15:20 (JST) / 6:20 – 7:20 (CET) Tomoyuki Kakehi (University of Tsukuba)
Surjectivity of convolution operators
16:00 – 17:00 (JST) / 8:00 – 9:00 (CET) Elmar Schrohe (Leibniz University Hannover)Degenerate Elliptic Boundary Value Problems with Non-smooth Coeﬃcients
17:20 – 18:20 (JST) / 9:20 – 10:20 (CET) Abdellah Laaroussi (Leibniz University Hannover)Heat kernel asymptotics for quaternionic contact manifolds

4Thursday, 17/March/2022
13:00 – 14:00 (JST) / 5:00 – 6:00 (CET) Shoya Motonaga (Kyoto University)
Obstructions to integrability of nearly integrable dynamical systems near regular level sets
14:20 – 15:20 (JST) / 6:20 – 7:20 (CET) Hiroyuki Chihara (University of the Ryukyus)
Microlocal analysis of d-plane transform on the Euclidean space
16:00 – 17:00 (JST) / 8:00 – 9:00 (CET) Ryszard Nest (University of Copenhagen)
Automorphisms of Boutet de Monvel algebra
17:20 – 18:20 (JST) / 9:20 – 10:20 (CET) Irina Markina (University of Bergen)
A unified approach to extremal curves on Stiefel manifolds

Friday, 18/March/2022
13:00 – 14:00 (JST) / 5:00 – 6:00 (CET) Genki Ishikawa (Ritsumeikan University)
An elliptic fibration arising from the Lagrange top
14:20 – 15:20 (JST) / 6:20 – 7:20 (CET) Setsuro Fujiié (Ritsumeikan University)
WKB and microlocal approach to various Bohr-Sommerfeld quantization rules
15:20 – 15:30 (JST) / 7:20 – 7:30 (CET) Closing

5(2) Abstracts

Title: Higher spin Dirac operators
Takuma Tomihisa
Waseda University, Japan
Abstract:
Higher spin Dirac operators are generalizations of the Dirac operator and the RaritaSchwinger operator. They are geometric first order diﬀerential operators on the spinor
fields with spin j + 1/2 over Riemannian spin manifolds. In this talk, I will introduce
higher spin Dirac operators and give some formulas for higher spin Dirac operators on
spaces of constant curvature. Furthermore, we calculate spectra of higher spin Dirac operators on the sphere by using formulas. This gives Branson’s result (2002) in a diﬀerent way of
calculation. This talk is based on a joint work with Yasushi Homma (Waseda University).
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Title: Birkhoﬀ normalization for a family of superintegrable symplectic maps
and its application
Hidekazu Ito
Kanagawa University, Japan
Abstract:
Integrability of a Hamiltonian system with n degrees of freedom is defined by existence ofn Poisson-commuting integrals. The Liouville-Arnold’s theorem states that, if the level setof those integrals is regular, compact and connected, then it is an n dimensional torus andthere exists a system of special symplectic coordinates called “action-angle variables” inwhich the system can be solved explicitly.
The aim of this talk is to clarify the orbit structure of an (super-)integrable system nearsingularities of those integrals. The special feature is that the singularities are resonant in
the sense of dynamical systems. For this purpose, we focus on Birkhoﬀ normalization fora family of 2n dimensional analytic symplectic maps near resonant fixed points and provethe existence of a convergent Birkhoﬀ transformation provided that the maps have n + q
integrals near a fixed point of resonance degree q. It leads to the existence of a special
coordinate system in which the given Hamiltonian system can be solved explicitly nearsingularities of its integrals.
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Title: Integrable Systems and their discretization
Jean-Pierre Françoise
Sorbonne University, France
Abstract:
We consider discretizations of integrable systems in dimension two, which are defined bybirational maps. We recall that in the case of two dimensional cubic planar vector fields,such discretizations are associated with automorphisms of elliptic fibrations. In contrast,
the discrete Lotka-Volterra system integrability is not yet established. This is the same withthe “canard” system and we further suggest similar analysis of discrete quadratic double
centers. Finally, we revisit the 3-dimensional discrete Euler Top.
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Title: S1 -actions, semitoric systems, and hyperbolic singularities
Sonja Hohloch
University of Antwerp, Belgium
Abstract:
In this talk, we will discuss recent progress towards a symplectic classification of 2-degreeof freedom integrable Hamiltonian systems with S1 -symmetry and mildly degenerate singularities. We give an overview over existing symplectic classifications of toric and semitoricsystems and explain the impact of an underlying global S1 -action on a symplectic classification. In particular, we will sketch how to extend an arbitrary eﬀective Hamiltonian
S1 -action on a compact 4-dimensional manifold to a 2-degree of freedom integrable systems
with only mild degeneracies (if any).
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Title: Ray-Singer torsion and the Rumin Laplacian on lens spaces
Akira Kitaoka
University of Tokyo, Japan
Abstract:
The Rumin is the Bernstein-Gelfand-Gelfand complex (BGG complex) of the twisted deRham complex of a flat vector bundle with respect to contact manifolds. As a typicaltheorem, the cohomology of the BGG complex coincides with the cohomology of the de
Rham complex of a flat vector bundle. Moreover, the Rumin complex arises when we takethe sub-Riemannian limit.
Let us consider what happens when we replace a concept defined using the de Rham complex
with the Rumin complex. In this talk, we adapt this idea to analytic torsion. On flat vector
bundles with a unimodular holonomy over lens spaces, we express explicitly the analytic
torsion functions associated with the Rumin complex in terms of the Hurwitz zeta function.
In particular, we determine the analytic torsions, and it is written using the Betti numbers
and the Ray-Singer torsion.
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Title: Some recent results on nonintegrability of dynamical systems
Kazuyuki Yagasaki
Kyoto University, Japan
Abstract:
The problem of nonintegrability is very classical and important in the field of dynamicalsystems. Its history dates back to the time of Bruns and Poincare in the nineteenth century. In this talk I review my recent results on this topic. First, I describe an outlineof the Morales-Ramis theory and illustrate the theory for the SEIR epidemic model as an
example. Secondly, I explain a theory developed recently for nonintegrability of nearly integrable systems, and briefly describe its application to the restricted three-body problem
and Duﬃng oscillator. The theory is based on the Morales-Ramis theory. Thirdly, I state
my very recent result on the nonintegrability of general three- and four-dimensional systems
near degenerate equilibria: The Jacobian matrices of the vector fields have a zero and apair of purely imaginary eigenvalues for the former systems, and have two pairs of purely
imaginary eigenvalues without resonance for the latter ones. Finally, I give some comments
on future work.

