Osaka City University Advanced Mathematical Institute (OCAMI)MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics
OCAMI Reports Vol. 1 (2021)
doi: 10.24544/ocu.20210605-001

The 27th Osaka City University
International Academic Symposium“Mathematical Science of Visualization,
and Deepening of Symmetry and Moduli”
Organized by
Yoshihiro Ohnita (OCAMI)
Masashi Yasumoto (IMI Kyushu University & OCAMI)
Hiroshi Tamaru (Osaka City University)
Venkatesh Raghavan (Osaka City University)
Chihiro Matsuoka (Osaka City University)
Martin Guest (Waseda University & OCAMI)
Miyuki Koiso (IMI Kyushu University)
Seiichi Kamada (Osaka University)
Wayne Rossman (Kobe University & OCAMI)
Takashi Sakai (Tokyo Metropolitan University & OCAMI)
Franz Pedit (UMass Amherst, USA)
Udo Hertrich-Jeromin (TU Wien, Austria)
March 21 - 26, 2021
Abstract. This international academic symposium focuses on recent progressin mathematical visualization, and will promote discussions between researchers
in this field from abroad and in Japan, aiming to encourage future collaborations.
2020 Mathematics Subject Classification.
53-06, 52-06, 57-06
Key words and Phrases.
mathematical visualization, differential geometry, discrete differential geometry,
information geometry, conformal geometry, knot theory, low dimensional topology© 2021 OCAMI.
OCAMI. The 27th Osaka City University International Academic Symposium “Mathematical Science of Visualization, and Deepening of Symmetry and Moduli”. OCAMI Reports. Vol. 1, Osaka
City University Advanced Mathematical Institute. 2021, 96pp. doi: 10.24544/ocu.20210605-001

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OCAMI Reports Vol. 1 (2021)

PrefaceThis is the collection of abstracts for
The 27th Osaka City University International Academic Symposium“Mathematical Science of Visualization, and Deepening of Symmetry and Moduli”
held on March 21-26, 2021. In this research project, we promoted current researchdevelopments related to the concept of “mathematical visualization”, aiming to create
new interdisciplinary joint research by integrating the topics. In particular, our goal
is to promote research on “Symmetry and Moduli” that inhabit various research fields
in mathematics and mathematical science. Examples include
1. smooth and discrete minimal surfaces, surfaces of constant mean curvature,and constrained Willmore conjecture,
2. discrete geometry such as circle packings and circle patters, and related discrete
complex analysis and discrete conformal geometry,
3. mathematical foundations of information geometry and their application tomachine learning and deep learning,
4. moduli of nonlinear PDE and gauge theory,
5. knot theory and low dimensional topology, and
6. interplay of mathematical visualization and architectural design with the above
research fields.
For this purpose, we originally planned to gather world leaders in the above research
fields together on March 4-9, 2020. However, due to the COVID-19 pandemic, theorganizing committee had decided to hold the symposium with the above reschedulingin a hybrid format, that is, some of the talks were given on-site while most talks
were given online. Replacing direct discussion among participants, in order to makeour symposium a platform for the aformentioned research fields, all the talks wererecorded and uploaded online. This provided a framework for collaboration between
participants, and based on this activity, much collaborative work will be produced
that extends beyond the present borders of these research fields in the near future.
Apart from promoting research activities, our mission was also to encourage
younger researchers by giving them the opportunity to give presentations about
their research. Furthermore, among 15 young speakers, we had selected one Best
Presentation Award winner, and four Excellent Presentation Award winners by
online anonymous voting. We are confident that this selection was fair, and equalchances were given to all the young speakers.
One difficulty of online activities is the time differences amongst many countries.
In spite of this difficulty, we were able to welcome 14 international speakers online.
Moreover, 27 domestic speakers, including 15 younger speakers, gave presentations
about their works related to our concept. Independently of research talks, 2 public
lectures were given to both researchers and the general public. These lectures revealed
clear applications of recent mathematics to real world situations.
Finally, the organizing committee would like to represent our sincere gratitude toThe 27th OCU Symposium3

all the speakers for their contributions to our project. We are also grateful to the staff
at OCAMI, as well as the graduate student staff (Luis Pedro Castellanos Moscoso,
Yuta Taniguchi, Satoshi Ogawa, Ryo Takenaka at Osaka City University, and Jumpei
Yasuda at Osaka University), for their generous support. This project was supported
by JSPS KAKENHI Grant Numbers JP18H04489, JP19J02034, JP20K14314 (Principal Investigator: Masashi Yasumoto), JP17H06127 (Principal Investigator: MasaHiko Saito, Co-Investigator: Yoshihiro Ohnita), JP18H03668 (Principal Investigator:
Martin Guest, Co-Investigator: Yoshihiro Ohnita), and internal financial support
by Osaka City University and Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics
JPMXP0619217849).

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Organizers
Yoshihiro Ohnita
Osaka City University Advanced Mathematical Institute, 3-3-138, Sugimoto, Sumiyoshi-ku,
Osaka 558-8585, Japan

E-mail address: ohnita@sci.osaka-cu.ac.jp

Masashi Yasumoto
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka
819-0395, Japan

E-mail address: m-yasumoto@imi.kyushu-u.ac.jp

Hiroshi Tamaru
Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138,
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

E-mail address: tamaru@sci.osaka-cu.ac.jp

Venkatesh Raghavan
Graduate School of Engineering, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku,
Osaka 558-8585, Japan

E-mail address: raghavan@osaka-cu.ac.jp

Chihiro Matsuoka
Graduate School of Engineering, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku,
Osaka 558-8585, Japan

E-mail address: cmatsuoka@osaka-cu.ac.jp

Martin Guest
Department of Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1
Okubo, Shinjuku, Tokyo 169-8555, Japan

E-mail address: martin@waseda.jp

Miyuki Koiso
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka
819-0395, Japan

E-mail address: koiso@imi.kyushu-u.ac.jp

Seiichi Kamada
Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0043, Japan

E-mail address: kamada@math.sci.osaka-u.ac.jp

Wayne Rossman
Department of Mathematics, Graduate School of Science, Kobe University, 1-1 Rokkodaicho, Nada-ku, Kobe 657-8501, Japan

E-mail address: wayne@math.kobe-u.ac.jp

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Takashi Sakai
Department of Mathematical Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa,
Hachioji-shi, Tokyo 192-0397, Japan

E-mail address: sakai-t@tmu.ac.jp

Franz Pedit
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA
01003, USA

E-mail address: pedit@math.umass.edu

Udo Hertrich-Jeromin
Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstrasse 8-10/104,
1040 Wien, Austria

E-mail address: udo.hertrich-jeromin@tuwien.ac.atp

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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Organizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2
4

Shun-ichi Amari
Geometry of Random Deep Neural Networks: Information Geometry Pointof View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9Benjamin Bode
Visualization of link fibrations and critical points . . . . . . . . . . . . . . . 10
Luis Pedro Castellanos Moscoso
Degenerations of orbits and left-invariant symplectic structures on Lie groups 12
Hang Thi Do*, Venkatesh Raghavan, Go Yonezawa
Land use/land cover classification using object and pixel based classification
approaches with medium resolution satellite images: A case study in
Lao Cai Providence, Vietnam . . . . . . . . . . . . . . . . . . . . . . 14
Tatsuyoshi Hamada
MathLibre: Mathematical Software Environment . . . . . . . . . . . . . . . 19
Shin Hayashi
Topological invariants and corner states for Hamiltonians on a lattice . . . 20
Hiroki Ito
A family of index polynomial invariants for virtual string links . . . . . . . 22
Pavithra Jayasinghe*, Venkatesh Raghavan, Go Yonezawa
Evaluating spatio-temporal expansion patterns and urban growth models: A
case study of Colombo City, Sri Lanka . . . . . . . . . . . . . . . . . 24
Akio Kawauchi
A spatial arc and the knotting probability . . . . . . . . . . . . . . . . . . . . 29
Naoki Kitazawa
Geometric and constructive understanding of higher dimensional differentiable manifolds via explicit Morse functions and fold maps . . . . . 30
Hiroto Kusano
Solution to the reflection equation in the symmetric tensor representation . 32
Wai Yeung Lam
Circle patterns on surfaces with complex projective structures . . . . . . . . 33
Na LeiMesh Generation Based on Computational Conformal Geometry . . . . . . 34
Katrin Leschke
Darboux transforms via a Sym-type formula . . . . . . . . . . . . . . . . . . 35
Feng Luo
Discrete conformal geometry of polyhedral surfaces and its application . . . 37

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Hiroshi Matsuzoe
Geometry of quasi statistical manifolds . . . . . . . . . . . . . . . . . . . . . 38
Masahiro Morimoto
On minimal PF submanifolds in Hilbert spaces with symmetries . . . . . . . 40
Sanefumi Moriyama
M2-branes: Integrability and Weyl group . . . . . . . . . . . . . . . . . . . . 42
Christian Müller
Geometric structures motivated by architecture . . . . . . . . . . . . . . . . 44
Hisashi Naito
Carbon Structures and a Discrete Surface Theory . . . . . . . . . . . . . . . 46
Naomichi Nakajima
The dually flat structures for singular models with degenerate potentials . . 50
Satoshi Ogawa
CR-funtions of suspension foliations and Dynamical systems . . . . . . . . . 51
Kento Okuda*, Miyuki Koiso
Uniqueness of local minimizers for crystalline variational problems . . . . . 52
Franz Pedit
Isometric Immersions via Variational Calculus . . . . . . . . . . . . . . . . 55
Joseph Cho, Mason Pember*, Gudrun Szewieczek
Spherical curvature lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Ulrich Pinkall
Computing Constrained Willmore Surfaces . . . . . . . . . . . . . . . . . . . 61
Konrad Polthier
Branched Covering Surfaces - New Shapes, New Materials and New Processes 63
Helmut Pottmann
Smoothness of polyhedral surfaces motivated by applications in computational design and fabrication . . . . . . . . . . . . . . . . . . . . . . . 64
Naoki Sakata
3D nets and handlebody decompositions of 3-manifolds . . . . . . . . . . . . 65
Mako Sato
Beltrami fields with a non-constant proportionality factor . . . . . . . . . . 66
Yuichiro Sato
Totally umbilical submanifolds in pseudo-Riemannian space forms. . . . . 67

Ayaka Shimizu
Region crossing change and Region Select game . . . . . . . . . . . . . . . . 70
Koya Shimokawa
3-dimensional topology and poly-continuous pattern . . . . . . . . . . . . . . 73

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Gudrun Szewieczek
Discrete flat fronts in hyperbolic space via discrete cyclic circle congruences

75

Asuka Takatsu
Relaxation of optimal transport problem via Bregman divergence . . . . . . . 77
Ryo Takenaka
Parafermionic bases of standard modules for twisted affine Lie algebras . . . 78
Yuta Taniguchi
An f -twisted Alexander matrix for knot quandles . . . . . . . . . . . . . . . 80
Ryotaro Ueda
Quandle coloring quivers of virtual links using virtual quandles . . . . . . . 82
Jumpei Yasuda
Presentation of surface links using plat forms . . . . . . . . . . . . . . . . . 84
Jun Zhang
Statistical Mirror Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
伊師 英之
情報と幾何 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

巴山 竜来
数学 × 工芸：テンソル演算と西陣織 . . . . . . . . . . . . . . . . . . . . . . .

88

Albert Chern
Presentation of surface links using plat forms . . . . . . . . . . . . . . . . . 89
Sebastian Heller
CMC surfaces via DPW and xlab . . . . . . . . . . . . . . . . . . . . . . . . 91
Naoto Satoh
Statistical structure of constant statistical sectional curvature on hyperbolic
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Kazuma Tateiri
Computing Cut Locus of Convex Polyhedron using Wavefront Propagation . 95
Tadashi Udagawa
Classification of Smyth potentials . . . . . . . . . . . . . . . . . . . . . . . . 96

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.9

GEOMETRY OF RANDOM DEEP NEURAL NETWORKS:INFORMATION GEOMETRY POINT OF VIEW
SHUN-ICHI AMARI

Information geometry provides tools for analyzing information systems. In this talk,I will give a short summary of history of information geometry [1]. I will then focus
on recent researches on deep neural networks. First, Fisher information matrix is
calculated for a randomly connected neural network with suﬃciently large numbers ofneurons. It is shown that Fisher information matrix is asymptotically neuron-wisediagonal. It is shown recently that there exists a true target function in a smallneighborhood any network of random connections. This is a surprising fact shown by
the theory of neural tangent kernel [2]. We give a geometrical perspective of this fact[3]: When a very high-dimensional sphere of radius 1 is projected to a low-dimensional
space, the resultant is a 0-mean Gaussian distribution with suﬃciently small variances.
This explains the above fact.
References[1] S. Amari, Information Geometry and Its Applications. Springer, 2016.
[2] A. Jacot, F. Gabriel and C. Hongler, Neural tangent kernel: Convergence and generalization inneural networks, Neurips 2018, arXiv:1806.07572v3, (2018).
[3] S. Amari, Any target function exists in small neighborhoods of any randomly connected networks:
A geometric perspective, (2020), in preparation.
(S. Amari) RIKEN Center for Brain Science
Email address: amari@brain.riken.jp

Date: Received on January 9, 2020.

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.10–11

VISUALIZATION OF LINK FIBRATIONS AND CRITICAL POINTSBENJAMIN BODE

A link is the disjoint union of embedded circles in∪the 3-sphere S 3 . A braid on n
strands is the disjoint union of n parametric curves nj=1 (zj (t), t) ⊂ C × [0, 2π], t ∈[0, 2π], such that for each index i there is a unique j ∈ {1, 2, . . . , n} with zi (0) = zj (2π).
It is a well-known result by Alexander that every link (up to isotopy) can be realised
as the closure of a braid by identifying the t = 0 and t = 2π plane and embedding theresulting C × S 1 in R3 ⊂ S 3 using the standard embedding of the solid torus in R3 [1].
I am going to consider a special class of links, the so-called ﬁbered links. A link L isﬁbered if its complement S 3 \L ﬁbers over the circle, i.e., if there is a map ϕ : S 3 \L → S 1(in a certain homotopy class) that has no critical points.
Such critical points often have a physical interpretation, for example as the points
where a magnetic ﬁeld that is induced by an electric current through a knotted wire
vanishes [2], or as point defects of liquid crystals with knotted defect lines [4].
A good measure of how far a link L is from being ﬁbered is its Morse-Novikov number
MN (L). It is deﬁned as the minimal number of critical points of any S 1 -valued Morsefunction on S 3 \L in the relevant homotopy class [5]. In particular, MN (L) = 0 if andonly if L is ﬁbered.
In general, it is very diﬃcult to compute MN (L) or even give (good) bounds forthis link invariant. In my talk I am going to present a construction of S 1 -valued maps
on link complements, which can be used to give upper bounds for MN (L) [3]. This
construction is based on an interpretationof a braid as a loop in the space of monic∏complex polynomials gt (z) = nj=1 (z − zj (t)). Note that the roots of gt trace out the∪
braid nj=1 (zj (t), t), as t varies from 0 to 2π. Similarly, we will see that the criticalpoints and critical values of gt also can form braids and their topology not only tells
us something about the number of critical points of the eventual S 1 -valued map, butalso helps us visualise where the critical points occur and how the level sets (surfaces)
behave in a neighbourhood of a critical point.
References[1] J. Alexander, A lemma on a system of knotted curves, Proc. Nat. Acad. Sci. USA 9 (1923) 93–95.
[2] J. Binysh and G. P. Alexander, Maxwell’s theory of solid angle and the construction of knottedﬁelds, J. Phys. A: Math. Theor. 51 (2018), 385202.
[3] B. Bode and M. R. Dennis, Constructing a polynomial whose nodal set is any prescribed knot orlink, Journal of Knot Theory and Its Ramiﬁcations 28 (2019), 1850082.
[4] T. Machon, H. Aharoni, Y. Hu and R. D. Kamien, Aspects of defect topology in smectic liquid
crystals. Comm. Math. Phys. 372 (2019), 525–542.
Date: Received on February 14, 2020.
Key words and phrases. ﬁbered link, braid, critical point, Morse-Novikov number.

The 27th OCU Symposium[5] A. Pajitnov, L. Rudolph and L. K. Weber, The Morse-Novikov number for knots and links, Algebra
i Analiz 13 (2001) 105–118. Translation in St Petersburg Math J 13 (2002) 417–426.
Department of Mathematics, Osaka University, Toyonaka, Osaka 560-0043, Japan
Email address: ben.bode.2013@my.bristol.ac.uk

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.12–13

DEGENERATIONS OF ORBITS AND LEFT-INVARIANTSYMPLECTIC STRUCTURES ON LIE GROUPS
LUIS PEDRO CASTELLANOS MOSCOSO
Abstract. We are interested in the classiﬁcation or ﬁnding conditions for the existence of left-invariant symplectic structures on Lie groups. Some classiﬁcations are
known, especially in low dimensions. For studying the existence problem we provethat it is enough to study the closed orbits in the space of left-invariant nondegenerate 2-forms. As an application of this we ﬁnd a condition for the existence of such
structures in some particular almost abelian Lie groups.

1. Introduction
In the setting of Lie groups is very natural to ask about the existence of left-invariant
structures. A symplectic Lie group is a Lie group G endowed with a left-invariant
symplectic form ω (i.e. a nondegenerate closed 2-form). There are many interesting
results in the structure of symplectic Lie groups and considerable classiﬁcation eﬀorts
in low dimensions, but the general picture is far from complete.
Previously we approached this problem by studying the “moduli space of left-invariant
nondegenerate 2-forms” which is a certain orbit space in the set of all nondegenerate
2-forms on a Lie algebra g. However in most cases this moduli space is too big to be
useful, so we focus on a subset of the moduli space.
2. Main results
Let use the notation [ω] to denote an orbit in the moduli space and [ω] to denote thetopological closure.
Theorem 1 (Main Result). If ω1 is symplectic and [ω2 ] ⊂ [ω1 ], then ω2 is also symplectic.
So to study symplectic forms it is enough to look at closed orbits.
Then we apply this result to a particular family of almost abelian Lie algebras, obtaining
conditions for the existence of symplectic structures, improving a result by Ovando([2]).
References[1] Baues, O., Cortés V. : Symplectic Lie groups. Astérisque 379 (2016).
[2] Ovando, G. Four dimensional symplectic Lie algebras. Beiträge Algebra Geom. 47 (2006), 419–
434.
[3] Castellanos Moscoso, L. P. and Tamaru, H. A classiﬁcation of left-invariant symplectic structures
on some Lie groups. arXiv e-prints, 2020.
Date: Received on March 18, 2021.

The 27th OCU Symposium[4] Castellanos Moscoso, L. P. Degenerations of orbits and left-invariant symplectic structures on Liegroups. In progress.
Department of Mathematics, Graduate School of Science, Osaka City University
Email address: caste3.1416@gmail.com

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.14–18

LAND USE/LAND COVER CLASSIFICATION USING OBJECT AND
PIXEL BASED CLASSIFICATION APPROACHES WITH MEDIUMRESOLUTION SATELLITE IMAGES: A CASE STUDY IN LAO CAIPROVINCE, VIETNAM
HANG THI DO, VENKATESH RAGHAVAN, AND GO YONEZAWA
Abstract. This study aims to classify land use/ land cover of a part of Lao Cai
province in Vietnam using object and pixel based machine learning classiﬁcations.
The research investigates the capability of medium resolution satellite images such as
RapidEye, Sentinel-2 and Landsat-8 on rice terrace class classiﬁcation and land use/land cover mapping. Also, the comparison of pixel and object based approaches, aswell as the capability of the classiﬁcation methods is evaluated.

1. Introduction
Monitoring Land Use/Land Cover (LULC) situation is important in mitigating problems related biogeochemical cycles, loss of productive ecosystems and degradation ofagricultural lands. Several approaches to obtain LULC maps from Remote Sensing (RS)images have been applied with varying degree of success. Diverse resolution of remotelysensed data facilitates mapping LULC characteristics on local, regional and global
scales. Medium resolution optical RS data have been used to generate LULC maps
using Pixel Based Image Analysis (PBIA) or Object Based Image Analysis (OBIA)
approaches. However, there are only few studies on quantitative or comparative evaluation of either approaches in generating LULC maps of acceptable accuracy. The focusof this research is to apply PBIA and OBIA to medium resolution RapidEye, Sentinel-2
and Landsat-8 data for test area in Lao Cai province of Vietnam and evaluate theireﬃcacy to produce reliable rice terrace as well as LULC maps.
First of all, the study is aimed to classify a single rice terrace class in PBIA and OBIA.
Rice terrace is an important land cover which reduces soil erosion, mitigates landslide
hazard, controls water resources and oﬀers scenic beauty for attracting tourism. The
rice terrace class could be extracted from optical RS based on high brightness signatures(Galletti et al., 2013; Zhang et al., 2017). Most of studies about rice terrace classiﬁcation
have focused on edge characteristics and a distinct contrast with the surrounding naturalobjects obtained from very high-resolution RS data (Zhang et al., 2017). However,these data sets are costly and computationally intensive in monitoring rice terrace overlarge areas. Medium resolution RS images available as Open Data oﬀer a valuable
data source for regional monitoring. In this research, three Machine Learning (ML)Key words and phrases. Rice Terrace, Land Use Land Cover, Machine Learning, CNN.

