# Talks (slides)

## (click titles for the slides)

(click titles for the slides)

- Topological characterisation of Interstitial Lung Disease
- "医学と数理”（第２回 京大―ハイデルベルク大―理研 ワークショップ ）, Kyoto University, 21 Mar. 2020 (postponed due to coronavirus outbreak)

- 数学で形をデザインする
- 数学と諸分野の連携に向けた若手数学者交流会2020, JST, Tokyo, 15 Mar. 2020 (postponed due to coronavirus outbreak)

- Discrete surface deformation with a specified Gaussian curvature
- Closed kinematic chains and discrete space curves
- Closing Workshop of Joint Project between Austria (FWF) and Japan (JSPS): Geometric shape generation, Tokyo Institute of Technology, 20 Feb. 2020

- Flood prediction by geographic data analysis with GANs and tailored loss functions
- Forum-Math-for-Industry 2019, Massey University, New Zealand, 21 Nov. 2019
- In recent years, Japan has suffered from severe damage caused by floods. There is an urgent need for accurate flood forecast. Flood prediction usually takes the form of time series analysis in which future water level is regressed from that of the observed past and additionally precipitation, geography, and so on. There are standard machine learning models such as ARIMAX models to perform this type of analysis. Here, we investigate the use of neural network-based models for the problem. The two key factors are (1) use of generative adversarial networks; this is to cope with the lack of data (2) a loss function which is tailored for the problem; in flood prediction, usual losses such as the mean squared error are not optimal, since, for example, undershooting in prediction is much more harmful than overshooting.

- Forum-Math-for-Industry 2019, Massey University, New Zealand, 21 Nov. 2019
- Geometry of hinged linkage systems
- International Congress on Industrial and Applied Mathematics (ICIAM2019), Valencia, 17 Jul 2019.
- We study a certain class of spacial linkage mechanisms consisting of hinges by identifying them as discrete space curves and discuss their motion in terms of isoperimetric and torsion preserving deformation of the curves. In particular, we construct explicit motions corresponding to closed hinged linkages consisting of congruent tetrahedra, governed by the semi-discrete mKdV and sine-Gordon equations. This is joint work with K. Kajiwara and H. Park.

- International Congress on Industrial and Applied Mathematics (ICIAM2019), Valencia, 17 Jul 2019.
- Homology assisted neural networks for images
- International Congress on Industrial and Applied Mathematics (ICIAM2019), Valencia, 16 Jul 2019.
- Tools in topological data analysis (TDA) look at global features of the data while convolutional neural networks (CNNs) are good at detecting local features. The question is how we can combine the power of these two. We report our attempt to make CNNs utilise the global information obtained by TDA for a practical task on non-destructive inspection using sensor images. Then, we will investigate if CNNs can be trained to compute (persistent) homology. Finally, we describe how a CNN approximating persistent homology of the distance transformation of binary images may be used to "optimise" the topology of images.

- International Congress on Industrial and Applied Mathematics (ICIAM2019), Valencia, 16 Jul 2019.
- かたちの線型代数・微積分
- 東京大学数理情報学談話会, 3 Jul 2019
- 画像処理では、画像を二次元の格子グラフ上の関数とみなし、その関数を線型代数や微積の道具立てを用いて料理する。同様に、３次元空間内の形状は、グラフやより一般に単体複体上の関数とみなすことで、やはり線型代数やベクトル解析を用いて処理することができる。本講演では、形状処理、特にかたちの変形について、この観点からの研究を紹介したい。

- 東京大学数理情報学談話会, 3 Jul 2019
- やわらかい幾何の拡がり
- マス・フォア・イノベーション シンポジウム, 九州大学, 6 Jun 2019