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Title: Teichmüller space and diﬀerential character valued momentum maps
Tudor S. Ratiu
Shanghai Jiao Tong University, China / École Polytechnique Fédérale de Lausanne, Switzerland
Abstract:
TBA

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Title: Trivializable subriemannian structures on spheres and subelliptic heat
equation
Wolfram Bauer
Leibniz University Hannover, Germany
Abstract:
Subriemannian geometry provides a mathematical framework for the study of motions under non-holonomic constraints. Generalizing the Beltrami-Laplace operator in Riemannian
geometry a subriemannian structure induces in many examples geometrically defined second order subelliptic diﬀerential operators such as the sublaplacian. In recent years, thestudy of such operators and their corresponding heat equation has reveals interesting links
between analysis and geometry. In this talk we consider a specific class of subriemannian
structures on Euclidean spheres in [1] which we call trivializable. In particular, on theseven dimensional sphere S7 we compare a rank 4 trivializable subriemannian structure
and a more standard structure induced by the quaterionic Hopf fibration under diﬀerent
geometric and analytic aspects. We consider the corresponding intrinsic sublaplacians andtheir heat kernel. Explizit formulas and the small time heat kernel asymptotic will be discussed. By choosing appropriate coordinates we can derive the heat kernel of a rank fivetrivializable structure on S7 which is then applied to spectral theory and leads to the inverseof the conformal sublaplacian. This talk is based on joint work with Abdellah Laaroussi(Leibniz University Hannover) and Daisuke Tarama (Ritsumeikan University).
[1] W. Bauer, K. Furutani, C. Iwasaki, Trivializable sub-Riemannian structures on
spheres, Bull. Sci. Math. 137 (2013), no. 3, 361-385.
[2] W. Bauer, A. Laaroussi, Trivializable and quaternionic subriemannian structure on
S 7 and subelliptic heat kernel, available on arXiv:2102.04784, 2021.
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Title: Dirac structures over the space of connections on 3 and 4 dimensional
Riemannian manifolds
Yuji Hirota
Azabu University, Japan
Abstract:
The study of the space of connections over manifolds, which arose from gauge theory, hasbeen developed involving topology and symplectic geometry. As is well known, the spaceof connections over Riemannian surfaces was shown to be a symplectic manifold of infinite
dimension by M. Atiyah and R. Bott. Afterwards, the space of irreducible connections overfour dimensional manifolds with boundary was discussed by T. Kori, and was shown to bea presymplectic manifold. Like these, the space of connections provides us with examples
of geometric structures in infinite dimensional cases.
In this talk, we focus on the space of irreducible connections over three and four dimensional
manifolds and show that it has a certain geometric structure, so called a Dirac structure.
the notion of Dirac structure was first introduced by T. Courant and A. Weinstein, which
gives the unified framework of presymplectic geometry and Poisson geometry. We describe
explicitly some Dirac structures by using bundle homomorphisms. This is a joint work with
Emeritus Professor Tosiaki Kori, Waseda University.
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Title:Surjectivity of convolution operators
Tomoyuki Kakehi
University of Tsukuba, Japan
Abstract:
If µ is a compactly supported distribution on Rn , the convolution operator Cµ : f 7→ f ∗ µ
is well defined as a map from C ∞ (Rn ) to itself. Then we arrive at the following question.
When is Cµ surjective? Ehrenpreis gave an answer to this question by showing that Cµ issurjective if and only if the Fourier-Laplace transform µb(λ) of µ is slowly decreasing. In thistalk, we start with the above result by Ehrenpreis, and next, we explain that an analogous
result holds for convolution operators on noncompact symmetric spaces.
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Title: Degenerate Elliptic Boundary Value Problems with Non-smooth Coeﬃcients
Elmar Schrohe
Leibniz University Hannover, Germany
Abstract:
On a manifold X with boundary and bounded geometry we consider a uniformly strongly
elliptic second order operator A that locally is of the form∑∑A=−ajk ∂xj ∂xk +bj ∂xj + c,j,kj