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classiﬁers, namely, Multilayer Perceptron (MLP), Random Forest (RF) and Support
Vector Machine (SVM) are applied for the rice terrace classiﬁcation.
Secondly, LULC map of the study area is classiﬁed using PBIA and OBIA Light
Convolutional Neural Network (LCNN). The LCNNs use only three convolutional layers
and oﬀers a simpliﬁed workﬂow to produce accurate LULC classiﬁcation (Song et al.,2019). PBIA-LCNN and OBIA-LCNN models are applied on RapidEye, Sentinel-2
and Landsat-8 images covering the study area to generate seven LULC classes (water,
built-up, mining/bare land, rice ﬁeld, rice terrace, non-forest vegetation and forest).
2. Methodology
2.1. OBIA and machine learning classiﬁcations. In this study, rice terrace class
is identiﬁed using MLP, RF and SVM algorithms in PBIA and OBIA. In OBIA, image
segmentation techniques have been used frequently to process satellite sensor data.
This is the process of grouping neighbor pixels which have similar digital values toimage objects using threshold value (i) (Blaschke et al., 2004). Optimal threshold was
deﬁned as the value when the image is not over-segmented or under-segmented (Kim etal.,2008), obtained by using ROC-LV measurement (Bauer et al., 1998; Dragut et al.,2014).
The binary classiﬁcation is investigated to identify two classes: rice terrace andothers (non-rice terrace). In the case of MLP, 8 hidden layers which include 128 nodes
for each layer is setup. Rectiﬁed Linear unit (ReLu) is chosen as an activation function.
In order to prevent overﬁtting, dropout, L1 and L2 regularization methods are chosen.
Dropout ratio is set as 0.2 and L1, L2 at 10-5 , 10-5 , respectively. Log Loss metric isutilized to decide the most eﬃcient model. The network is implemented in R StatisticalComputing environment using Open Source machine learning platform H2O package.
RF classiﬁer is implemented using Caret (Classiﬁcation and Regression Training) R
package which includes a set of functions that attempt to streamline the process forcreating predictive models. Two parameters in RF method: the number of decision
trees (Ntree) is chosen to be 500, while the number of variables in the random subset
at each node (Mtry) is as default.
As same as RF, SVM SVM model is implemented using Caret. Radial Basic Function(RBF) kernel is chosen as it is commonly used. K-fold cross-validation is applied toestimate model accuracy. The size of the subset equals 9. It may not the best value ofsubset size, but good enough for the study.
2.2. PBIA-LCNN and OBIA-LCNN. Traditional Convolutional Neural Network is
widely applied for LULC classiﬁcation. However, a deep network can lead to overﬁtting
and time-consuming. To ignore these drawbacks, the application of a LCNN for land
cover classiﬁcation was introduced (Song et al., 2019).
In this study, PBIA-LCNN and OBIA-LCNN are proposed for LULC classiﬁcation.
The two models have three convolutional layers. The ﬁrst convolutional layer of PBIALCNN ﬁlters the 3D input with 20 ﬁlters of size 1 × 1 × number of bands of the RS
data. In case of OBIA-LCNN, the ﬁrst convolutional layer includes 20 ﬁlters of size 3× 3 × the number of bands of the RS data. The second and third convolutional layers
of PBIA-LCNN and OBIA-LCNN have 20 ﬁlters of size 2 × 2 × 20. Zero padding

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and stride equals 1 are employed. The last Softmax layer which provides a probability
distribution over 7 LULC classes. Fully connected layers and pooling layers are not
employed. Instead, ReLU activation function and Adam optimizer with a learning rateof 10-5 are used. The two networks are implemented for RapidEye, Sentinel-2 andLandsat-8 images to classify LULC maps for the study area. 5 bands of RapidEye,
4 bands of Sentinel-2 and 6 bands of Landsat-8 are used as inputs. The number of
epochs equals to 100 and early stopping technique is applied, the training processing
stops when the diﬀerent between two consecutive loss (r) is lower or equals to 10-6 . The
models are implemented using Google Colab.
3. Result and Discussion
3.1. Rice terrace classiﬁcation. Rice terrace class classiﬁcation achieves accurate
results using RapidEye image. In all three ML classiﬁers, PBIA provides the mostaccurate results, at 91.9%, 92.0%, 92.0% for MLP, RF and SVM classiﬁers, respectively.
In case of OBIA, the classiﬁcation accuracies achieve the highest values at the ﬁrst peakof the ROC-LV graph, at 90%, 90.3%, 89.8% of MLP, RF and SVM, respectively. In
general, the values show a slight decrease when i increases. In both PBIA and OBIA,
RF classiﬁer works better than SVM and MLP.
PBIA attains the most accurate results for Sentinel-2 image classiﬁcation, at 91.3%,
90.3%, 91.2% for MLP, RF and SVM, respectively. The ﬁrst peak of ROC-LV graphof OBIA presents the best result, at 86.1%, 85.2% of RF and SVM, respectively. MLP
produces the highest overall accuracy at the fourth peak (84.7%), slightly higher thanthe value at the ﬁrst peak (84.6%). In general, the classiﬁcation accuracies slightly ﬂuctuate in a downward trend when i increases. The diﬀerences in classiﬁcation accuracy
among three classiﬁers at the same approach are small.
Overall accuracies for Landsat-8 classiﬁcation of PBIA are at 89.7% for MLP, 89.2%
for RF and 89.9% for SVM, higher than all OBIA. OBIA yields high accuracy at the ﬁrst
peak of ROC-LV graph, at 83.3%, 83.4%, 83.9% of MLP, SVM and RF, respectively.
The classiﬁcation accuracies signiﬁcantly decline from PBIA to the second peak of
ROC-LV graph and subsequently show gradual decline of accuracies.
3.2. LULC classiﬁcation. The results of LULC classiﬁcation using PBIA-LCNN and
OBIA-LCNN show that, PBIA and OBIA produce the same overall classiﬁcation accuracy using RapidEye image, at 94%. Rice terrace and forest classes achieve excellent
accuracies, while mining/bare land, built-up, paddy ﬁeld and non-forest vegetation
attain lower accuracies but higher than 87%.
In the case of Sentinel-2, PBIA and OBIA achieve the same overall classiﬁcation
accuracy, at 91%. However, the accuracies of LULC classes are diﬀerent. In detail,water, built-up and rice terrace classes obtain higher than 90%, other LULC classes
gain lower accuracies, while OBIA produces higher than 90% classiﬁcation accuracy forbuilt-up, forest and rice terrace classes.
Both PBIA and OBIA gain accurate classiﬁcation result with 96% of overall accuracy
using Landsat-8 image. All classiﬁed classes in PBIA acquire high accuracy, higher thanor equal to 93%. Similar to PBIA, all LULC classes get accurate classiﬁcation result,higher or equal to 92%.

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3.3. Discussion. The results of this research show that MLP, RF, and SVM classiﬁers
produces higher classiﬁcation accuracy for rice terrace in PBIA than OBIA. RapidEyeachieves the most accurate result, follow by Sentinel-2 and Landsat-8. In comparison of
ML classiﬁer, SVM attains slightly higher classiﬁcation accuracies than MLP and RF.
In the case of LULC mapping, PBIA-LCNN and OBIA-LCNN models produce similar
LULC classiﬁcation accuracy. In comparison of RS images, while Landsat-8 achieves
the most accurate classiﬁcation result, following by RapidEye and Sentinel-2. The
spatial resolution of Sentinel-2 is ﬁner than Landsat-8 and coarser than RapidEye, but
its classiﬁcation result does not perform well when compared to other data sets. The
reason for this could be due to spectral bands used and/or because of diﬀerence inresolution of three RS datasets. Fewer bands used for Sentinel-2 as compared to with
RapidEye and Landsat-8 may be one of the reasons for lower classiﬁcation accuracy incase of Sentinel-2.
The research also provides a simple and eﬀective workﬂow for a single rice terrace
class as well as LULC mapping and provide useful inputs for decision making andplanning. The methodologies have been entirely implemented using Free and Open
Source Software and their eﬃcacies demonstrated with freely accessible Open Data withnear global coverage. Thereby, the workﬂow can be easily adopted and reﬁned to suita variety of geographical locations, data sets and user needs. Validating the proposed
methodologies with multi-spectral UAV (Unmanned Aerial Vehicle) imagery could help
in providing high resolution LULC information and overcome some of the limitationsof optical satellite data such as temporal resolution, cloud cover and timeliness.
References[1] Bauer, R. J. and Dahlquist, J. R. (1998) Technical market indicators: analysis and performance.
Wiley, New York.
[2] Blaschke, T., Burnett, C. and Pekkarinen, A. (2004) Image segmentation methods for object-based
analysis and classiﬁcation. In: Jong S.M.D., Meer F.D.V. (eds) Remote sensing image analysis:
Including the spatial domain. Remote Sensing and Digital Image Processing, vol. 5. Springer,
Dordrecht.
[3] Dragut, L., Csillik, O., Eisank, C. and Tiede, D. (2014) Automated parameterization for multiscale image segmentation on multiple layers. ISPRS Journal of Photogrammetry and Remote
Sensing, vol. 88, pp. 119-127.
[4] Galletti, C. S., Ridder, E., Falconer, S. E. and Fall, P. L. (2013) Maxent modeling of ancient andmodern agricultural terraces in the Troodos foothills, Cyprus. Applied Geography, vol. 39, pp.
46–56.
[5] Kim, M., Madden, M. and Warner, T. (2008) Estimation of optimal image object size for thesegmentation of forest stands with multispectral IKONOS imagery. In: T. Blaschke, S. Lang,and G.J. Hay, eds. Object-based Image Analysis–spatial Concepts for Knowledge Driven Remote
Sensing Applications, Springer, Berlin, pp. 291–307.
[6] Knorn, J., Rabe, A., Radeloﬀ, V. C., Kuemmerle, T., Kozak, J. and Hostert, P. (2009) Land
cover mapping of large areas using chain classiﬁcation of neighboring Landsat-8 satellite images.
Remote Sensing of Environment, vol. 113, no. 5, pp. 957–964.
[7] Song, J., Gao. S., Zhu, Y. and Ma, C. (2019) A survey of remote sensing image classiﬁcation
based on CNNs. Big Earth Data, vol. 3, no. 3, pp. 232-254.
[8] Zhang, Y., Shi, M., Zhao, X., Wang, X., Luo, Y. and Zhao, Y. (2017) Methods for automatic
identiﬁcation and extraction of terraces from high spatial resolution satellite data (China-GF-1).
International Soil and Water Conservation Research, vol. 5, no. 1, pp. 17–25.

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OCAMI Reports Vol. 1 (2021)(Hang Thi Do) Graduate School of Engineering, Osaka City University
Email address: dohanghumg@gmail.com(Venkatesh Raghavan) Graduate School of Engineering, Osaka City University
Email address: raghavan@osaka-cu.ac.jp(Go Yonezawa) Graduate School of Engineering, Osaka City University
Email address: yonezawa@osaka-cu.ac.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.19

MATHLIBRE: MATHEMATICAL SOFTWARE ENVIRONMENTTATSUYOSHI HAMADA
Abstract. MathLibre is a project to archive open source mathematical software andfree mathematical documents and oﬀer them on Live Linux system.

MathLibre is a project to archive open source mathematical software and free mathematical documents and oﬀer them on Live Linux system. MathLibre Project is the
direct descendant of KNOPPIX/Math Project. It provides a desktop for mathematicians that can be set up easily and quickly. If your machine is not DVD bootable, or
has very special hardware devices which MathLibre cannot drive, we recommend you
download virtual machine like Virtual Box . Once you have installed the virtual machine, you can start our live system from DVD or from ISO image ﬁle. The instructions
for installing and using the virtual machine can be found in our DVD.
MathLibre includes over 100 mathematical software packages, i.e. computer algebra
systems, programming environment, visualization libraries for research and educationaluse. Anyone can explore the world of mathematical software systems.
References[1] Tatsuyoshi Hamada, “MathLibre: modiﬁable desktop environment for mathematics”, Mathematical Software – ICMS2014, Lecture Notes in Computer Science 8592, Springer, p702–705.
(T. Hamada) Nihon University
Email address: hamada.tatsuyoshi@nihon-u.ac.jp

Date: Received on February 15, 2020.
Key words and phrases. Mathematical Software, Open Source Software.
This work was supported by JSPS KAKENHI Grant Number JP16K05139.

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.20–21

TOPOLOGICAL INVARIANTS AND CORNER STATES FOR
HAMILTONIANS ON A LATTICE
SHIN HAYASHI

Abstract
In condensed matter physics, Topology takes much interest. It is known that there
exist some materials that, although their interior (bulk) is insulating, its (codimensionone) edge behaves like a metal (see Figure 1). This can be explained as follows: insulating bulk possesses some topology and, corresponding to it, there appear edge localizedstates. The existence of edge states explains its metallic behavior around the edge.
This correspondence of bulk and edge is called the bulk-edge correspondence and wasﬁrst proved by Hatsugai, and such materials are called topological insulators. A mathematical aspect of the bulk-edge correspondence can be stated as a relation between two
topological invariants (one for bulk, and the other for edge) and can be shown by using
index theory: Kellendonk-Richter-Schulz-Baldes proved the bulk-edge correspondence
by using the index theory for Toeplitz operators.
In this poster, we mainly consider Hamiltonians on systems with (codimension-two)corners which appear as an intersection of two edges (see Figure 2). We show that, whenboth bulk and two edges are insulating, there exists some topology, and corresponding
to it, there exists (topologically protected) corner states [2]. For its proof, we use index
theory for quarter-plane Toeplitz operators developed by Simonenko, Douglas–Howe,
Park and others. A construction of non-trivial examples by using Hamiltonians of twoconventional topological insulators will also be presented.
Recently in condensed matter physics, such systems possessing topologically protected corner states are actively studied with its relation to higher-order topologicalinsulators [1]. A (variant of) our theory explains a two dimensional model of higherorder topological insulators proposed by Benalcazar–Bernevig–Hughes [1, 3]. In thissense, our discussion can be seen as a mathematical approach to higher-order topological insulators.
References[1] W.A. Benalcazar, B.A. Bernevig and T.L. Hughes, Quantized electric multipole insulators, Science
357 (2017), 61-66.
[2] S. Hayashi, Topological invariants and corner states for Hamiltonians on a three-dimensionallattice, Comm. Math. Phys. 364 (2018), no. 1, 343-356.
Date: Received on February 20, 2020.
Key words and phrases. Higher-order topological phases, K-theory and index theory.
This work was supported by JSPS KAKENHI (Grant Nos. JP17H06461, JP19K14545) and JST
PRESTO (Grant No. JPMJPR19L7).

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Figure 1. Bulk and edge

21

Figure 2. Bulk, edges
and corner[3] S. Hayashi, Toeplitz operators on concave corners and topologically protected corner states, Lett.
Math. Phys. 109 (2019) no. 10, 2223-2254.
(S. Hayashi) Mathematics for Advanced Materials-OIL, National Institute of Advanced Industrial Science and Technology/PRESTO, JST
Email address: shin-hayashi@aist.go.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.22–23

A FAMILY OF INDEX POLYNOMIAL INVARIANTS FOR VIRTUAL
STRING LINKSHIROKI ITO
Abstract. Virtual string links are extentions of virtual links and classical string
links. In this talk, we introduce a family of index polynomial invariants for ncomponent virtual string links. Moreover, we give a method of distinguishing virtual
string links from their vertical mirror images.

1. Virtual string links and index polynomials
Fix n distinct point 0 < x1 < · · · < xn < 1. Let [0, 1]1 , . . . , [0, 1]n be n copies of [0, 1].
Then, an n-component virtual string link diagram is the image of an immersionn⊔[0, 1]i → [0, 1] × [0, 1]
i=1such that the image of [0, 1]i runs from (xi , 0) to (xi , 1), and its singular points are
positive, negative and virtual crossings.
An n-component virtual string link is an equivalence class of n-component virtual
string link diagrams under virtual isotopy.
Deﬁnition 1. Let A(D) := { semi arcs of D }. Then, a map c : A(D) → Z is a
Cheng coloring of D if the following two conditions are satisﬁed.
(1) Each crossing of D satisﬁes the condition as in Figure 1.
(2) The outgoing semi arc from (xi , 0) values 0.

Figure 1

Figure 2

Deﬁnition 2. Suppose that a Cheng coloring is given to a diagram D. For eachcrossing τ of D, W (τ, D) := b − a, where a, b are values of semi arcs depicted in Figure
3
Date: Received on March 15, 2021.
Key words and phrases. virtual link, string link, polynomial invariant.

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Figure 3
Deﬁnition 3. For any i, j ∈ {1, . . . , n},Tij (D) := { crossings∑of D that under arc is i-th component }
ϕD
(t)
:=
sign(τ )(tW (τ,D) − 1) ∈ Z[t±1 ]ijτ ∈Tij (D)2. Main results
We have the following theorem.
Theorem 4. If D, D′ are diagrams of an n-component virtual string link, then forD′
any i, j ∈ {1, . . . , n}, ϕDij (t) = ϕij (t)Therefore, ϕDij (t) is an invariant for virtual string links. Moreover, we obtain thefollowing proposition.
Proposition 5. Let D be an n-component virtual string link diagam and Dv be adiagram which is changed upper/lower information of all crossings of D. Then, for any
Dv −1
i, j ∈ {1, . . . , n}, ϕDij (t) = −ϕji (t )References[1] Louis H. Kauﬀman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663–690, DOI
10.1006/eujc.1999.0314.
[2] Louis. H. Kauﬀman, An aﬃne index polynomial invariant of virtual knots, J. Knot Theory Ramiﬁcations 22 (4) (2013) 1340007.
[3] Zhiyun Cheng. A polynomial invariant of virtual knots. Proc Amer Math Soc, 2014, 142[4] Zhiyun Cheng, Hongzhu Gao. A polynomial invariant of virtual links. J Knot Theory Ramiﬁcations, 2013, 22: 1341002.
(H. Ito) Osaka University
Email address: u063241a@ecs.osaka-u.ac.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.24–28

EVALUATING SPATIO-TEMPORAL EXPANSION PATTERNS AND
URBAN GROWTH MODELS: A CASE STUDY OF COLOMBO
CITY, SRI LANKAPAVITHRA JAYASINGHE, VENKATESH RAGHAVAN, AND GO YONEZAWA
Abstract. This research focuses on investigating urban expansion pattern of Colombo
City, Sri Lanka during 1997-2019 and simulating urban growth for 2030. The study
comparatively evaluates simulation results generated by three Open Source urban
growth Models namely, FUTURES, SLEUTH and MOLUSCE. Derived future urban
growth for Colombo City in 2030 is compared in light of proposed development plan.

1. Introduction
In broad context of investigating urban growth, numerous qualitative and quantitative techniques for detection of land change patterns have been proposed in numerous
studies. Intensity analysis is applied to decipher size and rate of change across timeintervals and spatial pattern analysis is used to understand urban growth dynamics considering outlying, edge-expansion and inﬁlling as main patterns to characterize urban
expansion.
In the ﬁrst stage of this research, intensity and spatial pattern analysis were applied tocharacterize the urban growth in Colombo City and its environs. Colombo is a rapidly
urbanizing coastal city in Sri Lanka. Land change intensity analysis [1] examines how
the size and speed of change vary across time intervals. Spatial pattern analysis is an
important topic in contemporary urban change studies [2,3]. Further, growth types
characterized by the spatial patterns such as outlying, edge-expansion, and inﬁlling[4] have been used to deﬁne the Landscape Expansion Index (LEI) as a quantitative
estimate of urban expansion [5]. These two methods; (i.e. intensity analysis and spatial
pattern analysis) provide an enhanced understanding of urban expansion of cities.
Although some similarities exist, these models vary in algorithms used, modelling
framework and assumptions made. Among deterministic and stochastic pattern based
spatial allocation models, stochastic models are gaining popularity due to their capability of accounting randomness and competency to test scenarios of alternative futures[6]. In the context of planning and decision making, particularly useful feature of landchange models is their ability to help explore diﬀerent scenarios and better understandtrade-oﬀs between diﬀerent planning interventions [7]. First, study uses the FUTURES
model as it bridges the gap between cell and object-based representation and aﬀords
exploration of diverse scenarios [6]. FUTURES is a stochastic patch-based, multilevel
Date: Received on March 6, 2021.
Key words and phrases. Urban growth, Simulation, FUTURES, SLEUTH, MOLUSCE.