- データから見える液晶の転移温度と秩序 -- QSPRの先へ
- 高分子学会九州支部フォーラム, 九州大学, 14 Mar 2019

- トポロジーに基づく形状設計法
- 芝浦工業大学 数理科学科 談話会, 19 Feb 2019
- ３次元形状をコンピューター上で扱う技術は、computer graphics (CG) や computer aided design (CAD) とともに発展してきた。また最近では 3D プリンタの発展により、digital fabrication も盛んである。これらの分野においては、トポロジーのアイデアが本質的な役割を果たす技術課題が散見される。計算機の上で"かたち”を扱うには、まず対象を有限の記号におとす必要があるが、代数トポロジーはまさにうってつけの道具である。この講演では、講演者がこれまで行ってきた共同研究の中から２つのトピックを取り上げ、単体複体で表された曲面の変形と、離散的な空間曲線としてモデル化されたリンク機構の設計についてお話ししたい。

- 芝浦工業大学 数理科学科 談話会, 19 Feb 2019
- 多数のヒンジからなる、自由度１の閉リンク機構
- 日本応用数理学会：応用数理ものづくり研究会, 政策研究大学院大学, 18 Feb 2019

- 微分形式を用いたメッシュ形状処理の基本(Introduction to discrete differential forms for mesh processing)
- IMI短期共同研究2018「造船工学における曲面幾何」, Kyushu univ., 25 Dec 2018

- A linkage mechanism that follows a discrete sine-Gordon equation
- SIDE13: Symmetries and Integrability of Difference Equations, Fukuoka, 15 Nov 2018
- We consider a family of linkage mechanisms which consist of $n$-copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous 3-fold symmetric Bricard6R linkage which exhibits a ``turning over'' motion. We can model such a linkage as a discrete closed curve in R^3 with a constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. In general, the degree of freedom of the motion, or equivalently, the dimension of the configuration space of such a linkage increases as n gets bigger. In this talk, we describe particular paths in the configuration space that are governed by semi-discrete sine-Gordor/mKdV equations. The infinitesimal motion is seen to be confined in the osculating plane. This is joint work with K. Kajiwara and H. Park at Kyushu university.

- SIDE13: Symmetries and Integrability of Difference Equations, Fukuoka, 15 Nov 2018
- Equivariant loop product (同変ループ積について)
- 日本数学会2018年度秋季総合分科会 トポロジー分科会 特別講演, 26 Sep 2018

- Image and Shape Manipulation (slides available upon request)
- 2018 NCTS Summer Course, 30 Jul - 17 Aug, 2018(with Chun-Chi Lin and Mei-Heng Yueh (NTNU))
- We will primarily be interested in altering shapes and images based on topological and geometrical techniques. Images are vector-valued functions on a rectangular domain, whose values are the color intensity of pixels. A mesh is an imbedding of a polygon into the three-space. Thus, we can manipulate images and shapes by defining operations on these maps. However, these maps are inevitably discrete to be dealt with on computer. Discrete differential geometry is a relatively new and active area in geometry to handle “smooth” shapes represented in a discrete way. We will see how basic notions in calculus are translated in a discrete language and can be applied to manipulate shapes and images. We will also discuss concrete algorithms and their implementations.

- 2018 NCTS Summer Course, 30 Jul - 17 Aug, 2018(with Chun-Chi Lin and Mei-Heng Yueh (NTNU))
- Representing the diagonal as the zero locus in a flag manifold
- Friday Topology Seminar, Kyushu University, 29 June 2018.
- The zero locus of a generic section of a vector bundle over a manifold defines a sub-manifold. A classical problem in geometry asks to realise the fundamental class of a specific sub-manifold in this way. We study the class of a point in a generalised flag manifold and of the diagonal in the direct product of two copies of a generalised flag manifold. These classes are important since they in a sense generate the cohomology of the flag manifold and the torus equivariant cohomology of the flag manifold, respectively.

