We organize a workshop in person and through zoom. Registration
is mandatory through the link below.

Mini-workshop：Global Analysis and Geometry●(1) Date (in Japan Standard Time (JST)) ：March 29, 2023 (Wednesday) 13:00 −→ 17:15 + (a social gathering)
March 30, 2023 (Thursday) 10:00 −→ 12:00

lunch break−→

13:30 −→ 17:35(2) Place : Osaka Central Advanced Mathematical Institute (OCAMI), Osaka
Metropolitan University
Auditorium Place : Room F415, Building F(3) Participation : On-site and on-line
In both cases, please make registration throughhttps://forms.gle/M8i32E6cGEdysUcy9
no later than March 27th, 2023. The Zoom link will be send March 28th.
(4) Organizers :Tarama Daisuke (Ritsumeikan Univ.), Furutani Kenro(OCAMI), Homma Yasushi (Waseda Univ.)(5) Supports :JSPS KAKENHI Grant Number 19K03480, 19K14540,

20K03662 and Osaka Central Advanced Mathematical Institute,Osaka

Metropolitan University1

Invited Speakers
NameUniversity

Country

on-line/on-site

Adachi Jiro
Wolfram Bauer
Erlend Grong
Iwasaki Chisato
Homma Yasushi
Irina Markina
Morimoto Tohru
Ohnita Yoshihiro
Ohno SomaTamaru Hiroshi

Hokkaido UniversityJapan

on-siteLeibniz University Hannover

Germany

on-lineUniversity of Bergen

Norwayon-lineUniversity of HyogoJapan

on-siteWaseda UniversityJapan

on-siteUniversity of Bergen

Norwayon-siteOka Mathematical Institute, Nara Women’s Univ.

Japan

on-siteOCAMI, Osaka Metropolitan UniversityJapanon-siteWaseda UniversityJapanon-siteOCAMI, Osaka Metropolitan UniversityJapanon-site2

Program (in JST)
2023/03/29 (Wednesday)

Speaker & Title

13:00 -13:10

Opening

13:10 -14:00

Iwasaki Chisato (Univ. Hyogo)
Symbolic calculus of pseudo-diﬀerential operators on manifolds with conicalsingularities

14:10 -15:00

Ohnita Yoshihiro (OCAMI, Osaka Metropolitan Univ.)
Geometry of Lagrangian submanifolds and isoparametric submanifolds

15:00 -15:25

Tea time

15:25 -16:15

Wolfram Bauer (Leibniz Univ. Hannover)

8:25 -9:15 (CET)

Subriemannian structures on S7 and beyond

16:25 -17:15

Erlend Grong (Univ. Bergen)

9:25 -10:15 (CET)Sub-Riemannian geometry, most probable paths and transformations

18:30 –(social gathering)2023/03/30 (Thursday)
10:00-10:50

Speaker & Title
Irina Markina (Univ. Bergen/OCAMI)
On H-type algebras: history, progress, and perspectives11:00-11:50

Tamaru Hiroshi (OCAMI, Osaka Metropolitan Univ.)
On the moduli spaces of left-invariant Riemannian metrics on Lie groups

12:00-13:20

Lunch

13:20-14:10

Morimoto Tohru (Oka Mathematical Institute, Nara Women’s Univ.)
Three dimensional contact submanifolds of special interest in the seven dimensional
projective space and the associated systems of linear PDE’s of second order

14:20-15:10

Adachi Jiro (Hokkaido Univ.)
Existence and Classiﬁcation of certain non-integrable distributions

15:10-15:30

Tea time

15:30-16:20

Ohno Soma (Waseda Univ.)
Rarita-Schwinger ﬁelds on nearly parallel G2 -manifolds

16:30-17:20

Homma Yasushi (Waseda Univ.)
Higgs algebra in harmonic analysis on the Grassmannian of 2-planes

17:25 -17:35

Closing3

Titles and Abstracts
2023/03/23 (Thursday)(1) 13:10–14:00 Chisato Iwasaki (岩崎千里), University of Hyogo
Title : Symbolic calculus of pseudo-diﬀerential operators on manifolds with conical singularities
Abstract : Symbolic calculus of pseudo-diﬀerential operators on Rn is an essential tool for constructionof the fundamental solution to the heat equation on smooth manifolds. In this talk I will show the modiﬁed
Bessel function plays an important role in asymptotic form of the fundamental solution to the heat equation onmanifolds with conical singularities.
This talk is based on the joint study with Kenro Furutani and Wolfram Bauer.
(2) 14:10–15:00 Ohnita Yoshihiro (大仁田義裕), OCAMI, Osaka Metropolitan University
Title : Geometry of Lagrangian submanifolds and isoparametric submanifolds
Abstract :
15:00 – 15:25 Tea time(3) 15:25–16:15 Wolfram Bauer, Leibniz University Hannover
Title : Subriemannian structures on S7 and beyond
Abstract : In this talk we compare various subriemannian structures deﬁned on the seven dimensional
Euclidean sphere. These are interesting examples in subriemannian geometry, which - under geometric andanalytic aspects - can be analyzed rather explicitly. In particular, we study their subriemannian isometrygroups and address the question of complete integrability of the subriemannian geodesic ﬂow. From an analytic
perspective we discuss the spectra of the induced hypoelliptic sublaplacians as well as asymptotic propertiesof the heat kernel. If time permits related results on the heat kernel asymptotic for H-type foliations will bementioned. This is a class of manifolds generalizing some of the above cases. Our presentation surveys results
obtained in former and ongoing joint work with K. Furutani, C. Iwasaki, A. Laaroussi, I. Markina, D. Tarama
and G. Vega-Molino.
(4) 16:25-17:15 Erlend Grong, University of Bergen
Title : Sub-Riemannian geometry, most probable paths and transformations.
Abstract : Doing statistics on a Riemannian manifold becomes very complicated for the reason that welack pluss to deﬁne such things as mean and variance. Using the Riemannian distance, we can deﬁne a mean
know as the Fréchet mean, but this gives no concept of asymmetry, also known as anisotropy. We introduce analternative deﬁnition of mean called the diﬀusion mean, which is able to both give a mean and the analogue of acovariance matrix for a dataset on a Riemannian manifolds. Surprisingly, computing this mean and covarianceis related to sub-Riemannian geometry. We describe how sub-Riemannian geometry can be applied in thissetting, and mention some ﬁnite dimensional and inﬁnite-dimensional applications.
The results are part of joint work with Stefan Sommer (Copenhagen, Denmark)4