together with a boundary operator
T = φ0 γ0 + φ1 γ1 ,where γ0 and γ1 denote the evaluation of a function and its exterior normal derivative,
respectively, at the boundary, and φ0 , φ1 are non-negative Cb∞ functions on the boundarywith φ0 + φ1 ≥ c0 > 0. This problem is not elliptic in the sense of Lopatinskij and Shapiro,
unless either φ1 ̸= 0 everywhere or φ1 = 0 everywhere.
The condition that φ0 and φ1 are smooth can be relaxed to φ0 = 1 and φ1 = ψ 2 for afunction ψ ∈ C 2+ϵ for some ϵ > 0.
Under the assumption that the ajk are Hölder continuous and bj as well as c are L∞ weshow• unique solvability for the boundary value problem (A − λ)u = f, T u = g for f ∈Lp (Ω) and g in a suitable trace space for all suﬃciently large λ ∈/ R>0• the existence of a bounded H∞ -calculus of arbitrarily small angle for the realization
AT of A in Lp (Ω)• short time solvability for the porous medium equation with the boundary condition.
Moreover, we sketch how these results possibly can be applied to a quasi-stationary Stefan
problem for the melting of ice against an inhomogeneous background.
(Joint work with Thorben Krietenstein, Hannover)
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Title: Heat kernel asymptotics for quaternionic contact manifolds
Abdellah Laaroussi
Leibniz University Hannover, Germany
Abstract:
We consider the small time asymptotics for the heat kernel associated to the intrinsic
sublaplacian on a quaternionic contact manifold considered as a subriemannian manifold.
More precisely, we explicitly compute the first two coeﬃcients c0 and c1 appearing in thesmall time asymptotics expansion of the heat kernel on the diagonal. We show that thesecond coeﬃcient c1 equals the scalar curvature κ (up to a constant multiple) associated tothe canonical connection defined on such a manifold.