- Friday Topology Seminar, Kyushu University, 29 June 2018.
- A closed linkage mechanism having a degenerate configuration space
- La Trobe-Kyushu Joint Seminar on Mathematics for Industry 34, 8 May 2018.
- Linkage is a mechanism consisting of rigid bars connected by revolute joints (hinges).Each hinge joins two adjacent bars and can rotate them around its axis. A linkage mechanism is said to be closed if it has the topology of a circle. We consider a closed linkage with n-hinges (hence, n-bars). Possible states of such a linkage form the configuration space, which can be thought of as a subspace of (S^1)^n. By dimension counting, the configuration space is generically n-6 dimensional. We discovered a closed linkage mechanism, which seems to have several interesting properties; most notably, the 1-dimensional configuration space regardless of n. Most of the results are only numerically confirmed and yet to be proved.

- Geometry seminar, Tokyo Institute of Technology, 24 May 2018.

- La Trobe-Kyushu Joint Seminar on Mathematics for Industry 34, 8 May 2018.
- 曲線の幾何学から生まれた閉リンク機構 (in Japanese)
- 精密工学会2018年春季大会 AIMaP数学応用シンポジウム: 精密工学と幾何学の新たな出会い, 中央大学理工学部, 17 Mar. 2018

- CGと数学 (in Japanese)
- 情報処理学会第８０回全国大会 イベント企画 「工芸から科学へ - CG技術の新たな挑戦 -」, 早稲田大学理工学部, 15 Mar. 2018
- 線形代数や流体力学を始めとして，CGには様々な数学が用いられていますが，数学を研究する立場からCG研究はどう映るのでしょうか．海外では意外にも古くから，また日本でも最近は，離散微分幾何学といった新しい数学の源泉としてCGが認識されてきています．数学には「役に立たないほど高等だ」という考え方と同時に，歴史的に物理や工学にアイデアの起源を見つけてきたという二面性があります．CGと数学の関わりについて，トポロジーの研究者である私個人の経験を中心にお話したいと思います．

- 情報処理学会第８０回全国大会 イベント企画 「工芸から科学へ - CG技術の新たな挑戦 -」, 早稲田大学理工学部, 15 Mar. 2018
- 柔らかい幾何の拡がり -トポロジーの応用- (in Japanese)
- AIMaP公開シンポジウム「数学と産業の協働ケーススタディ」, 日本橋ライフサイエンスビルディング, 20 Jan 2018.

- Interpolating Shapes
- Bilateral Mini-Workshop of NTNU and Yamaguchi University on Mathematics and its Applications, NTNU, Taiwan, 26 Dec. 2017
- Interpolation is one of the major principles in science and engineering. Curve fitting to observed data in experiments and image and audio up-scaling involve interpolation. Soap film naturally interpolates the boundary wire to a surface. In topology, we identify two shapes, such as a doughnut and a mag, which can be “interpolated" continuously. I will discuss a method to interpolate shapes based on topology to demonstrate possibilities of applying abstract mathematics to something “visible".

- Bilateral Mini-Workshop of NTNU and Yamaguchi University on Mathematics and its Applications, NTNU, Taiwan, 26 Dec. 2017
- ３次元形状モデリングへのトポロジーの応用 (in Japanese)
- 数学と諸分野の連携を通した知の創造, 東北大学, 9 Dec. 2017.

- PolyArc Fitter —An approximation of a curve using line segments and arcs —(with A. Hirakawa, Y. Onitsuka, C. Matsufuji, D. Yamaguchi)
- Poster at MEIS2017, Fukuoka, 16 Nov. 2017
- Given a sequence of points on the plane, we propose an algorithm to approximate them by a curve consisting of line segments and circular arcs. It has a practical use In computer aided manufacturing, and our algorithm has already been used in ship building. In a typical pipeline, ship parts are designed by a CAD software and numerical control machines (NCM) are used to actually cut steel plates. Many NCMs are capable of cutting only line segments and circular arcs, so the designed curves have to be converted in such forms. Moreover, it is desirable to have as few segments as possible due to efficiency and physical limitations of the machine. Given a sequence of points on the plane, our algorithm produces a curve consisting of a small number of line segments and circular arcs which passes within a user specified neighbour from every point.