The 27th OCU Symposium

modeling framework for simulating spatial landscape in urbanizing regions. FUTURES
compatibility in developing countries context was tested by applying it in Colombo City
Sri Lanka. It was validated using Remote Sensing derived urban growth map 2019 andthen applied to predict 2030 urban expansion.
However, numerous urban growth simulation models with diverse sophistication havebeen developed and demonstrated the high competency in projecting urban growth[6,8,9,10]. Selecting an appropriate model that serves the purpose of the study isoften arbitrary or depends upon the availability of inputs required to run the modelor researcher s decision. There are considerable studies that provide guidance forresearchers in understanding how accurate a model can replicate the urban systems.
Lack of studies comparing models limits the ability to understand the trade-oﬀs amongmodel accuracies and therefore creates a critical research gap [11]. Consequently, acomparative evaluation of models to provides a guidance in selecting an urban growthmodel and highlight trade-oﬀs between selecting one model over another is of great
importance. Therefore, the FUTURES model was compared with two more Open
Source urban growth models namely, MOLUSCE and SLEUTH.
Finally, model simulation results were compared with existing development plan WRMMP, prepared for Western Province of Sri Lanka where Colombo is located. Recommendations and suggestions were made reﬂecting the simulation results generated bythree models for 2030 urban growth.
2. Methodology and Results
Supervised Classiﬁcation which groups satellite image pixels with the same or similar spectral reﬂectance characteristics into same information categories was performed
to derive Land Cover (LC) maps of the study area from Landsat TM/OLI imageries.
K-Nearest neighbor method was used to generate LC maps for the years 1997, 2005,
2008, 2014, 2019 with overall accuracies 86.56%, 88.77%, 90.07%, 93.03%, 93.22% respectively.
In the ﬁrst stage of this research, LC change detection was carried out using derived
LC maps in 1997, 2008, 2019 considering regular time gap. Intensity analysis andPattern analysis was performed under LC change detection for better understand thequantitative and qualitative measures of spatiotemporal urban expansion of Colombo
City. Intensity analysis revealed that urban lands in Colombo has grown in a faster rateduring 1997-2008 than in 2008-2019 period. Results of urban growth pattern analysis
shows that between 1997-2008, out of the 4,924 new patches of urban class in 2008,
58.14% were classiﬁed as outlying, while 18.81% were classiﬁed as inﬁlling. During 20082019, the number of new urban patches in 2019 increased to 10,750. Of these, 50.08%
were classiﬁed as edge-expansion, while outlying type has become much less dominant(22.39%). These results indicate that spatial patterns of urban land change have evolved
from being predominantly outlying during 1997-2008 to edge-expansion during 20082019. In the second stage, diverse modelling approaches were investigated to predict
future changes to urban systems in Colombo City and surrounding areas. Initially,
FUTURES model was applied to identify urban expansion patterns. FUTURES is a
patch-based, multilevel modeling framework for simulating the emergence of landscape

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OCAMI Reports Vol. 1 (2021)

spatial structure in urbanizing regions. Urban transitions are simulated based on threesub-models namely, POTENTIAL, DEMAND and Patch Growing Algorithm. In theinitial stage the model was applied to simulate the urban growth during 2014-2019
for model validation purpose. Simulation result and validation metrices obtained byapplying FUTURES are in close agreement with observed urban growth.
FUTURES was used to predict three diﬀerent growth scenarios, namely Business as
Usual (BaU), Inﬁll growth and Sprawl and results of all three scenarios show over 100
km2 increase in urban lands by 2030. According to the simulation results generated
under BaU which assumes the continuation of current growth trend, Colombo is posed
to experience inﬁlling type urban growth over the coming decade. The emerging urban
patches will concentrate around two major centers, namely, Kadawatha and Kottawa.
Based on simulation results, Kadawatha and Kottawa are recommended to be developed
as 2nd tier towns along with other 2nd tier proposed in 2030 Western Region Megapolis
Master Plan.
In the third stage of the research, the results of the simulations were compared with
two other Cellular Automata models namely, MOLUSCE and SLEUTH. The objective
was to not only compare the simulation results using diverse approaches but also toevaluate the performance, strengths, and weaknesses of FUTURES, MOLUSCE and
SLEUTH models. Three models were also assessed in terms of possibility of integrating
similar explanatory variables and policy scenarios.
The comparative evaluation FUTURES, MOLUSCE and SLEUTH models was carried out in two phases. In Phase One, the simplest state for models was considered andin Phase Two, identical inputs were used to generate urban growth potential surface
based on Logistic Regression (LR). The observed urban area expansion extracted fromsatellite images during 2008-2019 was 127.37 km2 . In Phase One, the simulated urban growth with FUTURES, MOLUSCE and SLEUTH were 148.91 km2 , 250.55 km2
and 77.10 km2 respectively. The results indicate that SLEUTH model showed an overestimation and MOLUSCE model showed an under estimation while FUTURE model
simulation closely corresponds with observed growth. Accuracy assessment and validation metrics evaluated for Phase One results indicate that FUTURES model provides
better results. In Phase Two of simulation, the LR was selected for potential surface
identiﬁcation in MOLUSCE and SLEUTH as it is also used in FUTURES model. As aresult, the overall performance of SLEUTH and MOLUSCE show signiﬁcant improvements. The SLEUTH and MOLUSCE model simulated urban growth as 156.97 km2
and 135.88 km2 respectively which correspond with observed growth. Accuracy assessment and validation metrices indicate that SLEUTH provides better prediction results
due to inclusion of LR in Phase Two. Spatial pattern analysis for simulated growth by
three models was carried out and compared with observed landscape expansion patterns in 2019. The results indicate that FUTURES and MOLUSCE models correctly
simulated the prominent pattern as edge expansion which was observed in 2019 while
SLEUTH model simulated it as inﬁlling.
With improved prediction accuracy with LR, three models were used to simulate
urban growth for 2030. Comparison of simulated urban growth for 2030 by threemodels show dissimilarities in terms of areal extent and spatial allocation. Total urban

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expansion during 2019-2030 is 119.59 km2 , 197.78 km2 , and 78.51 km2 as predictedby FUTURES, SLEUTH and MOLUSCE models respectively. Growth predicted by
FUTURES closely matches with the average annual urban expansion during the past
two decades (13.4 km2 ) while SLEUTH overestimates and MOLUSCE underestimates.
In simulation for 2030, inﬁlling is predicted as the prominent growth pattern by all threemodels. While FUTURES model allocates expansion around existing urban centers,
SLEUTH and MOLUSCE mainly indicate outer peripheral growth.
Considering the tradeoﬀ between computational overheads and results it is concluded
that FUTURES oﬀers a robust, easily customizable model with ﬂexibility to address
disparate scenarios. The simulation models applied in this research reveal new information on growth intensity as well as patterns and provide valuable inputs for sustainableurban development planning of Colombo City. Further, the successful application ofFree and Open Source software and Open Data in the present research facilitates easy
adoption of the methodology in other rapidly urbanizing areas.

References[1] Aldwaik, S. and Pontius, R. (2012) Intensity analysis to unify measurements of size and stationarity of land changes by interval, category, and transition. Landscape and Urban Planning, vol.
106, no. 1, pp. 103-114.
[2] Linard, C., Tatem, A. and Gilbert, M. (2013). Modelling spatial patterns of urban growth inAfrica. Applied Geography, vol. 44, pp. 23–32.
[3] Thapa, R. and Murayama, Y. (2012). Scenario based urban growth allocation in Kathmandu
Valley, Nepal. Landscape and Urban Planning, vol. 105, no.1–2, 140–148.
[4] Dietzel, C., Oguz, H., Hemphill, J., Clarke, K. and Gazulis, N. (2005) Diﬀusion and Coalescenceof the Houston Metropolitan Area: Evidence Supporting a New Urban Theory. Environment andPlanning B: Planning and Design, vol. 32, no. 2, pp. 231-246.
[5] Liu, X., Li, X., Chen, Y., Tan, Z., Li, S. and Ai, B. (2010) A new landscape index for quantifying
urban expansion using multi-temporal remotely sensed data. Landscape Ecology, vol. 25, no. 5,
pp. 671-682.
[6] Meentemeyer, R., Tang, W., Dorning, M., Vogler, J., Cunniﬀe, N. and Shoemaker, D. (2013)
FUTURES: Multilevel Simulations of Emerging Urban–Rural Landscape Structure Using a Stochastic Patch-Growing Algorithm. Annals of the Association of American Geographers, vol. 103,
no. 4, pp. 785-807.
[7] Dorning, M., Koch, J., Shoemaker, D. and Meentemeyer, R. (2015) Simulating urbanization scenarios reveals tradeoﬀs between conservation planning strategies. Landscape and Urban Planning,
vol. 136, pp. 28-39.
[8] Clarke, K., Hoppen, S. and Gaydos, L. (1997) A Self-Modifying Cellular Automaton Model ofHistorical Urbanization in the San Francisco Bay Area. Environment and Planning B: Planningand Design vol. 24, no. 2, pp. 247-261.
[9] Jin, J. and Lee H. Y. (2018) Understanding residential location choices: an application of the
UrbanSim residential location model on Suwon, Korea. International Journal of Urban Sciences
vol. 22, no. 2, pp. 216-235.
[10] Van Berkel, D., Shashidharan, A., Mordecai, R., Vatsavai, R., Petrasova, A., Petras, V., Mitasova,
H., Vogler, J.B. and Meentemeyer, R. (2019) Projecting Urbanization and Landscape Change at
Large Scale Using the FUTURES Model. Land, vol. 8, no.10, pp. 1-28.
[11] Pickard, B., Gray, J. and Meentemeyer, R. (2017) Comparing Quantity, Allocation and Conﬁguration Accuracy of Multiple Land Change Models. Land, vol. 6, no. 3, pp. 1-21.

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OCAMI Reports Vol. 1 (2021)(Pavithra Jayasinghe) Graduate School for Creative Cities, Osaka City University
Email address: d18ud508@yj.osaka-cu.ac.jp(Venkatesh Raghavan) Graduate School of Engineering, Osaka City University
Email address: raghavan@osaka-cu.ac.jp(Go Yonezawa) Graduate School of Engineering, Osaka City University
Email address: yonezawa@osaka-cu.ac.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.29

A SPATIAL ARC AND THE KNOTTING PROBABILITYAKIO KAWAUCHI
Abstract. It is explained how the knotting probability is deﬁned for an oriented
polygonal spatial arc.

1. Introduction
The knotting probability of an arc diagram is deﬁned in [1] as the quadruplet of fourkinds of ﬁnner knotting probabilities which are invariant under a reasonable deformation
containing an isomorphism on an arc diagram. The deﬁnition of the knotting probabilityof an arc diagram uses the fact that every arc diagram induces a unique chord diagram
representing a ribbon 2-knot in 4-space. Then the knotting probability of an arc diagram
is set to measure how many non-trivial ribbon genus 2 surface-knots occur from thechord diagram induced from the arc diagram.
It is also shown in [2] that the projection image of an oriented spatial polygonal arc
to any oriented plane is approximated by an arc diagram uniquely determined by thespatial arc and a projection direction (up to isomorphic arc diagrams). By using theknotting probability of an arc diagram, the knotting probability of a spatial polygonalarc is deﬁned.
The canonical 12 arc diagrams determined uniquely by the sptatial arc such that theorientation change of the spatial arc makes only a substitute for these arc diagrams (upto isomorphic arc diagrams) are highlighted for a computation.
References[1] A. Kawauchi, Knotting probability of an arc diagram, J. Knot Theory Ramiﬁcations (to appear).
https://doi.org/10.1142/S0218216520420043,
http://www.sci.osaka-cu.ac.jp/ kawauchi/diagramknottingprobability.pdf[2] A. Kawauchi, Unique Diagram of a spatial arc and the knotting probability.
https://arxiv.org/abs/1907.10194,
http://www.sci.osaka-cu.ac.jp/ kawauchi/arcknottingprobability.pdf(OCAMI) Osaka City University Advanced Mathematical Institute
Email address: kawauchi@sci.osaka-cu.ac.jp

Date: Received on March 3, 2021.
Key words and phrases. Arc diagram, Spatial arc, Knotting probability.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849) and JSPS
KAKENHI Grant Numbers JP16H03928,JP19H01788.

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.30–31

GEOMETRIC AND CONSTRUCTIVE UNDERSTANDING OFHIGHER DIMENSIONAL DIFFERENTIABLE MANIFOLDS VIAEXPLICIT MORSE FUNCTIONS AND FOLD MAPSNAOKI KITAZAWA
Abstract. We will introduce new studies on understanding higher dimensional differentiable manifolds in geometric and constructive ways. We will use fold maps, which
are regarded as higher dimensional versions of Morse functions. We construct explicit
fold maps on 7-dimensional and higher dimensional closed and simply-connected manifolds and see their topologies and diﬀerentiable structures.

Investigating and understanding the topologies and the diﬀerentiable structures ofdiﬀerentiable manifolds is fundamental and important in geometry and mathematics.
Morse functions, existing densely on closed smooth manifolds (diﬀerentiable manifolds of class C ∞ ), tell us homology groups and some information on homotopy of the
manifolds. More precisely, singular points, appearing discretely, have such information.
As a simplest case, Reeb’s theorem implies that Morse functions with exactly two singular points on closed manifolds characterize spheres topologically except 4-dimensional
cases: in these cases, such functions characterize 4-dimensional standard spheres. Thisis so called classical Morse theory. See [8] and [9] for example.
The author is interested in understanding higher dimensional smooth manifolds ingeometric and constructive ways. Via sophisticated methods such as algebraic topological methods and ones based on surgery theory, in a sense they were considerably
understood already in 1970s: the assumption that the manifolds are high dimensional
yielded such facts. However, this assumption makes the manifolds diﬃcult to understand in geometric and constructive ways.
The author believes that new development in higher dimensional versions of the classical theory of Morse functions will give us strong tools in the understanding. Related
studies were ﬁrst launched in 1950s by Thom ([14]) and Whitney ([15]). Since 1990s,
Saeki, Sakuma, and so on, have developed the theory and apply it to understanding
diﬀerentiable structures of homotopy spheres and various closed 4-dimensional manifolds. More precisely, they used fold maps, which are regarded as higher dimensionalversions of Morse functions including special generic maps, which are regarded as higher
dimensional versions of Morse functions on homotopy spheres with exactly two singularDate: Received on February 18, 2020.
Key words and phrases. fold maps, Reeb spaces, (co)homology, diﬀerentiable structures.
This work was partly supported by the project Grant-in-Aid for Scientiﬁc Research (S) (17H06128
Principal Investigator: Osamu Saeki) ”Innovative research of geometric topology and singularities ofdiﬀerentiable mappings”.

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points: canonical projections of unit spheres are also special generic. For example, exotic homotopy spheres do not admit special generic maps into Euclidean spaces whose
dimensions are larger than 2 in general. See also [10], [11], [12], [13], [16], and so on.
In this presentation, as related new studies, for example, we construct explicit foldmaps on 7-dimensional or higher dimensional closed and simply-connected manifolds
such as manifolds in [6], [7] and so on. We also present their homology groups, cohomology rings, classical characteristic classes, diﬀerentiable structures, and so on. The
content is based on [1], [2], [3], [4], [5], and so on.
References[1] N. Kitazawa, On round fold maps (in Japanese), RIMS Kokyuroku Bessatsu B38 (2013), 45–59.
[2] N. Kitazawa, Fold maps with singular value sets of concentric spheres, Hokkaido Mathematical
Journal Vol.43, No.3 (2014), 327–359.
[3] N. Kitazawa, Constructing fold maps by surgery operations and homological information of their
Reeb spaces, submitted to a refereed journal, arxiv:1508.05630 (the title has been changed).
[4] N. Kitazawa, Notes on fold maps obtained by surgery operations and algebraic information of
their Reeb spaces, arxiv:1811.04080.
[5] N. Kitazawa, Notes on explicit smooth maps on 7-dimensional manifolds into the 4-dimensional
Euclidean space, arxiv:1911.11274.
[6] M. Kreck, On the classiﬁcation of 1-connected 7-manifolds with torsion free second homology, toappear in the Journal of Topology, arxiv:1805.02391.
[7] J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64 (1956), 399–405.
[8] J. Milnor, Lectures on the h-cobordism theorem, Math. Notes, Princeton Univ. Press, Princeton,
N.J. 1965.
[9] G. Reeb, Sur les points singuliers dùne forme de Pfaﬀ completement integrable ou d une fonction
numerique, -C. R. A. S. Paris 222 (1946), 847–849.
[10] O. Saeki, Notes on the topology of folds, J. Math. Soc. Japan Volume 44, Number 3 (1992),
551–566.
[11] O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl.
49 (1993), 265–293.
[12] O. Saeki and K. Sakuma, On special generic maps into R3 , Paciﬁc J. Math. 184 (1998), 175–193.
[13] O. Saeki and K. Sakuma, Special generic maps of 4-manifolds and compact complex analytic
surfaces, Math. Ann. 313, 617–633, 1999.
[14] R. Thom, Les singularites des applications diﬀerentiables, Ann. Inst. Fourier (Grenoble) 6 (195556), 43–87.
[15] H. Whitney, On singularities of mappings of Euclidean spaces: I, mappings of the plane into theplane, Ann. of Math. 62 (1955), 374–410.
[16] D. J. Wrazidlo, Standard special generic maps of homotopy spheres into Eucidean spaces, Topology
Appl. 234 (2018), 348–358, arxiv:1707.08646.
(Institute of Mathematics, Kyushu University) Institute of Mathematics, Kyushu University
Email address: n-kitazawa@imi.kyushu-u.ac.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.32

SOLUTION TO THE REFLECTION EQUATION IN THE
SYMMETRIC TENSOR REPRESENTATIONHIROTO KUSANO

The reﬂection equation appeared originally in the problem of the integrable systems
in mathematical physics. It reads as RKRK = KRKR, where R and K are matrices
encoding the interactions in the bulk and at the boundary of the integrable system.
The aim of this talk is to provide a solution to the reﬂection equation in the context ofthe mathematics, especially the representation theory of the quantum group.
(1)
Let Uq (A2n−1 ) be the Drinfeld-Jimbo quantum aﬃne algebra and (πl,x , Vl,x ) be the(1)l-fold symmetric tensor representation of Uq (A2n−1 ). The quantum R-matrix is deﬁned
as an intertwiner byR(x/y) : Vl,x ⊗ Vm,y → Vm,y ⊗ Vl,x ,(πm,y ⊗ πl,x )∆(u)R(x/y) = R(x/y)(πl,x ⊗ πm,y )∆(u)(1)

f or any u ∈ Uq (A2n−1 )(1)where ∆ is a coproduct of Uq (A2n−1 ). It is well-known that the quantum R-matrix
satisfy the Yang-Baxter equation.
(1)∗∗
Let U ′ be a right coideal subalgebra of Uq (A2n−1 ) and (πl,x, Vl,x) be the dual representation of (πl,x , Vl,x ). Then, the quantum K-matrix is also deﬁned as an intertwiner,
if it exists uniquely, by∗K(x) : Vl,x → Vl,x
−1 ,∗K(x)πl,x (a) = πl,x−1 (a)K(x)f or any a ∈ U ′ .

To obtain a solution to the reﬂection equation, we choose the aﬃne ıquantum groupof type AII as U ′ . Then, the quantum K-matrix is uniquely determined and we see thatthese R and K actually satisfy the reﬂection equation.
References[1] H. Kusano, M. Okado, Solution to the reﬂection equation related to the ιquantum group of type
AII, arXiv:2012.00967.
Department of Mathematics, Faculty of Science, Osaka City University
Email address: a17sa007@jf.osaka-cu.ac.jp

Date: Received on March 8, 2021.
Key words and phrases. reﬂection equation, symmetric tensor representation, ıquantum group.

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.33

CIRCLE PATTERNS ON SURFACES WITH COMPLEXPROJECTIVE STRUCTURESWAI YEUNG LAM
Abstract. We report progress on a conjecture of Kojima, Mizushima and Tan about
the deformation space of circle packings on surfaces with complex projective structures.

1. Introduction
In the classical theory, holomorphic functions are conformal, mapping inﬁnitesimal
circles to themselves. Instead of inﬁnitesimal size, a circle packing is a conﬁgurationof ﬁnite-size circles in the plane where certain pairs are mutually tangent. William
Thurston proposed regarding the map induced from two circle packings with the same
tangency pattern as a discrete holomorphic function [3]. Using machinery from hyperbolic 3-manifolds, a discrete analogue of the Riemann mapping is deduced from
Koebe-Andreev-Thurston theorem. A natural question is how this theory can be extended to Riemann surfaces. It is known that on a surface there are varying conformal
structures, which form the so-called Teichmüller space.
Since circles are preserved under complex projective transformations (i.e. Möbiustransformations), it is natural to consider circle packings on surfaces with complex
projective structures. Kojima, Mizushima and Tan [1] conjectured that, for a given
combinatorics, the conﬁguration space of circle packings is a manifold whose projection
to the Teichmüller space is a diﬀeomorphism.
In the talk, we report progress on the torus case and discuss its connection to discrete
holomorphic quadratic diﬀerentials.
References[1] Sadayoshi Kojima, Shigeru Mizushima, and Ser Peow Tan. Circle packings on surfaces with projective structures. J. Diﬀerential Geom., 63(3):349–397, 2003.
[2] Wai Yeung Lam. Quadratic diﬀerentials and circle patterns on complex projective tori. 2019,arXiv:1909.07258.
[3] Kenneth Stephenson. Introduction to circle packing: The theory of discrete analytic functions.
Cambridge University Press, Cambridge, 2005.
(W.Y. Lam) University of Luxembourg
Email address: wai.lam@uni.lu

Date: Received on February 23, 2020.
Key words and phrases. circle packings, discrete conformal geometry.

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The 27th OCU International Academic Symposium
2021, pp.34

MESH GENERATION BASED ON COMPUTATIONAL CONFORMALGEOMETRY
NA LEI

Generating meshes with regular structure plays a fundamental role in isogeometricanalysis. Regular hexahedral mesh generation is called the holy grid problem in computational mechanics. Intensive research eﬀorts have been spent on it for tens of years.
Although there are many heuristic methods in practice, the theoretic foundation still
remains widely open.
Recently, we have established a theoretic framework for quadrilateral mesh generation based on conformal geometry. Basically, we have discovered the intrinsic relation
between quad-meshes and meromorphic diﬀerentials on Riemann surfaces. This framework is simple, elegant but powerful. It can answer many fundamental problems, thatno other methods could shed a light. For examples, it can show the existence of quadmeshes with special properties, estimate the dimension of quad-meshes with constraints,
specify the geometric relations among the singular vertices of quad-meshes. More importantly, it gives a simple algorithm for high quality quad-mesh generation based on
Riemann-Roch and Abel-Jacobi theorems. Furthermore, the quad-meshes based on
Strebel diﬀerential can leads to hexahedral mesh generation for volumes.
(N. Lei) Dalian Institute of Technology
Email address: nalei@dlut.edu.cn

Date: Received on March 20, 2021.

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.35–36

DARBOUX TRANSFORMS VIA A SYM-TYPE FORMULA
KATRIN LESCHKE
Abstract. Isothermic surfaces are surfaces which allow, away from umbilic points,
a conformal curvature line parametrisation. We discuss an extension of Bianchi permutability to describe all double Darboux transforms with respect to the same parameter in terms of a Sym-type formula. This is joint work with J. Cho and Y.
Ogata.

An isothermic surface can be seen as a surface f : M → R3 which locally allows a
non-trivial dual surface f d such that the Gauss map of f d is the negative of the Gaussmap of f , that is df ∧ df d = df d ∧ df = 0 where we identiﬁed R3 = Im H. Examples ofisothermic surfaces are minimal surfaces, CMC surfaces and surfaces of revolution. A
Darboux pair (f, fˆ) is given geometrically by a pair of distinct conformal immersions
into 3–space such that there exists a sphere congruence conformally enveloping both
surfaces [1]. In this case, both surfaces f and f ♯ are isothermic. Algebraically, one
obtains a classical Darboux transform fˆ = f + T of an isothermic surface by a solution
T : M → R3 to a Riccati equation [2]dT = −df + T df d ϱT, ϱ ∈ R .
Here the real parameter ϱ can be considered as the spectral parameter of an integrable
system, e.g., [3]. Using a conformal formulation, we write a Darboux transform off : M → R3 as fˆ = φH in S 4 = HP1 where φ is a dϱ –parallel section for ﬁxed ϱ ∈ R,
e.g. [4, 5]. Recall that the associated family of ﬂat connections dϱ is given on the trivial
H2 bundle H2 by()
f df d −f df d fdϱ = d + ϱ.
df d −df d f
Given two parallel sections for distinct spectral parameter ϱ1 ̸= ϱ2 ∈ R the associatedDarboux transforms f1 , f2 of f have by Bianchi permutability [6] a common Darboux
transform fˆ which is given algebraically by φ1 , φ2 . In this talk we discuss an extensionof Bianchi permutability to the case when the spectral parameter coincide ϱ := ϱ1 = ϱ2 .
In this case, a Sym-type formula gives a Darboux transform of f by diﬀerentiating a
λ–dependent family of sections φλ with respect to the parameter λ near λ = ϱ wheredλ φλ = 0. In particular, again one obtains Darboux transforms of f1 = φ1 H at ϱ byDate: Received on February 23, 2020.
Key words and phrases. Isothermic surfaces, Bianchi permutability, simple factor dressing, Sym
formula.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics) and the Leverhulme Trust
Network Grant IN-2016-019.