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Title: Obstructions to integrability of nearly integrable dynamical systems near
regular level sets
Shoya Motonaga
Kyoto University, Japan
Abstract:
We consider analytical, nearly integrable systems which may be non-Hamiltonian, anddiscuss their nonintegrability in the non-Hamiltonian sense. We give a suﬃcient condition
for them to be analytically nonintegrable such that the commutative vector fields and first
integrals depend on the perturbation parameter analytically. We compare our results withclassical results of Poincar and Kozlov for systems written in action and angle coordinatesand discuss their relationships with the subharmonic and homoclinic Melnikov methods forperiodic perturbations of single-degree-of-freedom Hamiltonian systems. This is joint workwith Kazuyuki Yagasaki (Kyoto University).
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Title: Microlocal analysis of d-plane transform on the Euclidean space
Hiroyuki Chihara
University of the Ryukyus, Japan
Abstract:
The d-plane transform is defined by integrals of functions on the n-dimensional Euclidean
space over all the d-dimensional planes. where 0 < d < n. This maps functions on the
Euclidean space to those on the aﬃne Grassmannian G(d, n). This is said to be X-ray
transform if d = 1 and Radon transform if d = n − 1. When n = 2 the X-ray transform isthought to be measurements of CT scanners. In this talk we begin with the basic propertiesof the d-plane transform, and talk about concrete expression of the canonical relation of
the d-plane transform and quantitative properties of the product of the image of the dplane transforms. The latter one is related to the streaking artifact of CT image, andsome generalization of recent results of Park-Choi-Seo (2017) and Palacios-Uhlmann-Wang(2018) for the X-ray transform on the plane.
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Title: Automorphisms of Boutet de Monvel algebra
Ryszard Nest
University of Copenhagen, Denmark
Abstract:
In a remarkable work, Duistermaat and Singer in 1976 studied the algebras of all classicalpseudodiﬀerential operators on smooth (boundaryless) manifolds. They gave a descriptionof order preserving algebra isomorphism between the algebras of classical pseudodiﬀerential
operators of two manifolds under a cohomological assumption pertaining the first manifold.
‘Order preserving’ here means that the isomorphism preserves the order of the operators.
Surprisingly no continuity assumption is necessary; continuity is automatic. In this talk wewill describe constructions involved in generalising their results to the Boutet de Monvel
algebra of operators on manifolds with boundary. The main fact of life about manifold withboundary is that vector fields do not define global flows and the ”boundary conditions” area way of dealing with this problem. The Boutet de Monvel algebra corresponds to the
choice of local boundary conditions and is, eﬀectively, a non-commutative completion of
the manifold. One can think of it as a parametrised version of the classical Toeplitz algebra
as a completion of the half-space. Once this is explained, the analysis that we need reduces
to a high degree to relatively classical results about automorphisms and homology of the
Toeplitz algebra and we will explain what those are and how they are used. This is workin progress joint with Elmar Schrohe.
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Title: A unified approach to extremal curves on Stiefel manifolds
Irina Markina
University of Bergen, Norway
Abstract:
We present a unified framework for studying extremal curves on real Stiefel manifolds.
We start with a smooth one-parameter family of pseudo-Riemannian metrics on a productof orthogonal groups acting transitively on Stiefel manifolds. We find Euler-Langrange
equations for a class of extremal curves that includes geodesics with respect to diﬀerent
Riemannian metrics and smooth curves of constant geodesic curvature. For some specificvalues of the parameter in the family of pseudo-Riemannian metrics, we recover certain
well-known used in applied mathematics. This is a joint work with K. Hueper (Universityof Wurzburg) and F. Silva Leite (University of Coimbra).
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Title: An elliptic fibration arising from the Lagrange top
Genki Ishikawa
Ritsumeikan University, Japan
Abstract:
The Lagrange top is a typical example of integrable systems which describes the motion of arigid body about a fixed point with symmetric axis through the centre of mass and the fixed
point. The complexification of the energy-momentum mapping induces an elliptic fibration
over CP2 in the Weierstraß normal form. One investigates complex algebro-geometric aspects of this elliptic fibration. In particular, the discriminant locus of this elliptic fibration
is described in detail and the types of singular fibres are determined according to the classification of singular fibres for elliptic threefolds by R. Miranda.
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Title: WKB and microlocal approach to various Bohr-Sommerfeld quantization
rules
Setsuro FujiiéRitsumeikan University, Japan
Abstract:
The well-known Bohr-Sommerfeld quantization rule is the condition for an energy λ to bean eigenvalue of a 1D Schrödinger operator P = −h2 ∆ + V (x) with a simple well potential
V (x). It is an equation of the form−eiS(λ)/h = 1,∫
where S(λ) = γ(λ) ξdx is the action along the periodic classical trajectory γ(λ) in the phasespace T ∗ R ∼ Rx × Rξ associated with the simple well. A root of this equation gives anapproximation of an eigenvalue when the semiclassical parameter h is small.
This fact has been justified rigorously in various mathematical methods such Airy function
expansion or exact (complex) WKB method etc. In this talk, we propose another method
based on the microlocal analysis. The advantages of this method are the followings:• We need to construct solutions only along the periodic curve γ(λ) in the phasespace.
• We do not need to care about the divergence of the microlocal WKB expansion.
• This approach is in line of the intuitive interpretation of the rule that the quantum
wave should coincide after a tour around γ(λ) with the original wave.
• This method is naturally adapted to the quantization of quantum resonances in
multi-dimension or for systems.
In this talk, this microlocal method is explained with emphasize on the above features andsome recent applications are presented.
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(3) Organizers

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