- Poster at MEIS2017, Fukuoka, 16 Nov. 2017
- Image deformer on iPad [codes]
- Poster at MEIS2017, Fukuoka, 16 Nov. 2017
- We developed grid (2D-mesh) based image deformers which work on iOS devices. A loaded image or real-time input from the camera is converted to a texture placed on a triangulated grid, and the grid is deformed in real-time according to user’s touch gestures. A few different deformation algorithms are implemented. The source codes are available at my github under MIT license.

- Poster at MEIS2017, Fukuoka, 16 Nov. 2017
- Point and diagonal classes in flag varieties
- The 7th East Asian Conference on Algebraic Topology (EACAT 2017), IISER Mohali, India, 1 Dec. 2017
- The zero locus of a generic section of a vector bundle over a manifold defines a sub-manifold. A classical problem in geometry asks to realise the fundamental class of a specified sub-manifold in this way. We study the classes of a point in a generalised flag manifold and of the diagonal in the direct product of two copies of a generalised flag manifold. These classes are important since they are related to ordinary and equivariant Schubert polynomials.

- Topology Seminar, University of Southampton, 18 Dec. 2017

- The 7th East Asian Conference on Algebraic Topology (EACAT 2017), IISER Mohali, India, 1 Dec. 2017
- 画像関係の機械学習 Hands-on (Hands-on Machine Learning for Image analysis)
- IMI短期共同研究 三次元幾何モデリング評価手法の提案とソフトウェア開発, Kyushu university, Fukuoka, 5 Sep. 2017.

- Geometry of closed kinematic chain(** p. 21 the conjecture about a constant curvature is obviously wrong. I will fix it later. ** The slide with Possible Applications was not included in the talk due to patent application reasons.)
- IMI Workshop Mathematics in Interface, Dislocation and Structure of Crystals, Nishijin plaza, Fukuoka, 29 Aug. 2017
- Consider a system consisting of rigid bodies connected to each other. Such a system can be modelled by a graph with edges labelled by elements of the Euclidean group SE(3), where each cycle satisfies a certain closedness condition. We are particularly interested in a system consisting of hinges. To each vertex is assigned one degree-of-freedom, namely the rotation angle, and the configuration space of the system is described by the real solution to a system of polynomial equations. We found an interesting family of systems on cycle graphs, whose configuration spaces form positive dimensional real algebraic varieties. They are a type of so called Kaleidocycle, but exhibit intriguing properties such as anti-symmetry and constant bending energy. This is joint work with Eliot Fried, Michael Grunwald, and Johannes Schoenke at OIST.

- IMI Workshop Mathematics in Interface, Dislocation and Structure of Crystals, Nishijin plaza, Fukuoka, 29 Aug. 2017
- A topological view on shape deformation
- Applied Algebraic Topology 2017, Hokkaido University, 11 Aug. 2017.
- Manipulating shapes with computer has now become ubiquitous; Computer graphics (CG) is essential to film making, and so is Computer Aided Design (CAD) to industrial design. From a topological view point, a shape is a certain space (typically, a simplicial complex or a surface) and its deformation (animation) is a family of imbeddings into the ambient space (typically, $\R^3$). In this talk, I will review some of my recent work to demonstrate how this topological view can actually be made into algorithms of shape deformation.

- Applied Algebraic Topology 2017, Hokkaido University, 11 Aug. 2017.
- A secondary and equivariant string product
- Young Researchers in Homotopy theory and Toric topology 2017, Kyoto University, 4 Aug. 2017.
- A product on homology of the space of free loops LM over a closed manifold M is first defined by Chas and Sullivan and various similar constructions have been discovered since then. Among them is Chataur-Menichi’s product on homology of LBG where BG is the classifying space of a finite group G. We give a common generalisation to both by defining a product on homology of L(M_G), where M_G is the Borel construction of a compact (not necessarily connected) Lie group action on M. We also discuss a secondary product and show it is related to the cup product in negative Tate cohomology of G. This is joint work with Haggai Tene.