2023/03/30 (Thursday)(5) 10:00–10:50 Irina Markina, University of Bergen, Norway/OCAMI
Title : On H-type algebras: history, progress, and perspectives
Joint work with K.Furutani, Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University,Osaka, Japan.
Abstract : In this talk, I will concentrate on the joint long-lasting project with Professor Furutani. I startfrom historical notes: who and why introduced H-type Lie algebras, which are the natural generalization of theHeisenberg Lie algebra. We will reveal the relation of these Lie algebras with Cliﬀord algebras. After mentioningsome known properties, we will concentrate on the discussion of the existence of lattices on the corresponding
H-type Lie groups, and their possible classiﬁcation which is invariant under group automorphisms.
(6) 11:00–11:50 Tamaru Hiroshi (田丸博士), OCAMI, Osaka Metropolitan University
Title : On the moduli spaces of left-invariant Riemannian metrics on Lie groups
Abstract : For a given Lie group, a certain quotient space of the space of all left-invariant Riemannianmetrics is called the moduli space. Properties of the moduli space, such as the dimension or topological type,are invariants of the Lie group, and could be expected to reﬂect properties of left-invariant metrics. In thistalk, we introduce this framework, and mention some recent studies on Lie groups whose moduli spaces are
one-dimensional.
12:00–13:20 Lunch(7) 13:20–14:10 Morimoto Tohru (森本徹), Oka Mathematical Institute, Nara Women’s University
Title : Three dimensional contact submanifolds of special interest in the seven dimensional
projective space and the associated systems of linear PDE’s of second order
Abstract : We like to present some special concrete examples of contact submanifolds in projective space asa consequence of our general classiﬁcation program and discuss their symmetries and properties in relation to
the associated diﬀerential equations(8) 14:20–15:10 Adachi Jiro (足立二郎), Hokkaido University
Title : Existence and Classiﬁcation of certain non-integrable distributions
Abstract : Existence and Classiﬁcation of geometric structures on manifolds are fundamental interests in
diﬀerential topology. Sometimes, existence of geometric structures contribute to the global study of the basemanifold. In this talk, I would like to introduce some attempts to study geometric structures from the view
point of the h-principle. That method relates geometric structures some topological conditions. My struggle
will come to (3, 5)-distribution, Cartan’s (2, 3, 5)-distribution, and their generalizations.
15:10–15:30 Tea time5(9) 15:30–16:20 Ohno Soma (大野走馬), Waseda University
Title : Rarita-Schwinger ﬁelds on nearly parallel G2 -manifolds
Abstract : Rarita-Schwinger ﬁelds play an important role on Physics. On the other hand, in mathematics,Rarita-Schwinger ﬁelds can be considered as a ”spin-3/2 version” of the harmonic spinor. In this study, weﬁnd that the space of Rarita-Schwinger ﬁelds coincides with a subspace of an eigenspace of the Laplacian onnearly parallel G2 -manifolds. Applying the same technique to a deformation theory, we also identify the spaceof inﬁnitesimal deformations of a Killing spinor.
(10) 16:30–17:20 Homma Yasushi (本間泰史), Waseda UniversityTitle : Higgs algebra in harmonic analysis on the Grassmannian of 2-planes
Abstract : In the classical harmonic analysis on the sphere S m−1 , it is known that X = |x|2 /2, Y = −∆x /2,∑and H = Ex + m/2 =xi ∂i + m/2 constitute the Lie algebra sl(2, R), which acts on the space of polynomialmm
P(R , C) on R . Then we have a sl(2, R) × SO(m)-decomposition, P(Rm , C) ∼⊗ Hk , where V ∞is=V∞
k+m/2

k+m/2

an (∞-dimensional) irreducible Verma-module of sl(2, R) with the lowest weight k + m/2 and Hk = Pk ∩ ker ∆x
is an irreducible SO(m)-module of the harmonic polynomial with degree k. Note that the restriction P(Rm , C)
onto S m−1 gives L2 (S m−1 , C).
We consider a generalization from the sphere to the (oriented) Grassmannian manifold Gr0 (m, 2) of 2-planes
in Rm . The harmonic analysis on such manifolds has been investigated by many researchers. In this talk, I will
present a new aspect that an algebra called the Higgs algebra appears instead of sl(2, R). The algebra H3 is apolynomial deformation of sl(2, R) invented by physicist Higgs (but, no relation to Higgs ﬁleds) and developedin physics. By using H3 , we have a H3 × SO(m)-decomposition of the polynomial space P(Rm×2 , C)SL(2,R)
whose resection gives L2 (Gr0 (m, 2), C). If I have time, I will also talk about the Casimir of H3 and the Pizzetti
formula on Gr0 (m, 2) as applicaitons. This talk is based on a collaborative research with D. Eelbode in Antwerp
university.

Thank you for your cooporation and interest.

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