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parallel sections of the family of ﬂat connections of f , that is, without further integration. This result is obtained by expressing the associated family of ﬂat connections of
f1 in terms of the simple factor dressing, that is, by a gauge of dϱ with simple pole.
Moreover, all Darboux transforms of f1 are given by linear combinations of parallel
sections which are given by Bianchi permutability and by the Sym-type formula.
References[1] G. Darboux, Sur les surfaces isothermiques, C. R. Acad. Sci. Paris, Vol. 128, 1899, 1299-1305.
[2] U. Hertrich-Jeromin and F. Pedit, Remarks on the Darboux Transfoms of Isothermic Surfaces,
Doc. Math. J. DMV, Vol. 2, 1997 313-333.
[3] F. E. Burstall, U. Hertrich-Jeromin, F. Pedit and U. Pinkall, Curved ﬂats and isothermic surfaces,
Math. Z., Vol. 225, 1997, 199–209.
[4] U. Hertrich-Jeromin, Introduction to Möbius diﬀerential geometry, London Mathematical Society
Lecture Note Series 300, 2003, Cambridge: Cambridge University Press.
[5] F. Burstall, Isothermic surfaces: conformal geometry, Cliﬀord algebras and integrable systems, inIntegrable systems, geometry, and topology, AMS/IP Stud. Adv. Math., 36, 2006, 1–82.
[6] L. Bianchi, Ricerche sulle superﬁcie isoterme and sulla deformazione delle quadriche Annali diMatematica, Ser. III, 11, 1905, 93-157
School of Mathematics, University of Leicester
Email address: k.leschke@leicester.ac.uk

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The 27th OCU International Academic Symposium
2021, pp.37

DISCRETE CONFORMAL GEOMETRY OF POLYHEDRALSURFACES AND ITS APPLICATIONFENG LUO

Classical theory of Riemann surfaces is a pillar in mathematics and has many applications within and outside of mathematics. There have been many approaches to establish
discrete versions of conformal geometry for polyhedral surfaces since the pioneer workof W. Thurston in 1978. In this talk, we will report some of the recent developmentsin this area. These include a notion of discrete conformality, a discrete uniformization
theorem for polyhedral surfaces and their applications. Some open problems and theirrelationship to classical Koebe conjecture and Weyl problem will be discussed. This isa joint work with D. Gu, J. Sun and T. Wu.
References(F. Luo) Rutgers University, New Brunswick, NJ, USA
Email address: fluo@math.rutgers.edu

Date: Received on February 10, 2020.

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.38–39

GEOMETRY OF QUASI STATISTICAL MANIFOLDSHIROSHI MATSUZOE
Abstract. A quasi statistical manifold is a generalization of statistical manifolds.
In this talk, after summarizing the foundations of quasi statistical manifold, we will
discuss several applications of geometry of quasi statistical manifolds.

Let M be a manifold, h a nondegenerate (0, 2)-tensor ﬁled on M , and ∇ an aﬃneconnection on M . The quasi dual connection ∇∗ (or the left conjugate connection) of∇ with respect to h is deﬁned byXh(Y, Z) = h(∇∗X Y, Z) + h(Y, ∇X Z)

for X, Y, Z ∈ X(M ).

If ∇∗ is torsion-free, the triplet (M, ∇, h) is a quasi statistical manifold [7]. In particular,
if h is a semi-Riemannian metric, (M, ∇, h) is a statistical manifold admitting torsion[5]. This manifold naturally arises in quantum information geometry [5] and the theoryof non-conservative estimating functions [3, 4].
In this talk, we will discuss geometry of quasi statistical manifolds. This talk ismainly based on joint works with K. Haba [2] at Toyota Systems Corporation, and M.
Henmi [3, 4] at the Institute of Statistical Mathematics.
References[1] Haba, K., 1-Conformal geometry of quasi statistical manifolds, Inf. Geom., (2020).
[2] Haba, K. and Matsuzoe H., Complex aﬃne distributions. Diﬀerential Geom. Appl., 75 (2021),
101734.
[3] Henmi, M. and Matsuzoe, H., Geometry of pre-contrast functions and non-conservative estimating
functions, International Workshop on Complex Structures, Integrability and Vector Fields, AIP
Conf. Proc., 1340 (2011), 31–41.
[4] Henmi, M. and Matsuzoe, H., Statistical manifolds admitting torsion and partially ﬂat spaces,
Geometric structures of information, Signals Commun. Technol., Springer, Cham, 2019, 37–50.
[5] Kurose, T., Statistical manifolds admitting torsion. Geometry and Something, 2007, in Japanese.
[6] Matsuzoe, H., Statistical manifolds and aﬃne diﬀerential geometry. Adv. Stud. Pure Math. 57,
2010, 303–321.
[7] Matsuzoe, H., Quasi statistical manifolds, Pure and Applied Diﬀerential Geometry – PADGE
2012: In Memory of Franki Dillen (Berichte aus der Mathematik), J. Van der Veken, I. Van de
Woestyne, L. Verstraelen, L. Vrancken (Eds.), Shaker Verlag GmbH, Aachen, Germany, 2013,
208–214.
Date: Received on March 18, 2021.
Key words and phrases. quasi statistical manifold, statistical manifold admitting torsion, aﬃne
distribution.
This research was partially supported by JSPS Grant-in-Aid for Scientiﬁc Research (KAKENHI)
19K03489.

The 27th OCU Symposium(H. Matsuzoe) Department of Computer Science, Nagoya Institute of Technology
Email address: matsuzoe@nitech.ac.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.40–41

ON MINIMAL PF SUBMANIFOLDS IN HILBERT SPACES
WITH SYMMETRIESMASAHIRO MORIMOTO

Let M be a submanifold immersed in a ﬁnitie dimensional Riemannian manifold M̄ .
M is called austere ([1]) if for each p ∈ M and each ξ ∈ Tp⊥ M the set of eigenvalues
with multiplicities of the shape operator Aξ is invariant under the multiplication by(−1). M is called weakly reﬂective ([2]) if for each p ∈ M and each ξ ∈ Tp⊥ M there
exists an isometry νξ of M̄ which satisﬁes νξ (p) = p, dνξ (ξ) = −ξ and νξ (M ) = M . M
is called arid ([3]) if for each p ∈ M and each ξ ∈ Tp⊥ M \{0} there exists an isometryφξ which satisﬁes φξ (p) = p, dφξ (ξ) ̸= ξ and φξ (M ) = M . From these deﬁnitions wehave:austere ⇒⇒reﬂective ⇒ weakly reﬂective ⇒minimal⇒arid
It is an interesting problem to classify these submanifolds under suitable conditions.
In my poster session we introduce the concepts of these submanifolds into a classof proper Fredholm (PF) submanifolds in Hilbert spaces (Terng [4]) and show that
there exist many inﬁnite dimensional weakly reﬂective PF submanifolds, austere PF
submanifolds and arid PF submanifolds in Hilbert spaces. In particular each ﬁber ofthe parallel transport map is shown to be a weakly reﬂective PF submanifold. Those
imply that in inﬁnite dimensional Hilbert spaces there exist many homogeneous minimal
submanifolds which are not totally geodesic, unlike in the ﬁnite dimensional Euclidean
case.
References[1] R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math., 148 (1982), 47-157.
[2] O. Ikawa, T. Sakai, H. Tasaki, Weakly reﬂective submanifolds and austere submanifolds. J. Math.
Soc. Japan 61 (2009), no. 2, 437-481.
[3] Y. Taketomi, On a Riemannian submanifold whose slice representation has no nonzero ﬁxed points,
Hiroshima Math. J. 48 (2018), no. 1, 1-20.
[4] C.-L. Terng, Proper Fredholm submanifolds of Hilbert space. J. Diﬀerential Geom. 29 (1989), no.
1, 9-47.
[5] M. Morimoto, On weakly reﬂective PF submanifolds in Hilbert spaces, to appear in Tokyo J. Math.
[6] M. Morimoto, Austere and arid properties for PF submanifolds in Hilbert spaces, to appear inDiﬀerential Geom. Appl.
Date: Received on February 23, 2020.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics).

The 27th OCU Symposium(M. Morimoto) Department of Mathematics, Osaka City University,
Email address: mmasahiro0408@gmail.com

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.42–43

M2-BRANES: INTEGRABILITY AND WEYL GROUP
SANEFUMI MORIYAMA
Abstract. Physical backgrounds and Mathematical structures of the ABJM matrix
model and its generalizations will be explained.

We deﬁne two-point functions of the ABJM matrix model to be the multiple integrals
1
2
2 Z
P
i− 2 (N1 −N2 )dN1 µ dN2 ν ik (PNm=1
1 µ2 − N2 ν 2 )
m
n=1 n
4π
′
e
⟨sλ sλ ⟩k (N1 |N2 ) =N1 (2π)N2
N1 !N2 !
RN1 +N2 (2π)
QN1νn −νn′ 2µm −µm′ 2 QN2))m<m′ (2 sinhn<n′ (2 sinh2×sλ (eµ |eν )sλ′ (e−µ |e−ν )(1)
QN1 Q2N2µm −νn 2
(2
cosh
)
m=1
n=1
2
where sλ (x|y) = sλ (x1 , x2 , · · · , xN1 |y1 , y2 , · · · , yN2 ) is the super Schur polynomial in
representation λ. Although k is quantized in physical theories, all positive real values
are allowed in the above matrix model. We also consider the generating function⟨sλ sλ′ ⟩GCk,M (z) =∞X

z N ⟨sλ sλ′ ⟩k (N |N + M ),(2)

N =max(0,−M )by regarding the rank N as the particle number and introducing the dual fugacity z in
the grand canonical ensemble [1].
This matrix model originates from the ABJM theory [2, 3, 4], which is the N = 6
super Chern-Simons theory with gauge group U(N1 )k × U(N2 )−k (subscripts (k, −k)denoting the Chern-Simons levels) and two pairs of bifundamental matters. The ABJM
theory is proposed to describe min(N1 , N2 ) M2-branes and |N2 − N1 | fractional M2branes on the background C4 /Zk . The partition function and one-point functions ofhalf-BPS Wilson loop operators on S 3 (deﬁned originally by inﬁnite-dimensional path
integrals) reduce to the above ﬁnite-dimensional multiple integrals [5] after cancellations
between contributions from bosons and fermions. The two-point functions of the matrix
model are deﬁned in [6] from its mathematical structure.
Then, by trivializing one of the insertions sλ′ = 1, SλM = ⟨sλ ⟩GCk,M (z), we can provethe compatibility with the Giambelli identity [7] and the Jacobi-Trudi identity [8]
M +j−1M
SλM /S•M = det(S(α/S•M , SλM /S•M = det S[λ/S•M +j−1 ,(3)
i |βj )
i −i+j]for the Young diagram λ = [λ1 , λ2 , · · · ] = (α1 , α2 , · · · |β1 , β2 , · · · ). We can further
identify the integrable structure of the two-point functions as the two-dimensional Toda
lattice hierarchy [9].
Date: Received on March 17, 2021.

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Also we can rewrite the grand canonical partition function as a determinantb −1 ),⟨1⟩GC
k,M (z) = det(1 + z H(4)b is a spectral operator given by the exponentiated canonical operators (Q,b Pb) =
where Hqb pb(e , e ). Especially, for the case without rank deformations M = 0, the spectral operator
is given byb = (Qb 12 + Q
b− 12 )(Pb 12 + Pb− 12 ),H(5)which, after change of variables, is the deﬁning equation of P1 × P1 , the simplest delPezzo geometry. In this sense, the ABJM matrix model enjoys the exceptional Weylgroup of the del Pezzo geometry [10, 11, 12].
As above, the ABJM matrix model enjoys both the integrability and the exceptional
Weyl group symmetry. The physical backgrounds and these mathematical structures
will be explained.
References[1] M. Marino and P. Putrov, “ABJM theory as a Fermi gas,” J. Stat. Mech. 1203 (2012), P03001[arXiv:1110.4066 [hep-th]].
[2] O. Aharony, O. Bergman, D. L. Jaﬀeris and J. Maldacena, “N=6 superconformal Chern-Simonsmatter theories, M2-branes and their gravity duals,” JHEP 10 (2008), 091 [arXiv:0806.1218 [hepth]].
[3] K. Hosomichi, K. M. Lee, S. Lee, S. Lee and J. Park, “N=5,6 Superconformal Chern-Simons
Theories and M2-branes on Orbifolds,” JHEP 09 (2008), 002 [arXiv:0806.4977 [hep-th]].
[4] O. Aharony, O. Bergman and D. L. Jaﬀeris, “Fractional M2-branes,” JHEP 11 (2008), 043[arXiv:0807.4924 [hep-th]].
[5] A. Kapustin, B. Willett and I. Yaakov, “Exact Results for Wilson Loops in Superconformal
Chern-Simons Theories with Matter,” JHEP 03 (2010), 089 [arXiv:0909.4559 [hep-th]].
[6] N. Kubo and S. Moriyama, “Two-Point Functions in ABJM Matrix Model,” JHEP 05 (2018),
181 [arXiv:1803.07161 [hep-th]].
[7] S. Matsuno and S. Moriyama, “Giambelli Identity in Super Chern-Simons Matrix Model,” J.
Math. Phys. 58 (2017) no.3, 032301 [arXiv:1603.04124 [hep-th]].
[8] T. Furukawa and S. Moriyama, “Jacobi-Trudi Identity in Super Chern-Simons Matrix Model,”SIGMA 14 (2018), 049 [arXiv:1711.04893 [hep-th]].
[9] T. Furukawa and S. Moriyama, “ABJM Matrix Model and 2D Toda Lattice Hierarchy,” JHEP
03 (2019), 197 [arXiv:1901.00541 [hep-th]].
[10] M. Marino and P. Putrov, “Exact Results in ABJM Theory from Topological Strings,” JHEP 06(2010), 011 [arXiv:0912.3074 [hep-th]].
[11] A. Grassi, Y. Hatsuda and M. Marino, “Topological Strings from Quantum Mechanics,” Annales
Henri Poincare 17 (2016) no.11, 3177-3235 [arXiv:1410.3382 [hep-th]].
[12] S. Moriyama, “Spectral Theories and Topological Strings on del Pezzo Geometries,” JHEP 10(2020), 154 [arXiv:2007.05148 [hep-th]].
(S. Moriyama) Department of Physics; NITEP; OCAMI (Osaka City University)
Email address: moriyama@sci.osaka-cu.ac.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.44–45

GEOMETRIC STRUCTURES MOTIVATED BY ARCHITECTURE
CHRISTIAN MÜLLER
Abstract. We will study discrete structures from a discrete diﬀerential geometry
point of view that have been motivated by architecture. In particular we will study
discrete surfaces that are isometric to surfaces of revolution, tensegrities which areframeworks from cables and bars in static equilibrium, and a support structure formedby quadrics.

We address three discrete structures that have been motivated by architectural applications.
1. Discrete Geodesic Parallel Coordinates
In [2] we study a discretization of geodesic parallel coordinates which are orthogonal
nets on surfaces where one of the two families of parameter lines are geodesic curves.
We describe a discrete version of these special surface parameterizations and show thatthey are very useful for speciﬁc applications, most of which are related to the design
and fabrication of surfaces in architecture. With the new discrete surface model, it
is easy to control strip widths between neighboring geodesics. This facilitates tasks
such as cladding a surface with strips of originally straight ﬂat material or designing
geodesic gridshells and timber rib shells. It is also possible to model nearly developablesurfaces. These are characterized by geodesic strips with almost constant strip widths
and are used for generating shapes that can be manufactured from materials whichallow for some stretching or shrinking like felt, leather, or thin wooden boards. Most
importantly, we show how to constrain the strip width parameters to model a class of
intrinsically symmetric surfaces. These surfaces are isometric to surfaces of revolution
and can be covered with doubly-curved panels that are produced with only a few molds
when working with ﬂexible materials like metal sheets.
2. Geometric criteria for realizability of tensegrities in higher
dimensions
In [1] we study a classical Maxwell question on the existence of self-stresses for frameworks, which are called tensegrities. We give a complete answer on geometric conditionsof at most (d + 1)-valent tensegrities in Rd in terms of discrete multiplicative 1-forms.
Date: Received on February 15, 2020.
Key words and phrases. discrete diﬀerential geometry, discrete geodesics, geodesic parallel coordinates, tensegrities, Koenigs nets.
This work was supported by the Austrian Science Fund (FWF) through project P 29981.

The 27th OCU Symposium45

3. Support structures from quadrics
Furthermore, we will investigate support structures from ruled quadrics in tangential
contact. It leads to an interesting characterization of discrete Koenigs nets that has
also been independently be discovered in [3].
References[1] O. Karpenkov and C. Müller, Geometric criteria for realizability of tensegrities in higher dimensions, ArXiv e-prints, 1907.02830 (2019), 1-27.
[2] H. Wang, D. Pellis, F. Rist, H. Pottmann, and C. Müller, Discrete Geodesic Parallel Coordinates,
ACM Trans. Graphics 38(6), (Proc. SIGGRAPH ASIA) (2019), 1-13.
[3] A.I. Bobenko, and A.Y. Fairley, Nets of lines with the combinatorics of the square grid and withtouching inscribed conics, ArXiv e-prints, 1911.08477 (2019), 1-23.
(C. Müller) TU Wien
Email address: cmueller@geometrie.tuwien.ac.at

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.46–49

CARBON STRUCTURES AND A DISCRETE SURFACE THEORYHISASHI NAITO
Abstract. Carbon structures typiﬁed by fullerenes and carbon nanotubes are considered as discrete surfaces. We discuss such carbon structures from mathematical
viewpoints.

1. Crystal structure from viewpoints of discrete geometric analysis
Crystal structures are classically realized by using space groups in mathematics.
Space groups describe symmetries of atoms in crystals, but does not describe chemicalbonds between atoms in them. Since crystals consist of atoms and bonds between atoms.
a natural notion describing crystals is a graph, which consists of vertices and edges.
Describing crystal structures by using graph theory, we should consider realizations
Φ : X = (V, E) −→ R3 , which describes positions of atoms of the crystal (vertices of
the graph X).
In 2000, Kotani and Sunada [5] deﬁnes topological crystals and their standard realizations. A topological crystal is a locally ﬁnite graph X = (V, E) with a free action of anabelian group G, and they also deﬁne the energy E of realizations Φ of X by the sum ofsquared distances of edges in X/G. A standard realization of the topological crystal isa critical point of E under the ﬁxed volume of the fundamental region of the G-action.
Kotani–Sunada shows the existence of standard realizations for any topological crystals. Moreover, they show that standard realizations are the most symmetric structure
among all realizations for each topological crystal. Cubic, face-centric, body-centric,
and diamond crystals are described by a standard realization of suitable topologicalcrystals (see Sunada [12], also Naito [8]).
Sunada [11] also shows that a diamond crystal and the gyroid (K4 ) structure havevery higher symmetry, which are called strongly isotropic. Since diamonds (carbon
structures consist of the diamond structure) have nice physical properties, we may
expect carbon structures that consist of the gyroid structure also have nice properties.
In fact, by using density functional theory (DFT) calculation, we show that K4 carbon
is meta-stable and metallic (Itoh–Kotani–Naito–Sunada–Kawazoe–Adschiri [2]).
2. Carbon structures as discrete surfaces
Graphenes, carbon nanotubes, and fullerenes (for an example C60 ) are sp2 -carbon
crystal/molecular structure, and we may consider them as 3-valent discrete surfaces
Date: Received on February 23, 2020.
Key words and phrases. discrete surface, carbon structure, graph theory.
This work was partially supported by KAKENHI 17H06466 and 19K03488.

The 27th OCU Symposium

47(see Fig. 1). We may see that Gauss curvatures of such structures are zero/positive.
(a)

(b)

(c)

(d)

(e)

Figure 1. (a) Graphene, (b) nanotube with chiral index (12, 0), (c) nanotube with chiral index (12, 8), (d) nanotube with chiral index (12, 12),and (e) C60 .
Therefore we have a natural question: Does there exist negatively curved sp2 -carbon
structures?
In 1991, Mackay and Terrones [6] theoretically proposed a negatively curved sp2 carbon structure, which lies on the Schwarz P surface (minimal surface, negatively
curved, see Fig. 3). We also construct negatively curved sp2 -carbon structures by using
standard realizations as initial conditions for DFT calculations (Tagami–Liang–Naito–
Kawazoe–Kotani [10], see also Fig. 2 and Naito [7]). Such structures have same symmetries with the Mackay–Terrones structure, hence they are considered as negatively
curved.
(a)

(b)

(c)

(d)

Figure 2. Mackay-like structures, which are physically stable. Physical stabilities are calculated by DFT. (Tagami–Liang–Naito–Kawazoe–
Kotani [10]).

3. Curvatures of 3-valent discrete surfaces
We deﬁne a 3-valent discrete surface by Φ : X −→ R3 for a 3-valent graph X as amathematical model of fullerenes. If the 3-valent graph X is a∑surface graph, then we
may deﬁne “total curvature” (the Euler number) χ(X) of X by (1−k/6)Nk , where Nk
is the number of k-gons contained in X. This implies that if χ(X) < 0, then Nk > 0 (k ≥7). But usual Gauss/mean curvatures (for smooth surfaces) are pointwise functions,

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OCAMI Reports Vol. 1 (2021)

and there are no deﬁnitions of curvatures for 3-valent discrete surfaces. Hence, we do
not know whether structures in the previous section are negatively curved or not.
We deﬁne Gauss/mean curvatures for 3-valent discrete surfaces as functions on vertices of X, and we conﬁrm that Mackay structures (the ﬁrst column of Fig. 3) andstructures in Fig. 2.