- Young Researchers in Homotopy theory and Toric topology 2017, Kyoto University, 4 Aug. 2017.
- Topology of brain-wide dynamics in consciousness (Satohiro Tajima and Shizuo Kaji)
- Poster at
*ASSC 21*, Beijing, China, 13-17 Jun. 2017

- Poster at
- Representations on Real Toric Manifolds
- Princeton-Rider workshop on the Homotopy Theory of Polyhedral Products, Princeton Univ./Rider Univ., 29 May 2017
- When a group G acts on a manifold M, the (co)homology of M is equipped with a G–module structure. One of the central questions in representation theory asks to realise a given G-module geometrically in this manner. In this talk, we consider finite group actions on real toric manifolds combinatorially through the correspondence between real toric manifolds and simplicial complexes with characteristic matrices. In particular, we see interesting representations of (signed) permutations appear on the homology of certain real toric manifolds. This is joint work with Soojin Cho and Suyoung Choi.

- Princeton-Rider workshop on the Homotopy Theory of Polyhedral Products, Princeton Univ./Rider Univ., 29 May 2017
- A type-A Weyl group action on the associated real toric manifold
- The 4th Korea Toric Topology Workshop, Jeju unipark, 27 Dec. 2016
- A simplicial complex $K$ with vertex labelling in an F_2 vector space gives rise to a real toric space. When a finite group $G$ acts on $K$ satisfying a certain condition, it induces an action on the toric space. We investigate the case when $(K, \lambda)$ is associated to a root system, and $G$ is its Weyl group. In particular, we give a concrete description of the Weyl group representation on the cohomology of the associated real toric space for type-A root systems. This is joint work with Soojin Cho and Suyoung Choi.

- The 4th Korea Toric Topology Workshop, Jeju unipark, 27 Dec. 2016
- Polar Decomposition of Square matrices
- Mathematical methods and practice in cryptography, security and bigdata, Hokkaido Univ. 21 Dec. 2016
- Polar decomposition is in the standard toolbox to analyse and visualise point cloud data in the Euclidean space. In this talk, I will discuss how it is used, in particular, in computer graphics. I will also talk about an efficient algorithm to compute the polar decomposition of a regular square matrix. This is joint work with Hiroyuki Ochiai at Kyushu university.

- Mathematical methods and practice in cryptography, security and bigdata, Hokkaido Univ. 21 Dec. 2016
- かたちの平均 ( some gif animations do not work with the online version of Power Point...)
- 信州大学数理科学談話会, 22 Nov. 2016
- ３次元空間内に複数の形状が与えられた時、 その”平均”とは何であろうか。この講演では、コンピューターで計算・出力するという観点から、 この問に一つの答えを与える。扱う数学の道具自体は高度ではないが、リー群論や離散幾何の アイデアが応用されることで、ペンギンたちが平均される様子をお見せしたい。

- 信州大学数理科学談話会, 22 Nov. 2016
- Moving Least Square + As-rigid-as-possible Shape Deformation
- Poster at MEIS2016, Nishijin-plaza, Fukuoka, 11 Nov. 2016
- We demonstrate that two well-known deformation techniques, Moving-least square and As-rigid-as possible schemes, can be combined straightforwardly and produce nice results.

- Poster at MEIS2016, Nishijin-plaza, Fukuoka, 11 Nov. 2016
- Designing transformations with simple ingredients
- GEMS2016: Geometry and Materials Sciences 2016, OIST, 17 Oct. 2016
- When we deign curves and surfaces with computer, it is enough to specify a small number of control points and parameters. Can we design transformations just as simply? I will propose a method to generate non-linear transformations from small input data which can be easily specified by the user. I will demonstrate this technique by applying it to character animation and shape modeling. Most of the codes used in the demonstration are available at my github repository.