Figure 3. Mackay structure (negatively curved structure [6]) and its
subdivisions (Upper row: Gauss curvature, lower row: mean curvature).
(Kotani–Naito–Omori [3]).
4. Subdivisions of 3-valent discrete surfaces and their spectral
properties
We also deﬁne a notion of subdivisions of 3-valent discrete surfaces. Our method of
subdivisions is based on the Goldberg-Coxeter construction for 3-valent plane graphs deﬁned by Dutour–Deza [1]. GC-subdivisions have integer parameters (k, l), and preserves
the number of polygons except for hexagons, hence the Euler number of subdivided surfaces are unchanged (see Fig. 4). Numerical results of Gauss/mean curvatures of discrete surfaces of GC-subdivisions of Mackay structures shown in Kotani–Naito–Omori[3] (see also Fig. 3). We show Hausdorﬀ convergence of sequences of GC-subdivisionsof a 3-valent discrete surface with suitable realizations (Kotani–Naito–Tao [4]).
We also consider behavior of eigenvalues of the Laplacian of graphs of GC-subdivisionsof 3-valent graphs. We obtain large multiplicity of eigenvalues 2 and 4 for (2k, 0)subdivisions (Omori–Naito–Tate [9]).
References[1] M. Dutour and M. Deza, Goldberg-Coxeter construction for 3- and 4-valent plane graphs, Electron.
J. Combin., 11, ♯R20, (2004).
[2] M. Itoh, M. Kotani, H. Naito, T. Sunada, Y. Kawazoe, and T. Adschiri, New metallic carbon
crystal, Physical Review Letters, 102, 055703, (2009).
[3] M. Kotani, H. Naito, and T. Omori, Eigenvalues of the Laplacian on the Goldberg-Coxeter constructions for 3- and 4-valent graphs, Comput. Aided Geom. Design, 58, 24–54, (2017).
[4] M. Kotani, H. Naito, and C. Tao, Construction of continuum from a discrete surface by its iterated
subdivisions, arXiv:1806.03531[5] M. Kotani and T. Sunada. Standard realizations of crystal lattices via harmonic maps, Trans.
Amer. Math. Soc., 353, 1–20, (2001).
[6] A. Mackay and H. Terrones. Diamond from graphite, Nature, 352, 762, (1991).

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49

step 1step 2step 3
zω

q1
q2
p

z
q3

step 40

step 5step 6

Figure 4. In case of (k, l) = (3, 2), Procedure of Goldberg-Coxeter
construction for a 3-valent graph around a vertex p of the graph. theGoldberg-Coxeter construction for a 3-valent graph. (Omori–Naito–Tate[9]).
[7] H. Naito, Construction of negatively curved cubic carbon crystals via standard realizations,
Springer Proc. Math. Stat., 166, (2016), 83-100.
[8] H. Naito, A short lecture on topological crystallography and a discrete surface theory, Anam
Lecture Notes in Mathematics, An introduction to discrete diﬀerential geometry, 2, 83–147, (2020).
http://math.korea.ac.kr/_res/math/etc/AnamLectureNotesVol2_20200210.pdf.
[9] T. Omori, H. Naito, and T. Tate, Eigenvalues of the Laplacian on the Goldberg-Coxeter constructions for 3- and 4-valent graphs, Electron. J. Combin., 26, ♯P3.7, (2019).
[10] M. Tagami, Y. Liang, H. Naito, Y. Kawazoe, and M. Kotani. Negatively curved cubic carbon
crystals with octahedral symmetry, Carbon, 76, 266–274, (2014).
[11] T. Sunada, Crystals that nature might miss creating, Notices Amer. Math. Soc., 55, 208–215,(2008).
[12] T. Sunada, Topological crystallography, vol. 6 of Surveys and Tutorials in the Applied Mathematical
Sciences. Springer, Tokyo, 2013.
(H. Naito) Graduate School of Mathematics, Nagoya University
Email address: naito@math.nagoya-u.ac.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.50

THE DUALLY FLAT STRUCTURE FOR SINGULAR MODELS
WITH DEGENERATE POTENTIALSNAOMICHI NAKAJIMA

The dually ﬂat structure introduced by Amari-Nagaoka is highlighted in Information
Geometry [1] – it brings a united geometric insight on various aspects in statisticalscience and optimizations. In practice, however, the (pseudo-)Riemannian metric may
often be degenerate, and then, precisely saying, the dually ﬂat structure is not deﬁned.
In the present paper, we propose a simple generalization of the dually ﬂat structure
to such singular models, and show that the extended Pythagorean theorem of AmariNagaoka still holds in this singular setting. Consequently, most of applications of the
Pythagorean theorem to statistical inference, em-algorithms, machine learning, etc.
are automatically justiﬁed even for general singular models in a rigorous sense. In
particular, it works even for non-convex optimization in the sense of Ekeland [2], where
the potential function has inﬂectional points and then the Legendre transform of thepotential becomes a multi-valued function. Here several techniques from singularity
theory and diﬀerential geometry would be applicable. In fact, a key ingredient in ourconstruction is the notion of coherent tangent bundles, which has been introduced by
Umehara-Saji-Yamada [3] for studying diﬀerential geometry of wavefronts, and actually
we consider a pair of ﬂat aﬃne connections of those vector bundles, which are mutuallyrelated via the Legendre duality.
References[1] S. Amari and H. Nagaoka, Methods of Information Geometry, AMS (2007).
[2] I. Ekeland, Legendre duality in nonconvex optimization and calculus of varientions, SIAM J.
Control and Optimization, Vol. 15 (6) (1977), 905–934.
[3] 梅原雅顕，佐治健太郎，山田光太郎，特異点をもつ曲線と曲面の微分幾何学，丸善出版（2017)(N. Nakajima) GRADUATE SCHOOL OF INFORMATION SCIENCE AND TECHNOLOGY, HOKKAIDO UNIVERSITY
Email address: nakajima-n@ist.hokudai.ac.jp

Date: Received on February 21, 2020.
Key words and phrases. Dually ﬂat structure, Information Geometry, Contact Geometry, Legendre
submanifolds, Wavefronts, e/m-geodesics, extended Pythagorean theorem.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics) and Hokkaido University GICoRE-GSB..

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.51

CR-FUNTIONS OF SUSPENSION FOLIATIONS AND DYNAMICALSYSTEMSSATOSHI OGAWA
Abstract. In this talk, I introduce suspension construction by some Riemann surface
and a group of circle diﬀeomorphisms and discuss the existence of non-constant CRfunction.

1. Suspension Foliations and CR-functions
Let M be a manifold of dimension n + m and {Uα , (xα , yα )} be an atlas of M , asxα = (x1α , · · · , xnα ) and yα = (yα1 , · · · , yαm ). Then, {Uα , (xα , yα )} is said to be a foliatedchart of codimension m if and only if the following relation about a transition of Uα ∩Uβholds.
xβ = φβα (xα , yα ) , yβ = ψβα (yα )When M has a foliation structure, we can give a decomposition of M as a disjoint
union of connected subsets called leaves.
1.1. Settings. Let τ be a complex number in the upper half plane, Diﬀ ω+ (S 1 ) be a groupof orientation preserving C ω -class circle diﬀeomorphisms and σ be a homomorphismfrom ⟨1, τ ⟩ to Diﬀ ω+ (S 1 ). We denote σ(1) = f1 , σ(τ ) = fτ . By considering the following
equivalent relation ∼σ over C × S 1 , we construct manifold X = C × S 1 / ∼σ .
(z, ζ) ∼σ (z + n + mτ, (f1n ◦ fτm )(ζ))
Then, we can give a foliated structure to X, which is called the suspensions foliation.
We say a function from X to C is a CR-function if it is leafwise holomolphic.
1.2. Main result. By focusing on dynamical propaties of f1 and fτ , we classify σ fromthe view point of the existence of non-constant CR-functions.
References
1 M. Adachi,On the ampleness of positive CR line bundles over Levi- at manifolds. Publ. Res.Inst.
Math. Sci.50(2014), 153-167.
2 K. Diederich and T. Ohsawa,Harmonic mappings and disc bundles over compact Kähler manifolds.Publ. Res. Inst. Math. Sci.21(1985), 819-833
3 Anatole Katok and Boris Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,
Cambridge University PressEncyclopedia Math. Appl., 54, Cambridge Univ. Press,1995(S.Ogawa) Department of Mathematics, Graduate School of Science, Osaka City University
Email address: m20sa006@xr.osaka-cu.ac.jp
Date: Received on March 17, 2021.

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.52–54

UNIQUENESS OF LOCAL MINIMIZERS FOR CRYSTALLINEVARIATIONAL PROBLEMSKENTO OKUDA AND MIYUKI KOISO
Abstract. An anisotropic surface energy is the integral of an energy density that
depends on the normal at each point over the considered surface, which is a generalization of the area of surfaces. The minimizer of any such an energy among all closed
surfaces enclosing the same volume is unique and it is (up to rescaling) so-called theWulﬀ shape. We show that, if the Wulﬀ shape is a regular polyhedron and the energy
density function is convex, then any convex local minimizer for volume-preserving
variations is, up to homothety and translation, the Wulﬀ shape.

1. Introduction
Anisotropic substances, such as crystals and some kinds of liquid crystals are considered to form a local minimizer of an anisotropic surface energy with volume constraint.
The anisotropic energy of a surface is the integral of an energy density that depends
on the normal direction of the surface. For a given energy density function, the energy
minimizer among all closed surfaces enclosing the same volume is unique. It is a convex
closed surface called the Wulﬀ shape ([4]). In this talk, more generally, we study local
minimizers of the anisotropic energy and their uniqueness. However, when the Wulﬀ
shape has a ﬂat part with dimension larger than or equal to one like a polyhedron,the energy density function has non-diﬀerentiable points and the classical variational
method cannot be applied, which makes it diﬃcult to study local minimizers of theenergy. In fact, only for curves in R2 , the uniqueness of local minimizers has been
proved ([3]). In this talk, we restrict ourselves to the case where the Wulﬀ shape isa polyhedron. Studying local minimizers in such a case is called a crystalline variational problem. We show that if the Wulﬀ shape is a regular polyhedron in R3 andthe energy density function is convex (see §2 for deﬁnition), then any local minimizerof the anisotropic energy for all variations that preserve the enclosed volume is, up tohomothety and translation, the Wulﬀ shape (see §2 for the precise statement). We alsomention an application of our main theorem to material science (cf. [1]).
Date: Received on March 17, 2021.
Key words and phrases. crystalline variational problem, anisotropic energy, Wulﬀ shape.
The second author is partially supported by JSPS KAKENHI Grant Number 20H01801 and JST
CREST GrantNumber JPMJCR1911. She is partially supported also by JSPS Grant-in-Aid for Scientiﬁc Research on Innovative Areas “Discrete Geometric Analysis for Materials Design”: Grant Numbers
18H04487 and 20H04642.

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2. Basic definitions and the precise statement of the main theorem
First let us recall the deﬁnition of a piecewise-C 1 hypersurface in Rn+1 . Let M =∪ki=1 Mi be an n-dimensional oriented compact connected C ∞ manifold (with or withoutboundary), where each Mi is an n-dimensional connected compact submanifold of M
with piecewise-C ∞ boundary, and Mi ∩ Mj = ∂Mi ∩ ∂Mj , (i, j ∈ {1, · · · , k}, i ̸= j).
We call a map X : M → Rn+1 a piecewise-C 1 hypersurface if X satisﬁes the following
conditions (I) and (II) (i ∈ {1, · · · , k}).
(I) X is continuous, and each Xi := X|Mi : Mi → Rn+1 is a C 1 -immersion.
(II) The unit normal vector ﬁeld νi : Mi → S n along Xi satisﬁes the following: If(u1 , · · · , un ) is a local coordinate system in Mi , then {∂/∂u1 , · · · , ∂/∂un , νi } gives the
canonical orientation in Rn+1 .
X is said to be closed if M does not have boundary.
Next let us introduce our energy density function. Let γ : S n → R>0 be a positive
continuous function on the unit sphere S n = {ν ∈ Rn+1 ; ∥ν∥ = 1} in Rn+1 , which is theenergy density function for piecewise-C 1 hypersurfaces in Rn+1 . γ is said to be convex
if the homogeneous extension γ : Rn+1 → R≥0 of γ that is deﬁned by γ(rX) := rγ(X),(∀X ∈ S n , ∀r ≥ 0), is a convex function.
Now let us deﬁne the anisotropic energy of a piecewise-C 1 hypersurface. Let X :M = ∪ki=1 Mi → Rn+1 be a piecewise-C 1 hypersurface as above. The anisotropic energyFγ (X) of X is deﬁned asFγ (X) :=

k ∫
∑
i=1

γ(νi ) dA,

(1)

Mi

where dA is the n-dimensional volume form of Mi induced by Xi . If γ ≡ 1, Fγ (X) isthe usual n-dimensional volume of the hypersurface X.
It is known that, for any positive number V > 0, among all closed hypersurfaces in
Rn+1 enclosing the same (n + 1)-dimensional volume V , there exists a unique (up totranslation in Rn+1 ) minimizer W (V ) of Fγ ([4]). Here a closed hypersurface means thatthe boundary (having tangent space almost everywhere) of a set of positive Lebesgue
measure. The minimizer Wγ (V0 ) for a speciﬁc volume V0 is called the Wulﬀ shape forγ, and{ we denote it by Wγ . More
} precisely, the boundary of the convex set W̃ [γ] :=∩ν∈S n X ∈ Rn+1 |⟨X, ν⟩ ≤ γ(ν) is the Wulﬀ shape for γ, where ⟨ , ⟩ stands for the
standard inner product in Rn+1 . If γ ≡ 1, Wγ is the unit sphere S n . All Wγ (V ) arehomothetic to Wγ . The Wulﬀ shape Wγ is convex, but, unlike the isotropic case, it
is not necessarily smooth. On the other hand, for any given convex set W̃ having theorigin of Rn+1 inside, there exists a Lipschitz continuous function γ : S n → R>0 suchthat the boundary W := ∂ W̃ of W̃ is the Wulﬀ shape for γ. In general, such γ is notunique. The “smallest” γ is uniquely-determined and it is called the convex integrandfor W .
Theorem 1 ([2]). Let W be a regular polyhedron in R3 , and let γ be the convex integrandfor W . Let X be a piecewise-C 1 convex closed surface. Then, X is a local minimizer

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OCAMI Reports Vol. 1 (2021)of the anisotropic energy Fγ (X) for all variations that preserve the enclosed volume if
and only if X(M ) = W (up to homothety and translation).
References[1] S. Asahina, S. Takami, T. Otsuka, T. Adschiri and O. Terasaki, Exploitation of surface-sensitiveelectrons in scanning electron microscopy reveals the formation mechanism of new cubic andtruncated octahedral CeO2 nanoparticles, ChemCatChem 3 (2011), 1038–1044.
[2] M. Koiso and K. Okuda, Uniqueness of local minimizers for crystalline variational problems, in
preparation.
[3] F. Morgan, Planar Wulﬀ shape is unique equilibrium, Proc. Amer. Math. Soc. 133 (2005), 809–
813.
[4] J. E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), 568–588.
(Kento Okuda) Graduate School of Mathematics, Kyushu University
Email address: k-okuda@math.kyushu-u.ac.jp(Miyuki Koiso) Institute of Mathematics for Industry, Kyushu University
Email address: koiso@imi.kyushu-u.ac.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.55–58

ISOMETRIC IMMERSIONS VIA VARIATIONAL CALCULUS
FRANZ PEDIT
Abstract. We will discuss a variational problem with parameters whose minimizers
are isometric immersions of a given Riemannian surface in 3-space. One of the parameters controls the Willmore energy, which can be used as a regularizer preventing
excessive creasing/crumpling of the resulting isometric immersion in 3-space.

1. Isometric immersions
The Nash-Moser theorem [9] assures that every Riemannian manifold can be isometrically embedded into Euclidean space Rn for large enough n. But it is unknown, andprobably false, whether a smooth, 2-dimensional metric can be isometrically realizedin Euclidean 3-space, even locally. There are obvious global obstructions: Hilbert’sclassical result that the hyperbolic plane does not admit an isometric immersion into
R3 , or the fact that a compact, non-positively curved, 2-dimensional Riemannian manifold cannot be isometrically immersed into R3 . If one relaxes the smoothness of theimmersion, every 2-dimensional Riemannian manifold (M, g) admits a C 1 -isometric immersion f : M → R3 into Euclidean space [8, 7, 6]. Neither the original existence proofs
nor the recent explicit constructions of such isometric immersions [1, 2] reﬂect much of
the underlying geometry of (M, g), as shown in Figure 1.
On the other hand, there are piecewise linear ﬂat tori and one seemingly can folda napkin into a ﬂat torus with hardly any creasing, Figure 2. Such observations, andexperiments carried out with the numerical algorithm [4], support the following conjecture discussed in [3]:Given a Riemannian surface (M, g), there exists a piecewise smooth isometric immersion f : M → R3 in each regular homotopy class.
2. The variational problem
A regular homotopy class of an immersion f : M → R3 is described by a quaternionicspin bundle L → M . If M is a Riemann surface, then L carries a unique, conformallyinvariant Dirac operator ∂¯ : Γ(L) → Γ(K̄L), where K̄ denotes the anti-canonical bundleof M . A conformal immersion f : M → R3 then induces a nowhere vanishing solutionψ ∈ Γ(L) of the non-linear Dirac equation¯ + 1 HJψ(ψ, ψ) = 0 .
∂ψ2

Date: Received on February 17, 2020.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics).

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Figure 1. A square ﬂat torus can be isometrically C 1 -embedded into
Euclidean 3-space [2]. Pictures by the Hévéa project.
Here H : M → R is the mean curvature of f with respect to the induced metric |df |2
and df = (ψ, ψ), where (·, ·) denotes the spin pairing. Conversely, given a quaternionicspin bundle, that is, a regular homotopy class, and a nowhere vanishing solution ψ ofthe non-linear Dirac equation for some H : M → R, one ∫obtains [10, 5] a conformal
immersion f : M → R3 (with translation periods) via f = (ψ, ψ).
The non-linear Dirac equation can be given a variational characterization: for nonnegative coupling constants ϵ = (ϵ1 , ϵ2 , ϵ3 ), consider the family of variational problems
Eϵ : Γ(L× ) → R on nowhere vanishing sections of L given by
∫
∫
∫
2
2
⟨∗ ∂¯ ψ∧∂¯ ψ⟩⟨∗ ∂¯ ψ∧ψ(ψ,ψ)⟩⟨∗ ∂¯ ψ∧Jψ(ψ,ψ)⟩Eϵ (ψ) = ϵ1
+
(ϵ
−
ϵ
)
+
(ϵ
−
ϵ
)
.
2
1
3
1
|ψ|2
|ψ|4
|ψ|4
M

M

M

Here | · | : L → |K| denotes the half-density valued quadratic form |ψ|2 = |(ψ, ψ)| on Land ⟨·, ·⟩ : L → |K| is the half-density valued inner product obtained via polarization.
The complex structure ∗ on T M ∗ is the negative of the Hodge-star on 1-forms.
It is worth noting that the functional Eϵ is conformally invariant, that is, well-deﬁnedonsurface M . The last integral turns out to be the Willmore functional∫ the2 Riemann
H |ψ|4 and the ﬁrst two integrals measure, in L2 , the failure of the non-linear Dirac
M
2

Figure 2. The angular defects of the embedded, piecewise linear torus
shown on the left are the same at all vertices and their sum is zero,
implying that the induced metric is ﬂat. The right image shows a smooth,almost isometric immersion of a ﬂat torus found by the algorithm in [4].

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equation to hold. Thus, for ϵ3 = 0 and ϵ1 , ϵ2 > 0, the functional attains its minimum
value Eϵ (ψ) = 0 at nowhere vanishing sections ψ which correspond to—in generalrather singular—conformal immersions. It is therefore conceivable that minimizers ofEϵ for ϵ3 > 0, which has the eﬀect of keeping the Willmore energy as a regularizer,
will converge as ϵ3 tends to zero to smooth conformal immersions of M minimizing theWillmore energy, that is, constrained Willmore surfaces.
Since the Dirac equation only guarantees the closedness of (ψ, ψ), the resulting conformal immersion given by df = (ψ, ψ) generally will have translation∫ periods which
are controlled by adding the squared lengths of the period integrals | γ (ψ, ψ)|2 to thefunctional Eϵ .

Figure 3. A smooth, almost isometric immersion of the Riemanniansurface of genus 2 with constant curvature shown on the left found by
the algorithm in [4].
As it turns out, this strategy works surprisingly well [4] when searching for isometricimmersions f : M → R3 of a compact, oriented Riemannian surface (M, g). Since theconformal class of g gives M the structure of a Riemann surface, one can consider thefamily of functionals Eϵ with the additional isometric constraint|ψ|4 = g .
Then the resulting minimizers under the above described procedure will be isometricimmersions f : M → R3 whose Willmore energy is “small”. The resulting surfaces
provide examples of how piecewise smooth, and sometimes even smooth, isometricimmersions of compact Riemannian surfaces might look, as can be seen in Figure 2 andFigure 3.
References[1] Borrelli, V., Jabrane, S., Lazarus, F., and Thibert, B. Flat Tori in Three-Dimensional
Space and Convex Integration. Proc. Nat. Acad. Sci. 109, 19 (2012), 7218–7223.
[2] Borrelli, V., Jabrane, S., Lazarus, F., and Thibert, B. Isometric Embeddings of the
Square Flat Torus in Ambient Space. Ensaios Matemáticos 24 (2013), 1–91.
[3] Chern, A., Knöppel, F., Pedit, F., Pinkall, U., and Schröder, P. Finding Conformal
and Isometric Immersions of Surfaces. ArXiv:1901.09432 [math.DG]

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OCAMI Reports Vol. 1 (2021)[4] Chern, A., Knöppel, F., Pinkall, U., and Schröder, P. Shape from Metric. ACM Trans.
Graph. 37, 4 (2018), 63.
[5] Ferus, D., Leschke, K., Pedit, F., and Pinkall, U. Quaternionic Holomorphic Geometry: Plücker Formula, Dirac Eigenvalue Estimates and Energy Estimates of Harmonic 2-Tori.
Inventiones mathematicae 146, 3 (Dec 2001), 507–593.
[6] Gromov, M. Partial Diﬀerential Relations. Springer, 1986.
[7] Kuiper, N. On C 1 -isometric Imbeddings Indag. Math. 58 (1955), 545–556, 683–689.
[8] Nash, J. C 1 Isometric Imbeddings. Annals of mathematics 60, 3 (1954), 383–396.
[9] Nash, J. The Imbedding Problem for Riemannian Manifolds. Annals of Mathematics 63, 1 (1956),
20–63.
[10] Pedit, F., and Pinkall, U. Quaternionic Analysis on Riemann Surfaces and Diﬀerential Geometry. Documenta Mathematica (1998), 389–400.