- GEMS2016: Geometry and Materials Sciences 2016, OIST, 17 Oct. 2016
- トポロジーを応用した計算機による形状処理
- 京都大学応用数学セミナー, 11 Oct. 2016
- ３次元空間内の形状をコンピューター内であらわす際，一般的には，点群の座標+単体複体といった構造情報を用いる．この構造情報を保ったまま，点の座標のみを動かす変換を作用させることで，基本形状を変形してデザインしたり，キャラクターをアニメーションさせたりすることができる．この様な視点から，様々な場面で，それぞれの入力に応じて実際に変換を構成する手法を紹介する．扱う数学の道具自体は高度ではないが，リー群論や離散幾何のアイデアが応用され，ペンギン計算が行われる様子をお見せしたい．

- 京都大学応用数学セミナー, 11 Oct. 2016
- ３次元アフィン写像と形状変形ライブラリ (A C++ library for 3D affine transformation and shape deformation)
- 数学ソフトウェアとフリードキュメント XXIII, 関西大学, 14 Sep. 2016
- ３次元空間のアファイン写像は，計算機で形状変形やアニメーションを扱う上で基本構成要素となる．アフィン写像は行列で表すことができるが，これは効率は良い反面，例えば正則行列の平均が必ずしも正則にはならないなど，用途によっては使い勝手が良くない．講演者は落合啓之氏と共同で，リー環のカルタン分解を用いてアフィン写像をパラメトライズする方法を考案した．また，アフィン変換をベースに，より複雑な非線形変換を構成することができるが，これを形状変形とアニメーション生成に利用する例を示す．全ての実装はC++のソースコードも含めて MIT License で公開している．
- Affine transformation serves as one of the fundamental tools in computer graphics. As is well-known, affine transforms are represented by matrices. This is computationally efficient, but not always convenient. H. Ochiai and I developed a parametrisation method of affine transforms using the Cartan decomposition of their Lie algebra. Based on this method, we see how one can construct more complex non-linear transformation with applications to shape deformation and animation. The codes used are available at github with the MIT license.

- 数学ソフトウェアとフリードキュメント XXIII, 関西大学, 14 Sep. 2016
- Equivariant string products
- The 5th GeToPhyMa: Summer school "Rational homotopy theory and its interactions" celebrating Jim Stasheff and Dennis Sullivan for their 80th and 75 anniversary, UIR, Rabat, Morocco, 20 Jul 2016.
- A product in the homology of the free loop space LM of a manifold was discovered by Chas and Sullivan together with other structures. Chataur and Menichi defined among other things a similar product in the homology of the free loop space LBG of the classifying space of a topological group G. We define a product in the homology of the free loop space L(M_G) of the Borel construction of a manifold with a Lie group action. We show it vanishes under a certain degree condition, and in that case, we define a secondary version of the product. This is joint work with Haggai Tene.

- The 5th GeToPhyMa: Summer school "Rational homotopy theory and its interactions" celebrating Jim Stasheff and Dennis Sullivan for their 80th and 75 anniversary, UIR, Rabat, Morocco, 20 Jul 2016.
- Homotopy decomposition of a suspended real toric space
- Toric Topology 2016 in Kagoshima, Kagoshima University, 20 Apr 2016.
- Choi and Park gave a formula for the homology of real topological toric manifolds with coefficients in a ring in which two is invertible, based on the work by Cai on the homology of real moment-angle complexes. On the other hand, Bahri, Bendersky, Cohen, and Gitler gave homotopical decompositions of the suspensions of real moment-angle complexes. We combine these results to give p-local homotopy decompositions of the suspensions of real topological toric manifolds for odd primes p. This is joint work with S. Choi and S. Theriault.

- Toric Topology 2016 in Kagoshima, Kagoshima University, 20 Apr 2016.
- トポロジーのCGへの応用
- サロン時間学, 時間学研究所, 山口大学, 29 Jan 2016
- コンピューターグラフィックス(CG)と伝統的な絵画や映像表現との違いの一つは、前者は時間変化を伴い表現に大きな自由度がある、と言えるのではないか。絵画は自由に視覚効果を創造できるが、動きを表現することは難しい。一方で、実写映像で可能な表現は、物理的な制約を受ける。宇宙空間での大爆発や、恐竜が闊歩する世界は、コンピューター無しでは実現が難しいだろう。CGは、お金と時間のかかる実写の代替として、シミュレーションを用いて映像を作るという方向とは別に、全く物理的な制約を離れて、自由に仮想的な表現を生み出すという可能性も拓いた。その一つの例として、モーフィングと呼ばれる特殊効果がある。少年がカレーを食べるとサッカー選手に変化するCM(古いですが)のあれである。今回は、トポロジーという数学を用いたモーフィングの実現方法について紹介したい。 モノの形状の時間変化を数学モデルを用いて定式化するのだが、実験結果をいかによく説明するかという物理モデルの話と違い、人間の目に心地よければ良いという、正解のない世界で自由にモデルを設計できる。そんなところにどこか純粋数学に通じるものがありおもしろい。