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The 27th OCU International Academic Symposium
2021, pp.59–60

SPHERICAL CURVATURE LINESJOSEPH CHO, MASON PEMBER*, AND GUDRUN SZEWIECZEKAbstract. In this talk we shall investigate surfaces that possess a family of curvaturelines each lying on a sphere. Motivated by a result of Blaschke, we can view such
surfaces as a Lie geometric evolution of an initial spherical curve. We investigate thecase when such a surface is also an Ω-surface and determine necessary and suﬃcientconditions on the evolution and initial curve that yield such a surface.

1. Spherical curvature lines
In Blaschke’s book [1] it is shown that surfaces with one family of spherical curvaturelines satisfy a remarkable property: any two curvature lines of this family are related bya Lie sphere transformation. In the hexaspherical model R4,2 of Lie sphere geometry,
this can be understood using a notion of parallel transport along a curve in
S 3,2 = {y ∈ R4,2 : (y, y) = 1}.
In such a way, surfaces with spherical curvature lines are determined by a curve in S 3,2
and an initial spherical curve.
Demoulin [3, 4] discovered a class of surfaces called Ω-surfaces that are invariant
under Lie sphere transformations. This class contains many well known classes of
surfaces such as isothermic surfaces and linear Weingarten surfaces in 3-dimensional
space forms. This class is given a gauge theoretic interpretation in [2, 5].
One can then ask what conditions are required for a surface with one family of
spherical curvature lines to be an Ω-surface. We obtain one condition on the curvein S 3,2 , requiring that it lies in a 4-dimensional subspace of R4,2 . A second condition
restricts our choice of initial curve, requiring that this curve project to a constrainedelastic curve in a certain Riemannian or Lorentzian 2-dimensional space form. We maythen use this characterisation to construct examples of such surfaces.
References[1] W. Blaschke, Vorlesungen über Diﬀerentialgeometrie III, Springer Grundlehren XXIX (1929).
[2] D. Clarke, Integrability in submanifold geometry, University of Bath, 2012.
Date: Received on 03/03/2021.
2010 Mathematics Subject Classiﬁcation. Primary 53A40, Secondary 53B25, 53C42.
Key words and phrases. Spherical curvature lines, Ω-surfaces, Lie sphere geometry.
This work was supported by the FWF research grant P28427-N35 “Non-rigidity and Symmetry
breaking”, the joint FWF and JSPS research grants I1671-N26: “Transformations and Singularities”
and I3809-N32: “Geometric shape generation”, the Grant-in-Aid for JSPS Fellows 19J10679 and the
MIUR grant “Dipartimenti di Eccellenza” 20182022, CUP: E11G18000350001, DISMA, Politecnico di
Torino.

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OCAMI Reports Vol. 1 (2021)[3] A. Demoulin, Sur les surfaces Ω, C. R. Acad. Sci. Paris 153, (1911), 927–929.
[4] A. Demoulin, Sur les surfaces R et les surfaces Ω, C. R. Acad. Sci. Paris 153, (1911), 590–593
and 705–709.
[5] M. Pember, Lie applicable surfaces, to appear in Communications in Analysis and Geometry.
(J. Cho) Institut für Diskrete Mathematik und Geometrie, Vienna University ofTechnology
Email address: jcho@geometrie.tuwien.ac.at(M. Pember) Dipartimento di Matematica, Politecnico di Torino
Email address: mason.pember@polito.it(G. Szewieczek) Institut für Diskrete Mathematik und Geometrie, Vienna Universityof Technology
Email address: gudrun@geometrie.tuwien.ac.at

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The 27th OCU International Academic Symposium
2021, pp.61–62

COMPUTING CONSTRAINED WILLMORE SURFACESULRICH PINKALL
Abstract. An immersion f : M → R3 of a surface M is called constrained Willmore
if f minimizes the Willmore functional W (f ) compared to local variations of f thatchange the induced metric only conformally. We present a version of this variational
problem in the context of Discrete Diﬀerential Geometry and use it to develop an
eﬃcient numerical algorithm. This algorithm allows us to experimentally explore thepossible shapes of Willmore minimizers within a given conformal class.

1. Smooth Constrained Willmore Surfaces
Let M be a Riemann surface, i.e. an oriented surface with a conformal structure.
Then a conformal immersion f : M → R3 is called a constrainedWillmore immersion∫
if f is a critical point of the Willmore functional W (f ) = M H 2 among all conformal
immersions of M into R3 [1].
Constrained Willmore surfaces are solutions of a certain variational problem underconstraints. The Lagrange multiplier corresponding to the conformality constraint turns
out to be a holomorphic quadratic diﬀerential on M .
2. Discrete Constrained Willmore Surfaces
Let M be a simplicial surface with vertex set V . A metric on M assigns a length
to each edge of M . If we are given conformal factors at the vertices and multiply thelength of each edge by the two factors at its end points, the resulting new metric is said
to be conformally equivalent to the old one [3]. Suppose now we have a discrete version
W of the Willmore functional, deﬁned on maps f : M → R3 . We want to minimize
W while allowing only variations of f that preserve the conformal class of the metricinduced on M by f . For a critical point of this variational problem, the Lagrange
multiplier corresponding to the conformality constraint can then be interpreted as adiscrete analog of a holomorphic quadratic diﬀerential [2].
3. Numerical Strategy
We use a novel approach to constrained variational problems, called Competive Gradient Descent [4]. This method allows us to eﬃciently minimize Willmore energy while
preserving conformality up to machine precision.

Date: Received on March 22, 2021.
Key words and phrases. Constrained Willmore, Discrete Riemann Surfaces.
This research was partially supported by SFB Transregio 109.

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Figure 1. Constrained Willmore surfaces (blue) obtained by minimizing the Willmore functional while preserving the conformal type. The
corresponding initial surfaces are shown in gray.
4. Applications
Minimizers of geometric energies under a conformality constraint can be practicallyuseful for a variety of modelling tasks. They are a two-dimensional analog for elastic
curves (minimizers of bending energy under a length constraint).
References[1] Christoph Bohle, G. Paul Peters and Ulrich Pinkall, Constrained Willmore surfaces. Calculus of
Variations and Partial Diﬀerential Equations 32 (2008).
[2] Richard Kenyon and Wai Yeung Lam, Holomorphic quadratic diﬀerentials on graphs and thechromatic polynomial. Journal of Combinatorial Theory, Series A (2020).
[3] Ulrich Pinkall and Boris Springborn, A discrete version of Liouville s theorem on conformal maps.
To appear in Geometriae Dedicata.
[4] Florian Schäfer and Anima Anandkumar, Competitive gradient descent. arXiv:1905.12103v3(2020).
(U. Pinkall) Technical University of Berlin
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2021, pp.63

BRANCHED COVERING SURFACES - NEW SHAPES, NEW
MATERIALS AND NEW PROCESSESKONRAD POLTHIER

The classic geometric view on smooth surfaces hardly ﬁts to the complex and often multiscale physical surface shapes in nature and, nowadays, in industrial applications.In this talk we will introduce a new class of multi-layered surface shapes derived
from recent algorithms in geometry processing and related to classic complex analysis.
Multivalued functions and diﬀerential forms naturally lead to the concept of branchedcovering surfaces and more generally of branched covering manifolds in the spirit ofHermann Weyl’s book “The Idea of a Riemann Surface” from 1913. We will illustrateand discretize basic concepts of branched (simplicial) covering surfaces starting fromcomplex analysis and surface theory up to their recent appearance in geometry processing algorithms and artistic mathematical designs. Applications will touch discrete anddiﬀerential surface modeling, image and geometry retargeting, optimal surfaces, andnovel weaved geometry representations.

(K. Polthier) Freie Universität Berlin, Germany
Email address: Konrad.Polthier@fu-berlin.de

Date: Received on March 1, 2021.

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The 27th OCU International Academic Symposium
2021, pp.64

SMOOTHNESS OF POLYHEDRAL SURFACES MOTIVATED BYAPPLICATIONS IN COMPUTATIONAL DESIGN AND
FABRICATIONHELMUT POTTMANN

We present recent work on smoothness concepts for polyhedral surfaces and on theintentional deviation from smoothness. This research that has been motivated by applications in computational design and fabrication and concerns(i) the visual regularity of polyhedral surfaces whose faces follow given patterns,(ii) the smoothest visual appearance of polyhedral surfaces and the closely related
problem of(iii) ﬁnding material-minimizing forms and structures, and(iv) the appearance of conical meshes as discrete principal curvature parameterizations and as discrete versions of remarkable pleated structures generated byfolding paper along curved creases.
(H. Pottmann) KAUST, Saudi Arabia
Email address: helmut.pottmann@kaust.edu.sa

Date: Received on March 22, 2021.

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The 27th OCU International Academic Symposium
2021, pp.65

3D NETS AND HANDLEBODY DECOMPOSITIONS OF
3-MANIFOLDSNAOKI SAKATA
Abstract. In this talk, we will introduce handlebody decompsitions of 3-manifolds.
We will also discuss a relation between handlebody decompositions of the 3-dimensional
torus and 3D nets.

The self-assembly of block copolymers produce periodic microphase separation structures: spherical, cylindrical, lamellar, and bicontinuous structures. A mathematicalmodel of a bicontinuous structure is a triply periodic non-compact surface embedded inthe 3-dimensional space which separates the space into precisely two regions. On theother hand, three-arm star molecules may self-assemble to form triply periodic polycontinuous structures. A mathematical model of a poly-continuous structure is a triply
periodic non-compact “branched surface” which separates the space into some regions.
Hence, each triply periodic poly-continuous structure gives a handlebody decompositionof the 3-dimensional torus.
In this talk, we will introduce handlebody decompositions of 3-manifolds. Furthermore, we will generalize the Reidemeister-Singer theorem on Heegaard splittings tohandlebody decompositions. We will also discuss a relation between poly-continuous
structures and nets (i.e., non-compact graph) in the 3-dimensional space.
This is joint work with Kai Ishihara, Yuya Koda, Masaki Ogawa, Makoto Ozawa,
and Koya Shimokawa.
References[1] K. Ishihara, Y. Koda, M. Ozawa, and K. Shimokawa, Neighborhood equivalence for multibranchedsurfaces in 3-manifolds, Topology Appl. 257 (2019), 11–21.
[2] D. Koenig, Trisections of 3-manifolds, arXiv:1805.11276.
(N. Sakata) Saitama University
Email address: sakata@casis.sakura.ne.jp

Date: Received on February 15, 2020.
This work was supported by JSPS KAKENHI Grant number 17H06463.

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The 27th OCU International Academic Symposium
2021, pp.66

BELTRAMI FIELDS WITH A NON-CONSTANTPROPORTIONALITY FACTOR
MAKO SATO

A Beltrami ﬁeld is the vector ﬁeld u in R3 such that{∇ × u = f u,
∇ · u = 0,
where f is a smooth function. A Beltrami ﬁeld is a solution of the steady Euler equationin ﬂuid mechanics. It also appears in plasma physics as a force-free vector ﬁeld. Whenthe proportional factor f is constant, we call the vector u a strong beltrami ﬁeld. The
existence of strong Beltrami ﬁelds is well known such as the ABC ﬂow, while nontrivial
Beltrami ﬁelds have not been found with a non-constant factor. In this talk we consider
the non-existence result of Beltrami ﬁelds with f having local extrema. This fact wasproved by A. Enciso and D. Peralta-Salas.
Theorem. ([1])
Suppose that the function f is smooth in R3 . If a regular level set f −1 (c) has a connectedcomponent diﬀeomorphic to S2 , then any solution u is identically zero.
The key idea of the proof is to write the equations as a diﬀerential form,{
du = f ∗ u,d ∗ u = 0,
for u = u1 dx1 +u2 dx2 +u3 dx3 , where d is the exterior diﬀerential and ∗ is the Hodge star
operator. We transform the coordinate from Cartesian system x to spherical systemξ − t by introducing a map Φ and deﬁne the diﬀerential form β by a pull-back as,
β = (u1 ◦ Φ)dx1 + (u2 ◦ Φ)dx2 + (u3 ◦ Φ)dx3 .
The diﬀerential form β is a closed 1-form and solves an elliptic equation on S2 .
References[1] A. Enciso and D. Peralta-Salas. Beltrami ﬁelds with a nonconstant proportionality factor are rare.
Archive for Rational Mechanics and Analysis, pages 243–260, (2016).
(M. Sato) Graduate School of Science, Osaka City University
Email address: m20sa020@tx.osaka-cu.ac.jp

Date: Received on March 16, 2021.

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The 27th OCU International Academic Symposium
2021, pp.67–69

TOTALLY UMBILICAL SUBMANIFOLDS IN
PSEUDO-RIEMANNIAN SPACE FORMS
YUICHIRO SATO
Abstract. Totally umbilical submanifolds in pseudo-Riemannian manifolds are deﬁned by the traceless part of the second fundamental form vanishing identically. We
classify the congruent class of full totally umbilical submanifolds in non-ﬂat pseudoRiemannian space forms.

1. Overview
We deﬁne an m-dimensional pseudo-Euclidean space with index p
(
)
p
m
∑
∑
m
m
2
2
Ep := R , h·, ·ip = −
dxi +
dxj ,
i=1

j=p+1

where (x1 , · · · , xm ) are canonical coordinates in Rm . For a constant c ∈ R, we alsodeﬁne hypersurfaces in Em+1p
m+1
Qm
| hx, xip = c}p,c := {x ∈ Ep.
and give them the induced metric by the inclusion into Em+1
p
m+1
2
m
,
Sm
p (r ) := Qp, 1 ⊂ Ep
r2

2
m
m+1
Hm
p (−r ) := Qp+1,− 1 ⊂ Ep+1 .
r2

They are called to be an m-dimensional pseudo-sphere and pseudo-hyperbolic spacewith index p, and of constant curvature r2 , −r2 and geodesically complete. Combining
pseudo-Euclidean spaces to be ﬂat, we call them pseudo-Riemannian space forms. Whenp = 0, simply Riemannian space forms, when p = 1, Lorentzian space forms, that is,
Minkowski spaces, de-Sitter spaces and anti de-Sitter spaces.
For a submanifold in pseudo-Riemannian space forms, we call the submanifold totally
umbilical if the traceless part of its second fundamental form identically vanishes. Totally umbilical submanifolds in pseudo-Riemannian space forms are all extrinsic spheres,that is, their mean curvature vector ﬁelds are parallel with respect to the normal connection. As recent studies, in ﬂat cases, Magid [5] and Ahn–Kim–Kim [1] investigatedin detail. In non-ﬂat cases, there exist studies by Chen [4]. However, its classiﬁcation
is incomplete.
Date: Received on February 23, 2020.
Key words and phrases. totally umbilical submanifolds, pseudo-Riemannian geometry, moduli
spaces.

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2. main result
Let ϕ : Msm→ Snp (1) be a full totally umbilical isometric immersion.

Theorem 1.
Then, it is congruent to an open portion of one of the following submanifolds:
m+1(1) Sm(1) ; x 7→ (x, 0);s (1) → Ss
m+1m(2) Ss (1)→S(1) ; x 7→ (0, x);
s+1
( )
√
1
m
m+1
(3) Ss→ Ss (1) ; x 7→ (x, 1 − r2 ) (0 < r < 1);
2
(r )
√
1
2
(4) Sm→ Sm+1s+1 (1) ; x 7→ ( r − 1, x) (r > 1);sr2
m+2(5) Sms (1)( → S)s+1 (1) ; x 7→ (1, x, 1);√
1(6) Hm− 2 → Sm+1
1 + r2 ) (r > 0);s+1 (1) ; x 7→ (x,
s
r
(
)
3
5
m+1
x
hx,
→
hx,
(7) Em
S
(1)
;
→
7
xi
−
,
x,
xi
−
.
s
s
s+1
s
4
4
Moreover, when Msm is geodesically complete, it globally coincides with one of theabove.

Theorem 2. Let ϕ : Msm → Hnp (−1) be a full totally umbilical isometric immersion.
Then, it is congruent to an open portion of one of the following submanifolds:
m+1(−1) ; x 7→ (x, 0);(1) Hms (−1) → Hs
m+1m(2) Hs (−1)→H(
) s+1 (−1) ; x 7→ (0, x);√
12(3) Hm− 2 → Hm+1
s+1 (−1) ; x 7→ ( 1 − r , x) (0 < r < 1);
s
r
(
)
√
1
(−1) ; x 7→ (x, r2 − 1) (r > 1);(4) Hm−→ Hm+1ss2r(−1))→ Hm+2(5) Hms+1 (−1) ; x 7→ (1, x, 1);s (√1(6) Sm→ Hm+1(−1) ; x 7→ ( 1 + r2 , x) (r > 0);
s
s
2
r
(
)
5
3
m+1
m
(7) Es → Hs (−1) ; x 7→ hx, xis + , x, hx, xis +.
44Moreover, when Msm is geodesically complete, it globally coincides with one of theabove.
As applications, we consider moduli spaces of isometric immersions between space
forms and non-zero mean curvature parallel submanifolds in pseudo-spheres.
(i) We obtain that moduli spaces of isometric immersions between space forms
which are of the same constant curvature (possibly ﬂat) are non-Hausdorﬀ.
(ii) A full parallel isometric immersion of an irreducible Riemannian symmetric Rspace into an unit sphere is rigid, i.e. it is homothetic to a standard embedding.
However, in the indeﬁnite case, there exist full parallel isometric immersionsof irreducible indeﬁnite symmetric R-spaces into unit pseudo-spheres which are
not homothetic to standard embeddings. See also Blomstrom’s rigidity theorem[3, Theorem 3].

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References[1] S. S. Ahn, D. S. Kim and Y. H. Kim, Totally umbilical Lorentzian submanifolds, J. Korean Math.
Soc. 33 (1996), 507–512.
[2] H. Anciaux, Minimal submanifolds in pseudo-Riemannian geometry, World Scientiﬁc (2011).
[3] C. Blomstrom, Symmetric immersions in pseudo-Riemannian space forms, in: Global Diﬀerential
Geometry and Global Analysis, in: Lecture Notes in Math., vol. 1156, Springer, Berlin, 1985,
pp. 30–45.
[4] B. Y. Chen, Pseudo-Riemannian geometry, δ-invariants and applications, World Scientiﬁc, (2011).
[5] M. A. Magid, Isometric immersions of Lorentz space with parallel second fundamental forms,
Tsukuba J. Math. 8 (1982), 31–54.
(Y. Sato) Tokyo Metropolitan University
Email address: satou-yuuichirou@ed.tmu.ac.jp

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OCAMI Reports
The 27th OCU International Academic Symposium
2021, pp.70–72

REGION CROSSING CHANGE AND REGION SELECT GAMEAYAKA SHIMIZU
Abstract. A region crossing change is a local transformation on knot, link or spatialgraph diagrams switching all the crossings on the boundary of a region. In this talk,we will see studies of region crossing changes and a game “Region Select”.

1. Region crossing change
A knot is an embedding of a circle to S 3 and a link is an embedding of some circles to
S . A spatial graph is an embedding of a graph to S 3 . For a diagram of a knot, link or
spatial graph on S 2 , a region crossing change is a local transformation which switches
the over/under relationship at all the crossings on the boundary of a region as shownin Figure 1. The study of region crossing changes started in Osaka City University in
3

Figure 1. A region crossing change on a spatial-graph diagram.
the early 2010s and the following theorem was shown in [15] 1.
Theorem 1. Any single crossing of a knot diagram can be switched by a ﬁnite numberof region crossing changes.
(See Figure 2.) Hence we can create any status of crossings by region crossing changes
for a knot diagram 2. On the other hand, for link diagrams, we can not switch any
Date: Received on March 15, 2021.
Key words and phrases. game, knot diagram, local move, region crossing change.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics).
1We note that an alternative proof of Theorem 1 was recently given graph-theoretically in [11]. We
also note that Theorem 1 was generalized in [9] to connected spatial-graph diagrams.
2This means that a region crossing change is an “unknotting operation” for knot diagrams, i.e.,
we can transform any knot diagram into a diagram of the trivial knot by a ﬁnite number of region
crossing changes. Then some kinds of “unknotting numbers” on region crossing changes for knots aredeﬁned and studied. See, for example, [8, 12, 13, 14, 15, 17]. We note here that it has been shownthat a region crossing change is an unknotting operation for knots in [1] showing that any knot has adiagram which is deformed into a diagram of the trivial knot by a single n-gon move.

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Figure 2. The crossing c is switched by region crossing changes in any
order at the shaded regions.
single crossing between diﬀerent knot-components by region crossing changes 3. See[4, 6] for region crossing changes on link diagrams 4.
2. Region SelectRegion Select is the following game using the property of region crossing changes([10]). At the beginning, there is a knot projection and each crossing has a lamp which
can be turned on or oﬀ. When a player chooses a region, the lamps on the boundary of
the region are switched. The goal of the game is to light up all the lamps (see Figure
3).

Figure 3. Region Select.
The strong point of this game is that we can clear the game for any knot projection
and any status of lamps at the initial setting by Theorem 1 5.
References[1] H. Aida, Unknotting operation for polygonal type, Tokyo J. Math. 15 (1992), 111–121.
[2] K. Ahara and M. Suzuki, An integral region choice problem on knot projection, J. Knot Theory
Ramiﬁcations 21 (2012), 1250119, 20 pp.
[3] S. Brown, F. Cabrera, R. Evans, G. Gibbs, A. Henrich and J. Kreinbihl, The region unknotting
game, Math. Mag. 90 (2017), 323–337.
3

Imagine, for example, the Olympic rings. The parity of the “linking number” of each component
and the other components is preserved by region crossing changes, whereas the value is deﬁnitelychanged by a single crossing change at a crossing of diﬀerent components.
4
See also [7, 16] for region crossing changes on spatial graphs. See [5] for region crossing changes
on some surfaces.
5
In [2], it is shown that a Region Select of “integer lamp” is also well-deﬁned. For two-player gameof region crossing changes, see [3].

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OCAMI Reports Vol. 1 (2021)[4] Z. Cheng, When is region crossing change an unknotting operation?, Math. Proc. Cambridge
Philos. Soc. 155 (2013), 257–269.
[5] J. Cheng, Z. Cheng, J. Xu and J. Zheng, Region crossing change on surfaces, preprint.
(arXiv:1908.06864)[6] Z. Cheng and H. Gao, On region crossing change and incidence matrix, Sci. China Math. 55(2012), 1487–1495.
[7] Y. Funakoshi, K. Noguchi and A. Shimizu, Unknottability of spatial graphs by region crossingchanges, preprint. (arXiv:2009.12119)[8] S. Goodhill, A. Laorance, V. Gonzales, J. Rattray and A. Zeh, The multi-region index of a knot,
J. Knot Theory Ramiﬁcations 29 (2020), 2050022.
[9] K. Hayano, A. Shimizu and R. Shinjo, Region crossing change on spatial-graph diagrams, J. Knot
Theory Ramiﬁcations 24 (2015), 1550045, 12 pp.
[10] A. Kawauchi, K. Kishimoto and A. Shimizu, Knot theory and game (in Japanese), Mathematicalworld viewing from the game “Region Select”, Asakura Publishing Co., Ltd. (2013).
[11] O. Dasbach and H. Russell, Equivalence of edge bicolored graphs on surfaces, Electron. J. Combin.
25 (2018), Paper 1.59, 15 pp.
[12] V. Siwach and P. Madeti, A sharp upper bound for region unknotting number of torus knots, J.
Knot Theory Ramiﬁcations 22 (2013), 1350019, 21 pp.
[13] V. Siwach and P. Madeti, Region unknotting number of 2-bridge knots, J. Knot Theory Ramiﬁcations 24 (2015), 1550053, 20 pp.
[14] A. Shimizu, On region unknotting numbers, RIMS Kokyuroku 1766 (2011), 15–22.
[15] A. Shimizu, Region crossing change is an unknotting operation, J. Math. Soc. Japan. 66 (2014),
693-708.
[16] A. Shimizu and R. Takahashi, Region crossing change on spatial theta-curves, J. Knot Theory
Ramiﬁcations 29 (2020), 2050028.
[17] T. Tanaka, The region index and the unknotting number of a knot, Topology Appl. 219 (2017),
141–151.
(A. Shimizu) National Institute of Technology (KOSEN), Gunma College, 580 Toribacho, Maebashi-shi, Gunma, 371-8530, Japan.
Email address: shimizu@gunma-ct.ac.jp, shimizu1984@gmail.com

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2021, pp.73–74

3-DIMENSIONAL TOPOLOGY AND POLY-CONTINUOUS PATTERNKOYA SHIMOKAWA
Abstract. In this talk we will discuss an application of 3-dimensional topology to
study of microphase-separated structures of block copolymers.

Block copolymers produce spherical, cylindrical, lamellar and bicontinuous patterns
as microphase-separated structures. Typical examples of bicontinuous patterns are Gyroid, D-surface, and P-surface. A mathematical model of such a structure is a triply
periodic non-compact surface embedded in the 3-dimensional space which divides it
into two possibly disconnected submanifolds. We will consider the case where submanifolds are the open neighborhood of networks. Here a network means an inﬁnite graphembedded in the 3-dimensional space. In this case, the bicontinuous pattern is uniquelydetermined by networks up to isotopy. We say such a bicontinuous pattern is associatedwith a network.
On the other hand, triblock-arm star-shaped molecules, for example, yield a tricontinuous pattern. One mathematical model of such a tricontinuous (resp. poly-continuous)
pattern is a triply periodic non-compact multibranched surface (or more generally polyhedron) dividing the 3-dimensional space into 3 (resp. several) possibly disconnected
non-compact submanifolds. We assume that each submanifold is the open neighborhood of one component of three (resp. several) entangled networks. We call such amultibranched surface a tricontinuous pattern (resp. poly-continuous pattern). See Figure 1. The relation between poly-continuous patterns and networks is not obvious inthis case. Two diﬀerent poly-continuous patterns are associated with one network andvice versa. In this talk, we will give a condition for poly-continuous patterns to give thesame network[1]. We will also show that two poly-continuous patterns can be related
by a ﬁnite sequence of moves and discuss further applications.
References[1] K. Ishihara, Y. Koda, M. Ozawa, and K. Shimokawa, Neighborhood equivalence for multibranchedsurfaces in 3-manifolds, Topology Appl. 257 (2019), 11-21.

Date: Received on February 16, 2020.
Key words and phrases. 3-manifold, handlebody decomposition, polymer.
This work was partly supported by JSPS KAKENHI Grant Numbers 16H03928, 17H06463,
19K21827 and JST CREST JPMJCR17J4.

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Figure 1. Tricontinuous pattern and entangled network(K. Shimokawa) Department of Mathematics, Saitama University
Email address: kshimoka@rimath.saitama-u.ac.jp

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2021, pp.75–76

DISCRETE FLAT FRONTS IN HYPERBOLIC SPACE VIADISCRETE CYCLIC CIRCLE CONGRUENCES
GUDRUN SZEWIECZEKAbstract. We shall discuss an integrable discretization of cyclic circle congruences
and their associated cyclic coordinate systems. It is then demonstrated that special
discrete cyclic circle congruences give rise to parallel families of discrete ﬂat fronts in
hyperbolic space.

1. Discrete cyclic circle congruences
In classical diﬀerential geometry, a 2-parameter family of circles is called cyclic if
it admits a 1-parameter family of orthogonal surfaces [7]. It is well-known that for
any Ribaucour pair of surfaces, the circle congruence that orthogonally intersects theenveloped spheres in the point pairs of the Ribaucour transformed surfaces is cyclic.
We shall discuss an integrable discrete notion for cyclic circle congruences and showthat any discrete Ribaucour pair gives rise to such a circle congruence [4]. Although thediscussed concepts are invariant under Möbius transformations, we will work in a Liesphere geometric setup, where we consider a ﬁxed Möbius geometry as subgeometry.
Enhancing methods developed in [8], this framework enables us to characterize discrete
cyclic circle congruences by the existence of M-Lie inversions that interchange adjacentcircles and induce a ﬂat connection for the congruence. Moreover, these M-Lie inversions will provide a way to eﬀectively construct the orthogonal surfaces as well as the
canonically associated discrete cyclic systems, that is, discrete orthogonal coordinate
systems with one family of circular trajectories.
1.1. Discrete ﬂat fronts in hyperbolic space. Smooth and discrete intrinsicallyﬂat surfaces in hyperbolic space (so-called ﬂat fronts) are obtained in various ways: forexample, as space form projections of Ω-surfaces spanned by two isothermic sphere congruences each lying in a parabolic linear sphere complex [1, 2] or they can be recoveredfrom holomorphic data by using Weierstrass’ and Bryant’s representations [3, 5, 9].
Furthermore, parallel families of smooth ﬂat fronts in hyperbolic space stem from aDarboux pair in S 2 (cf. [1, 6]) and are therefore orthogonal surfaces of a particularcyclic circle congruence.
We pursue the latter approach and shall demonstrate how discrete ﬂat fronts arise as
orthogonal surfaces of a discrete cyclic circle congruence that is associated to a totally
umbilic discrete Darboux pair.
Date: Received on March 17, 2021.
Key words and phrases. discrete diﬀerential geometry, discrete orthogonal coordinate systems, discrete cyclic systems, cyclic circle congruences, discrete ﬂat fronts.

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References[1] F.E. Burstall, U. Hertrich-Jeromin and W. Rossman, Lie geometry of ﬂat fronts in hyperbolic
space. C. R. Acad. Sci. Paris, Ser. I 348, 661-664 (2010)[2] F.E. Burstall, U. Hertrich-Jeromin and W. Rossman, Discrete linear Weingarten surfaces. Nagoya
Math. 231 (2018)[3] J.A. Gálvez, A. Martı́nez, F. Milán, Flat surfaces in the hyperbolic 3-space. Math. Ann. 316,
419–435 (2000)[4] U. Hertrich-Jeromin and G. Szewieczek, Discrete cyclic systems, in preparation.
[5] T. Hoﬀmann, W. Rossman, T. Sasaki, M. Yoshida, Discrete ﬂat surfaces and linear Weingarten
surfaces in hyperbolic 3-space. Trans. Amer. Math. Soc. 364 no. 11, 5605–5644 (2012)[6] M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of ﬂat fronts in
hyperbolic space. Paciﬁc J. Math. 221, 303 – 352 (2005)[7] A. Ribaucour, Mémoire sur la théorie générale des surfaces courbes. Journal de mathématiquepures et appliquées No.4 (1891)[8] T. Rörig and G. Szewieczek, The Ribaucour families of discrete R-congruences, Geometriae dedicata, https://doi.org/10.1007/s10711-021-00614-1 (2021).
[9] W. Rossman, M. Yasumoto, Discrete linear Weingarten surfaces with singularities in Riemannianand Lorentzian spaceforms. Adv. Stud. Pure Math. 78, 383–410, Math. Soc. Japan, Tokyo, 2018.
(G. Szewieczek) Institute of Discrete Mathematics and Geometry, TU Wien. Wiedner
Hauptstrasse 8-10/104, 1040 Vienna, Austria
Email address: gudrun.szewieczek@tuwien.ac.at

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The 27th OCU International Academic Symposium
2021, pp.77

RELAXATION OF OPTIMAL TRANSPORT PROBLEM
VIA BREGMAN DIVERGENCEASUKA TAKATSU
Abstract. My talk concerns with mathematical foundations and an iterative processof the relaxed optimal transport problem via Bregman divergences.

I discuss the following optimal transport problem: Given C = (cij )1≤i,j≤N ∈ MN (R)NN
and x = (xi )N
with
i=1 , y = (yj )j=1 ∈ R
xi , yj ≥ 0 for 1 ≤ i, j ≤ N

and

N
∑
i=1

xi =

N
∑

yi = 1,i=1

ﬁnd Π = (πij )1≤i,j≤N ∈ MN (R) minimizingN∑

cij πiji,j=1

under the constraints
πij ≥ 0,N∑k=1

πik = xi ,N∑

πkj = yj ,for 1 ≤ i, j ≤ N .

k=1

Since this is a linear program, a minimizer may lie on the boundary of the constraint set
and not be unique. Furthermore, a gradient descent is not useful to ﬁnd a minimizer.
To avoid this problem, a relaxation of the optimal transport problem via strictly convex
functions is useful. Actually, in data sciences, such a relaxation achieves substantial
success, where mathematical argument is sometimes not rigorous.
As a strictly convex function, focusing attention on a Bregman divergence, I provide
the mathematical foundations and an iterative process based on a gradient descent.
The advantage of this iterative process that the output is always transport plan even
if one stops the iteration at the ﬁnite step.
References[1] Takatsu, A.: Relaxation of optimal transport problem via strictly convex function, Preprint available at arXiv:2102.07336 (2021)(A. Takatsu) Tokyo Metropolitan University/Tohoku University
Email address: asuka@tmu.ac.jp
Date: Received on March 4, 2021.
Key words and phrases. optimal transport and entropic relaxation and Bregman divergences.
This work was partly supported by JSPS Grant-in-Aid for Scientiﬁc Research (KAKENHI)
19K03494, 19H01786.

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2021, pp.78–79

PARAFERMIONIC BASES OF STANDARD MODULES FORTWISTED AFFINE LIE ALGEBRASRYO TAKENAKA
Abstract. The parafermionic current appeared in the context of conformal ﬁeld
theories in two dimensions in mathematical physics. In this talk, we consider certain
combinatorial bases of the parafermionic space for twisted aﬃne Lie algebras which
are spanned by the monomials of the coeﬃcients of the parafermionic current. As aconsequence we obtain the character formula for the parafermionic space.

1. Introduction
For an aﬃne Lie algebras ĝ, we have an irreducible ĝ-module called the standard
module. The standard module has a natural subquotient called parafermionic space
related to conformal ﬁeld theory. A. Kuniba, T. Nakanishi and J. Suzuki [3] conjectured
character formulas of the parafermionic spaces associated to the untwisted aﬃne Liealgebras. These character formulas were proved by G. Georgiev [4] in the simply-lacedcases and M. Butorac, S. Koz̆ić, and M. Primc [1] in the untwisted non-simply-lacedcases.
2. main result
Let g be a simply-laced simple Lie algebra, namely, of type A, D, E. Then we havethe Dynkin diagram automorphism ν and the twisted aﬃne Lie algebra ĝ[ν] obtained
from ν. For the standard module L(kΛ0 ) of ĝ[ν], we can deﬁne the principal subspace
WL(kΛ0 ) (cf. [2]). By using the canonical projection, we obtain the parafermionic bases
from combinatorial bases of the principal subspace. Finally, we calculate the characterof the parafermionic space from the character of the principal subspace.
References[1] M. Butorac, S. Koz̆ić, and M. Primc, Parafermionic bases of standard modules for aﬃne Liealgebras, Math. Z (2020).
[2] M. Butorac, C. Sadwski, Combinatorial bases of principal subspaces of modules for twisted aﬃne
(2)
(2)
(2)
(3)
Lie algebras of type A2l−1 , Dl , E6 and D4 .
[3] A. Kuniba, T. Nakanishi, J. Suzuki, Characters in Conformal Field Theories from Thermodynamic
Bethe Ansatz, Modern Phys. Lett. A 08 (1993), 1649-1659.
[4] D. Gepner, New conformal ﬁeld theories associated with Lie algebras and their partition functions,
Nuclear Phys. B 290 (1987), 10-24.
Date: Received on March 17, 2021.
Key words and phrases. aﬃne Lie algebra, vertex algebra, parafermionic space.

The 27th OCU Symposium(R. Takenaka) Osaka City University
Email address: m20sa023@zp.osaka-cu.ac.jp

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2021, pp.80–81

AN f -TWISTED ALEXANDER MATRIX FOR KNOT QUANDLES.
YUTA TANIGUCHI
Abstract. In a knot theory, knot invariants play an important role. Recently, A.
Ishii and K. Oshiro introduced a new knot invariant, which is called an f -twisted
Alexander matrix. An f -twisted Alexander matrix is a qundle version of an Alexander
matrix for knot groups. In this talk, we show that the f -twisted Alexander matrix isa stronger knot invariant than the Alexander invariant.

1. Introduction
1.1. Knot invariant. A knot is a smooth submanifold of the Euclidian 3-space R3
which is homeomorphic to a circle. We say that two knots are equivalent when there
exists an autohomeomorphism of R3 carrying one to another. A knot invariant is a map
from the equivalence class of knots to a set. In a knot theory, knot invariants play animportant role.
A typical example of a knot invariant is the knot group. The knot group is thefundamental group of the complement of a knot. An another typical example of a knot
invariant is the Alexander invariant [1]. This invariant can be obtained from a knot
group via the Alexander matrix [2]. Since then Alexander invariants have been used as
the most fundamental tools in studying various problems.
1.2. Quandle. A quandle [4, 5] is an algebraic structure whose axioms was motivatedin a knot theory. D. Joyce and S. V. Matveev associated a quandle to a knot, whichis called the knot quandle. Since then many knot invariants using knot quandles havebeen introduced and studied.
2. Main result
2.1. f -twisted Alexander matrix. In [3], A. Ishii and K. Oshiro introduced thenotion of an f -twisted Alexander matrix of a knot quandle. This notion is a quandle
version of the Alexander matrix of a knot group. They showed that the Alexander
invariant can be obtained from the f -twisted Alexander matrix.
2.2. Main result. The following three facts are known.
• The knot quandle is a stronger invariant than the knot group ([4, 5]).
• The Alexander invariant depends on the knot group ([2]).
• The f -twisted Alexander matrix depends on the knot quandle ([3]).
By the above facts, we have the following Question.
Date: Received on March 15, 2021.
Key words and phrases. Alexander invariant, Quandle, f -twisted Alexander matrix.

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Question 1. Is the f -twisted Alexander matrix a stronger invariant than the Alexander
invariant?
The following Theorem is a main result of this talk.
Theorem 1. Let K1 and K2 be the knots as shown Figure 1. We can distinguish theseknots by using an f -twisted Alexander matrix.

Figure 1
It is known that the knot group of K1 and the knot group of K2 are group isomorphic.
Thus, the both of K1 and K2 are the same Alexander invariant.
Therefore, Theorem 1 gives a positive answer to Question 1.
References[1] J. W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928),
275-306[2] R. H. Fox, Free diﬀerential calculus. II. The isomorphism problem of groups, Ann. of Math. (2)59 (1954), 196-210.
[3] A. Ishii and K. Oshiro, Twisted derivatives with Alexander pairs for quandles. preprint.
[4] D. Joyce. A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra. 23 (1982),
no. 1, 37-65.
[5] S. Matveev. Distributive groupoids in knot theory. Math. USSR. Sbornik. 47 (1884), 73-83.
(Y. Taniguchi) Osaka City University
Email address: m19sa012@du.osaka-cu.ac.jp

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2021, pp.82–83

QUANDLE COLORING QUIVERS OF VIRTUAL LINKS USINGVIRTUAL QUANDLESRYOTARO UEDA
Abstract. S.Nelson and K.Cho introduced the notion of a quandle coloring quiver,which is a quiver-valued classical link invariant. In this talk, we extend Nelson-Cho’s
invariant to virtual links using virtual quandle colorings. Properties similar to classicallink’s quandle coloring quiver studied by Yuta Taniguchi are discussed.

1. Virtual quandle colorings
Deﬁnition 1. A virtual quandle is a pair (X, f ) of a quandle X and f ∈ Aut(X).
Let D be a virtual link diagram. A virtual arc of D is an oriented interval in D
divided by under and virtual crossings. We denote the set of the virtual arcs of D by
VA(D).
Deﬁnition 2. Let (X, f ) be a virtual quandle and D be a virtual link diagram. A
map c : VA(D) → X is an (X, f )-coloring if c satisﬁes the following condition at every
crossing of D as shown in Figure 1.

Figure 1. The coloring condition.
We denote by Col(X,f ) (D) the set of (X, f )-colorings of D.
2. Virtual quandle coloring quivers
K. Cho and S. Nelson [2] introduced the notion of a quandle coloring quiver. We
extend Nelson-Cho’s invariant to virtual links using virtual quandle colorings.
Deﬁnition 3. Let (X, f ) be a virtual quandle and D be a virtual link diagram. For
any subset S ⊂ Comf (X) := {g ∈ End(X) | f ◦ g = g ◦ f }, the virtual quandle coloring
quiver of D, which is denoted by V QS(X,f ) (D), is the quiver with a vertex for each
Date: Received on March 15, 2021.
Key words and phrases. virtual knot, quandle, quandle coloring quiver.

The 27th OCU Symposium

83(X, f )-coloring c ∈ Col(X,f ) (D) and an edge directed from v to w when w = g ◦ v foran element g ∈ S.
Example 4. Let D be a virtual Hopf link diagram in Figure 2. Let (R3 , f ) be a virtual
quandle deﬁned by the dihedral quandle R3 and f ∈ Aut(R3 ) deﬁned by f (x) = 2x.
We put g ∈ Comf (R3 ) deﬁned by g(x) = 0. Let S be a subset of Comf (R3 ) consistingof the element g. Then a virtual quandle coloring quiver V QS(R3 ,f ) (D) is the following
oriented graph.

D

V QS(R3 ,f ) (D)Figure 2. An example of virtual quandle coloring quiver.
3. Main results
Theorem 5. If two virtual link diagrams D and D′ are related by generalized Reidemeister moves, then the virtual quandle coloring quivers V QS(X,f ) (D) and V QS(X,f ) (D′ )are isomorphic for any ﬁnite virtual quandle and S ⊂ Comf (X).
By Theorem 5, the virtual quandle coloring quiver is a virtual link invariant.
Remark 6. By deﬁnition, if virtual quandle coloring quivers V QS(X,f ) (D) and V QS(X,f ) (D′ )are isomorphic, then we have |Col(X,f ) (D)| = |Col(X,f ) (D′ )|.
The converse is not true. However, in a certain case, the converse is true.
Theorem 7. Let D and D′ be virtual link diagrams, p be a prime, and (Rp , f ) bea virtual quandle deﬁned by the dihedral quandle Rp and for any f ∈ Aut(Rp ). If|Col(Rp ,f ) (D)| = |Col(Rp ,f ) (D′ )|, then for any S ⊂ Comf (Rp ), the virtual quandle coloring quivers V QS(Rp ,f ) (D) and V QS(Rp ,f ) (D′ ) are isomorphic.
References[1] D. Joyce, A classifying invariant of knots, the knot quandle. J. P ure Appl. Algebra. 23 (1982),
no.1, 37-65.
[2] K. Cho and S. Nelson, Quandle coloring quivers. J. Knot T heory Ramications.28 (2019), no. 1,
1950001, 12 pp.
[3] Louis H. Kauﬀman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663-690, DOI
10.1006/eujc.1999.0314.
[4] S. Matveev, Distributive groupoids in knot theory. M ath. U SSR. Sbornik. 47 (1884), 73-83.
[5] V. O . Manturov, On Invariants of Virtual Links, Acta Applicandae M athematicae 72, 295-309(2002). https://doi.org/10.1023/A:1016258728022[6] Y. Taniguchi, Quandle coloring quivers of links using dihedral quandles. preprint,
arXiv:2004.12437.
(R. Ueda) Osaka University
Email address: u081574k@ecs.osaka-u.ac.jp

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2021, pp.84–85

PRESENTATION OF SURFACE LINKS USING PLAT FORMSJUMPEI YASUDA
Abstract. A plat form for links is a presentation of classical links using braids, andthis is another aspect of the bridge decomposition for links. In this talk, we introduce
a method of presenting surface links using braided surfaces, which we call plat forms.
Our method can be applied to not only orientable surface links, but also non-orientablesurface links.

1. Preliminaries
Let m be a positive integer, I = [0, 1] be an interval, D = I 2 be a 2-dimensional disk
and Qm be a subset of D deﬁned as
)
(
k
1
,
∈ D | k = 1, 2, · · · , m}.
Qm := {
2 m+1
A braid of degree m (or simply called m-braid ) is a properly embedded compact curve
β in D × I satisfying following two conditions.
(1) A restriction map π := pr2 |β : β → I is a covering map of degree m.
(2) ∂β = Qm × ∂I.
A link L is in a plat form if L is a braid of even degree equipped with caps in the
picture below. This is an example of a plat form of the ﬁgure-eight knot, which is a
4-braid and four caps.

A braided surface is a generalization of a braid. Let D1 , D2 be 2-dimensional disks
and pri : D1 × D2 → Di (i = 1, 2) be the i-th factor projection, and y0 ∈ ∂D2 be a ﬁxedbase point. A (poinited ) braided surface of degree m is a properly embedded compact
surface S ⊂ D1 × D2 such that satisfying following conditions ([6],[7]).
(1) π := pr2 |S : S → D2 is a branched covering map of degree m.
(2) ∂S ⊂ D1 × ∂D2 is closed m-braid.
(3) pr1 (π −1 (y0 )) = Qm .
In particular, a braided surface with ∂S = Qm × ∂D2 is called a 2-dimensional braid orsurface braid (O.Viro, cf. [5]).
Date: Received on March 4, 2021.
Key words and phrases. surface link, braided surface, plat form, bridge index .

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2. A Plat form for Surface links
Suppose that S is decomposed into two 4-balls, S 4 = (D1 × D2 ) ∪∂ (D1 × D2′ ). An
annular surface system is a disjoint union of annuli or Möbius bands. A wicket annularsurface system in D1 × D2′ is a properly embedded annular surface system in D1 × D2′
satisfying a certain condition using the notion of wickets [2].
4Deﬁnition. A surface link F in S 4 is in a plat form if it is a braided surface of evendegree (in D1 × D2 ) equipped with a wickets annular surface system (in D1 × D2′ ):F = S ∪∂ A ⊂ S 4 .
In particular, F is a surface link in a genuine plat form if S is a 2-dimensional braid.
The following are main results in this talk.
Theorem 1. Every (not necessarily orientable) surface link is equivalent to a surface
link in a plat form.
Theorem 2. Every orientable surface link is equivalent to a surface link in a genuine
plat form.
Let Braid(F ) be the braid index of F [4], and Plat(F ) be a minimal half degree of abraided surface whose plat closure is equivalent to a surface link F .
Theorem 3. Then the following inequality is satisﬁed:
Plat(F ) ≤ Braid(F ).
This is analogue to the following inequality for any classical link L:
bridge(L) = plat(L) ≤ braid(L).
References[1] Joan S. Birman, on the stable equivalence of plat representations of knots and links, Can.J.Math.,Vol. XXVIII, No.2, 1976, pp.264-290.
[2] Tara Brendle and Allen Hatcher. Conﬁguration spaces of rings and wick-ets. Commentarii Mathematici Helvetici, 88, 05 2008.
[3] Hugh M. Hilden. Generations for two subgroups of the braid group. Paciﬁc J. Math., 59(2), 1975.
[4] Seiichi Kamada, Surfaces in R4 of braid index three are ribbon, J. Knot Theory Ramiﬁcations,
1(2):137–160, 1992.
[5] Seiichi Kamada, A characterization of groups of closed orientable surfaces in 4-space, Topology,
33(1):113-122, 1994.
[6] Lee Rudolph. Braided surfaces and Seifert ribbons for closed braids. Comment. Math. Helv.,
58(1):1–37, 1983.
[7] Lee Rudolph. Special positions for surfaces bounded by closed braids. Rev. Mat. Iberoamericana,
1(3):93–133, 1985.
(Osaka University Graduate School of Science) Osaka University Graduate School of Science
Email address: u444951d@ecs.osaka-u.ac.jp

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The 27th OCU International Academic Symposium
2021, pp.86

STATISTICAL MIRROR SYMMETRY
JUN ZHANG

A parametric statistical model is a family of probability density functions over a given
sample space, whereby each function is indexed by a parameter taking value in somesubset of Rn . Treating parameterization as a local coordinate chart, the family formsa manifold M endowed with a Riemannian metric g given by the Fisher-information(the well-known Fisher-Rao metric). The classical theory of information geometry
prescribes a family of dualistic, torsion-free alpha-connections constructed from AmariChensov tensor as deformation from the Levi-Civita connection associated with g. Here
we prescribe an alternative geometric framework of the manifold M by treating theparameter as an aﬃne parameter of a ﬂat connection and then prescribing its dual
connection (with respect to g) as one that is curvature-free but carries torsion. From
our new model, we are able to construct a pair of distinct objects on the tangent
bundle TM based on the Sasaki lift of g and a canonical split using data from the basemanifold M . The pair consists of a Hermitian structure and an almost Kahler structure
simultaneously constructed that are in “mirror correspondence.” In analogous to mirror
symmetry in string theory between a complex manifold on the one side and a symplectic
manifold on the other, we call this “statistical mirror-symmetry,” and speculate its
meaning in the context of statistical inference. (Joint work with Gabriel Khan).
The University of Michigan, Ann Arbor
Email address: junz@umich.edu

Date: Received on March 20, 2021.

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2021, pp.87

情報と幾何
伊師 英之

情報を幾何学的に表現するということは，様々なグラフを使うことで日常的に行われている．我々が実際に目で見えるのは３次元までであるが，数学的な認識能力を訓練することにより，データを高次元の幾何と結びつけて新たな洞察が得られるようになる．
たとえば，「１００個のデータを１００次元空間の１点と考える」という視点から見えてくるものがある．本講演の前半では，そのような考え方の例として，統計分析の基本的な道具である相関係数を説明する．
甘利俊一氏が創始した情報幾何学では，考えたい情報（正確には確率分布）を微分幾
何と結びつける．すなわち，確率分布がつくる集合を一つの空間と考えて，そこに双対接続構造なるものによって二種類の「空間の曲がり方（真っ直ぐさ）」を定める．これによって統計学・情報理論・機械学習理論などに統一的な理解と新しいアイディアが与
えられ，その理論は世界中で活発に発展している．講演の後半では，この情報幾何学の一端を簡単な例で紹介する．

References[1] 甘利俊一・長岡浩司「情報幾何の方法」 岩波講座 応用数学, 岩波書店, 1993.
[2] 赤穂昭太郎「情報幾何と機械学習」，計測と制御 44 巻, 299–306.
[3] 藤原彰夫「情報幾何学の基礎」 数理情報科学シリーズ 29, 牧野書店, 2015.
[4] 甘利俊一「新版 情報幾何学の新展開」 SGC ライブラリ 154, サイエンス社, 2019.
(H. Ishi) 大阪市立大学 大学院理学研究科
Email address: hideyuki@sci.osaka-cu.ac.jp

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2021, pp.88

数学×工芸：テンソル演算と西陣織
巴山 竜来
TATSUKI HAYAMA

数学と工芸は全く異なるもののように見えますが，近年の計算機やデジタル工作機械の発展は，その両者の橋渡しを可能にしています．例えば 3D 幾何形状を表す数式が与えられたとき，その式から 3D オブジェクトデータを生成し，3D プリンタで出力すれば，それを実物のものとして手に取ることができます．数式を操作することによって，人間の手で扱うには複雑すぎる形状や無数のバリエーションを生み出すこともできます．こういったデジタルデータの物質化はデジタルファブリケーションとも呼ばれ，ものづくりにも大きな影響を与えています．
この講演では，とくに織物に焦点を当て，講演者が最近取り組んできた西陣織企業
細尾との研究開発プロジェクトを紹介し，数学の織物への応用についてお話しします．
織物はタテ・ヨコに直交する糸の交差からなり，布をつくるための糸の交差情報は二次元バイナリ配列データと見なすことができます．つまり織物をつくることは，原始的なデジタルファブリケーションなのです．このプロジェクトでは，ジャカード織機と呼ばれる大型装置を使い，数理的な手法で作られたデータから織物をつくりました．とくにテンソル演算と呼ばれる計算がデータ生成において重要な役割を果たしており，大量のテンソル演算を高速に計算するためには，近年大きく進展している GPU 演算装置と計算実行システムが必要不可欠です．
専修大学 Senshu University
Email address: hayama@isc.senshu-u.ac.jp

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2021, pp.89–90

GEOMETRY IN VORTEX DYNAMICSALBERT CHERN
Abstract. Vorticity tends to concentrate in a nearly inviscid ﬂuid. This results insingular structures called vortex ﬁlaments, which evolve in the ﬂuid in an intriguing
fashion. As a dynamical system on the inﬁnite dimensional space of space curves, vortex dynamics reveals surprising relations between the geometric invariants of curves.
For example, in the inviscid and thin vortex limit, we ﬁnd a new isoperimetric characterization to Euler elastica. In the second part of the talk, I will explain a new
approach to model the general variable thickness, viscous vortex ﬁlaments. The underlying theory is an instance of the dissipative geodesic equation on a Kaluza–Klein
Riemannian manifold. The resulting equation can describe such varied phenomena as
underwater bubble rings or the intricate “chandeliers” formed by ink dropping into a
ﬂuid.

1. Integrable System of Da Rios Equation
In the classical theory of vortex dynamics in an inviscid ﬂuid, the dynamics of vortex
ﬁlament γ : S1 → R3 is governed by the integrodiﬀerential equation∂
γ = uBS ◦ γ∂t(1)where uBS is the (regularized) Biot–Savart ﬁeld generated by γ. Asymptotically, as the
thickness of the vortex ﬁlament approaches zero, (1) becomes the Da Rios equation∂
γ = γ ′ × γ ′′ .
∂t(2)Here (·)′ is the derivative with respect to arclength. The
traditional view of (2) is studied by Hashimoto, who foundthat the complex curvature of γ satisﬁes the nonlinear
Schrödinger equation, which is an integrable system. We
revisit this integrable system at the level of γ. This turns
out to be an elementary
integrable system with a symplectic∫
form σMW (γ̇,γ̊) = S1 det(γ̇,γ̊, dγ) introduced by Marsden
and Weinstein in the context of ﬂuids. The most common
geometric quantities of a space curve, including the length, the writhe, the bending
energy, the area projected onto a given plane, and the volume swept by the revolution
Date: Received on February 16, 2020.
Key words and phrases. Da Rios equation, Kaluza–Klein geometry, vortex dynamics.
The work was supported in part by the DFG Collaborative Research Center TRR 109 “Discretizationin Geometry and Dynamics,” the Caltech Center for Information Science & Technology, and theEinstein Foundation Berlin. Additional support was provided by SideFX software.

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about a given line, are all commuting Hamiltonians in the integrable system. This
theory leads to a remarkable 3D isoperimetric problem:
Theorem 1 ([1]). Curves that extremize their lengths subject to a ﬁxed projected area
and a ﬁxed revolved volume are 3D Euler elastica.
2. Riemannian Newtonian Mechanics
In reality, vortex ﬁlaments have ﬁnite and varying thickness, and are subject to viscosity and gravity; the latteris present when the vortices trap materials such as ink or
bubbles under water. Indeed, in these scenarios vortices
exhibit phenomena beyond what Eq. (1) can describe. To
incorporate the mentioned physical eﬀects, we ﬁrst returnto the space of curves with the Marsden–Weinstein symplectic structure. There, the inviscid vortex dynamics turns
out to be the inﬁnite dimensional version of the motion ofa charged particle under magnetic ﬁeld. By the machineryof Kaluza and Klein, these motions are the projection ofgeodesics on a space with one extra dimension. After arriving at such form, we can easily add dissipation and gravitational force, yielding a complete vortex ﬁlament model.
The ﬁnal equation captures phenomena such as ink chandeliers and describes the thickness distribution of an underwater bubble ring [2]. The
inset ﬁgure shows a footage of ink drop developed under water (left) and a simulation
based on our model (right).
References[1] A. Chern, F. Knöppel, F. Pedit and U. Pinkall (2020). Commuting Hamiltonian Flows of Curves inReal Space Forms. In R. Donagi and T. Shaska (Ed.), Integrable System and Algebraic Geometry,
Cambridge University Press, Vol. 1, No. 13. Preprint: arXiv:1809.01394 (2018).
[2] M. Padilla, A. Chern, F. Knöppel, U. Pinkall and P. Schröder (2019). On Bubble Rings and Ink
Chandeliers. ACM Transactions on Graphics, Vol. 38(4), Art. 129.
(TU Berlin) Institute of Mathematics, Technical University of Berlin, Germany
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CMC SURFACES VIA DPW AND XLABSEBASTIAN HELLER

CMC surfaces in space forms are described by their associated families of ﬂat connections. The associated family satisﬁes a reality condition depending on the ambient
space and on the mean curvature. Using loop group factorizations the associated family
can be constructed from a family of meromorphic connections - the DPW potential.
The challenge is to construct a DPW potential such that the loop group factorization
works globally on the underlying surface with non-trivial topology.
In this talk I will discuss examples of CMC surfaces with non-trivial topology constructed by the DPW method in the last ten years. Crucial in this theory has always
been the interplay between with experimental mathematics. The experiments are basedon the implementation of the DPW method in the xlab software by Nick Schmitt. The
talk is based on joint work with with A. Bobenko, L. Heller, N. Schmitt, M. Traizet.
(S. Heller) Institut für Differentialgeometrie, Gottfried Wilhelm Leibniz Universität Hannover
Email address: sheller@math.uni-hannover.de

Date: Received on Fabruary 4, 2020.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics).

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The 27th OCU International Academic Symposium
2021, pp.92–94

STATISTICAL STRUCTURE OF CONSTANT STATISTICALSECTIONAL CURVATURE ON HYPERBOLIC SPACENAOTO SATOH
Abstract. Statistical sectional curvature can be introduced on a statistical manifold
by using the statistical curvature tensor ﬁeld, which is deﬁned as the arithmetic meanof curvature tensor ﬁelds with respect to the pair of dual connections. We construct
the hyperbolic space of constant curvature for any constant c ∈ R. This is a joint
work with H. Furuhata, I. Hasegawa, T. Nakane, Y. Okuyama and K. Sato.

Throughout this paper, M denotes a smooth manifold of dimension m ≥ 2, and all
the objects are assumed to be smooth. Γ(E) denotes the set of sections of a vector
bundle E → M . For example, Γ(T M (p,q) ) means the set of all the tensor ﬁelds on M
of type (p, q).
Let ∇ be an aﬃne connection of M , and g ∈ Γ(T M (0,2) ) a Riemannian metric. We
denote the Levi-Civita connection of g by ∇g .
The pair (∇, g) is said to be a statistical structure on M if ∇ is of torsion free, andthe Codazzi equation (∇X g)(Y, Z) = (∇Y g)(X, Z) holds for any X, Y, Z ∈ Γ(T M ).
Then we can deﬁne another torsion free connection ∇∗ by Xg(Y, Z) = g(∇X Y, Z) +g(Y, ∇∗ X Z), and call ∇∗ the dual connection of ∇ with respect to g. If (∇, g) is astatistical structure, so is (∇∗ , g). We have ∇g = 12 (∇ + ∇∗ ) as well.
g
For a statistical structure (∇, g), we set K(X, Y ) = ∇X Y − ∇X
Y . Then K ∈
(1,2)
Γ(T M
) satisﬁes
K(X, Y ) = K(Y, X), g(K(X, Y ), Z) = g(Y, K(X, Z)).

(1)

Conversely, for a Riemannian metric g if K ∈ Γ(T M (1,2) ) satisﬁes (1), the pair (∇ =∇g + K, g) is a statistical structure.
The curvature tensor ﬁeld for ∇ is deﬁned as R∇ (X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z −∇[X,Y ] Z. For a statistical structure (∇, g) on M , we deﬁne
1∗
S(X, Y )Z = {R∇ (X, Y )Z + R∇ (X, Y )Z}2
and call S ∈ Γ(T M (1,3) ) the statistical curvature tensor ﬁeld of (∇, g) (See [1]).
Date: Received on February 23, 2020.
Key words and phrases. statistical manifold, statistical sectional curvature, hyperbolic space.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics) and Global Station for Big
Data and CyberSecurity, a project of Global Institution for Collaborative Research and Education at
Hokkaido University.

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We remark that a statistical structure is locally determined by R∇ and K (See [2]),ggand S can rewrite by using R∇ and K as S(X, Y )Z = R∇ (X, Y )Z + [K, K](X, Y )Z,
where [K, K](X, Y )Z = K(X, K(Y, Z)) − K(Y, K(X, Z)).
Let Π = X ∧ Y be a 2-dimensional section spanned by orthonormal vector ﬁelds{X, Y } with respect to g. We deﬁne the statistical sectional curvature for Π bygK(Π) := g(S(X, Y )Y, X).
This value is independent of the choice of the basis of Π. We say that (M, ∇, g) is ofconstant statistical sectional curvature c if there exists c ∈ R such that K = c for any
2-dimensional section. (M, ∇, g) is of constant statistical sectional curvature c if andonly if S(X, Y )Z = c{g(Y, Z)X − g(X, Z)Y } for any X, Y, Z ∈ Γ(T M ).
If (M, ∇, g) is of constant curvature c, that is, R∇ (X, Y )Z = k{g(Y, Z)X −g(X, Z)Y }for X, Y, Z ∈ Γ(T M ), then (M, ∇, g) is of constant statistical sectional curvature c.
In Riemannian geometry, the sign of curvature presents the most important characteristics of the space. However, the following statistical manifold suggests that it does
not work for statistical manifolds.
Let (Hm , g) be the hyperbolic space of constant curvature −κ (κ > 0) deﬁned by
Hm = {y = (y 1 , . . . , y m−1 , y m ) ∈ Rm | y m > 0},m∑
g = (κ)−1 (y m )−2(dy j )2 .
j=1

We denote ∂i = ∂y∂ i (i = 1, . . . , m). Deﬁne K ∈ Γ(T Hm (1,2) ) byK(∂i , ∂j ) = δij (α(y m )−1 ∂m + αi ∂i ),K(∂i , ∂m ) = K(∂m , ∂i ) = α(y m )−1 ∂i ,K(∂m , ∂m ) = 2α(y m )−1 ∂mfor α ∈ R, αi ∈ C ∞ (Hm ) and i, j = 1, . . . , m − 1. Then, (Hm , ∇ = ∇g + K, g) is astatistical manifold of constant statistical sectional curvature κ(α2 − 1). Furthermore,
if αi = 0 for all i = 1, . . . , n − 1, (Hm , ∇, g) is of constant curvature κ(α2 − 1).
Denote c = κ(α2 − 1) for short. This statistical manifold shows that for any constantc ∈ R, we can construct a statistical manifold of constant curvature c. See [4] in thecase where α = 1 and αi = 0.
References[1] H. Furuhata and I. Hasegawa, Submanifold theory in holomorphic statistical manifolds, Dragomir
S, Shahid M.H., and Al-Solamy F.R. (eds), Geometry of Cauchy-Riemann Submanifolds, Springer
Singapore (2016), 179-215.
[2] N. Satoh, Local existence of statistical diﬀeomorphisms, JP Journ. of Geom. and Topol., 22 (2019),
73-95.
[3] N. Satoh, H. Furuhata, I. Hasegawa, T. Nakane, Y. Okuyama, K. Sato, M.H. Shahid and A.N.
Siddiqui, Statistical submanifolds from a viewpoint of the Euler inequality, submitted.
[4] H. Shima, The Geometry of Hessian Structure , World Scientiﬁc, (2007).

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OCAMI Reports Vol. 1 (2021)(N. Satoh) Department of Mathematics, Faculty of Science, Hokkaido University,
Email address: n-satoh@math.sci.hokudai.ac.jp

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2021, pp.95

COMPUTING CUT LOCUS OF CONVEX POLYHEDRON USINGWAVEFRONT PROPAGATIONKAZUMA TATEIRIAbstract. We created an algorithm to compute cut loci on convex polyhedra incertain situations. This algorithm is based on a famous algorithm to compute shortest
paths on polyhedral surface, created by Mitchell, Mount and Papadimitriou.

We created an algorithm to compute cut loci on triangulated oriented convex polyhedra, with no internal points on the cut locus having local minima of the distancefunction. Our algorithm is based on a famous algorithm to compute shortest paths onpolyhedral surface, created by Mitchell, Mount and Papadimitriou (MMP) [1]. Ouralgorithm utilizes “intervals”, introduced in the MMP algorithm. Each interval in ouralgorithm is modiﬁed to be a node of a doubly circular linked list and express a circular
arc segment of a smooth wavefront. Our algorithm processes an “event” at each event
point, and there are ﬁve event types. Out poster explains the detail of our algorithm.
References[1] Joseph S. B. Mitchell, David Mount, Christos H. Papadimitriou, The Discrete Geodesic Problem”,
SIAM Journal on Computing, 16(4), 647-668, 1987.
(K. Tateiri) Graduate School of Information Science and Technology, Hokkaido University
Email address: kazuma.tateiri16@gmail.com

Date: Received on February 23, 2020.
Key words and phrases. cut locus, geodesic, polyhedron.
This work was partly supported by Hokkaido University GSB (Global Station for Big Data &Cybersecurity).

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2021, pp.96

CLASSIFICATION OF SMYTH POTENTIALSTADASHI UDAGAWA (WASEDA UNIVERSITY)

Abstract
In the DPW method [2], any holomorphic data (ξ, z0 , ϕ0 ) on a simply-connected
Riemann surface produces a CMC-immersion. Conversely, all CMC-immersions are
generated by holomorphic data. But this corrrespondence is not one-to-one. If two
holomorphic potentials ξ, ξ˜ are gauge equivalent by an element C of the positive loop
group of SL2 C,
ξ˜ = ξ · C := C −1 ξC + CdC,
then the resulting CMC-immersions are the same. In this talk, we consider normalized
potentials ξ of the form:
(
)
1
0
zk
ξ=
dz, n, k ∈ Z, c ∈ C∗ .
λ cz n−k 0
We call this type of potential a Smyth potential. For this potential, we consider thecondition that two holomorphic data give the same CMC-immersion, i.e. we investigate
convergence of the formal solution C of the above equation. As applications, following [1], [3] we construct CMC-immersions with one parameter group of isometries in
R3 , R2,1 .
References[1] J. F. Dorfmeister, M. A. Guest, and W. Rossman, The tt∗ structure of the quantum cohomologyof CP 1 from the viewpoint of diﬀerential geometry, Asian J. Math. 14 (2010) 417–438.
[2] J. Dorfmeister, F. Pedit, and H. Wu, Weierstrass type representation of harmonic maps into
symmetric spaces, Comm. Anal. Geom. 6 (1998), no. 4, 633–668.
[3] J.Dorfmeister, W.Rossman, The triviality for a Smyth-type potential and nonclosing of the resulting CMC surface, Geometry―Exploratory Workshop on Diﬀerential Geometry and its Applications, 61–70, Cluj Univ. Press (2011).
Email address: t udagawa@fuji.waseda.jp

Date: Received on February 17, 2020.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT
Joint Usage/Research Center on Mathematics and Theoretical Physics).