- サロン時間学, 時間学研究所, 山口大学, 29 Jan 2016
- Steenrod algebra and Leibniz-Hopf algebra, and their duals
- Topology Seminar, Fukuoka University, Japan, 18 Jan 2016.
- A famous theorem by Milnor tells a lot about the structure of the dual Steenrod algebra; it is a polynomial algebra. He also gave a formula for the antipode on the generators. In this talk, we regard the dual Steenrod algebra as a sub-Hopf algebra of the Leibniz-Hopf algebra and provide another proof and an extension to Milnor's result purely combinatorially. This is joint work with N.D. Turgay.

- Topology Seminar, Fukuoka University, Japan, 18 Jan 2016.
- A topological algorithm for shape deformation in computer graphics
- Topology and Combinatorics seminar, Ajou University, Korea, 31 Dec 2015.
- In computer graphics, mathematics has been a fundamental toolbox. The Navier-Stokes equations is indispensable for generating clouds and fires, reproducing kernel Hilbert space is used for mixing different facial expressions, and simplicial complex and piecewise linear map (PL-map, in short) provide a natural framework for shape manipulation. In this talk, I will discuss an algorithm to blend/deform shapes based on PL-map and Lie theory. A shape is represented by a simplicial complex and its deformation by a PL-map. Then, creating an animation boils down to finding a nice path in the space of 3-dimensional PL-maps.

- Topology and Combinatorics seminar, Ajou University, Korea, 31 Dec 2015.
- Products in equivariant homology
- The 6th (non)Commutative algebra and Topology, Shinshu Univ., 20-22 Feb 2016.
- The 3rd Korea Toric Topology Winter Workshop, The-K Hotel, Gyeongju, Korea, 29 Dec 2015.
- We discuss an external product associated with the homology of a compact manifold with a Lie group action. It is then used to define a product on $H_G^*(LM)$ the homology of the free loop space over the Borel construction. This product unifies two known constructions in string topology; Chas-Sullivan's string product for the free loop space $LM$ over a manifold and Chataur-Menichi's string product for the free loop space $LBG$ over the classifying space of a compact Lie group. We will also introduce its secondary product, which generalises Tate's cup product in the homology of $BG$. This is joint work with Haggai Tene.

- Tetrisation of triangular meshes and its application in shape blending
- MEIS2015, Nishijin plaza, Kyushu university, 26 Sep 2015.

- Products in Equivariant Homology
- Topology Seminar, CRM, Barcelona, 12 June 2015.
- Let G be a compact Lie group and M be a compact G-manifold. Denote its Borel construction by M_G. Given a homotopy pullback square with the base space M_G, we will introduce a generalised external product in the homology such that it reduces to the ordinary external product when M is a point and G is trivial. ( In this case, the homotopy pullback is just the direct product. ) It unifies two constructions in string topology; Chas-Sullivan's string product for the free loop space LM over a manifold and Chataur-Menichi's string product for the free loop space LBG over the classifying space of a compact Lie group. We will also introduce its secondary product, which generalises Tate's cup product in the homology of BG. This is joint work with Haggai Tene.

- Topology Seminar, CRM, Barcelona, 12 June 2015.
- The dual Steenrod algebra and the overlapping shuffle product
- Antalya Algebra Days XVII, Nesin Maths Village, Turkey, 23 May 2015.

- "Homotopy decomposition of a suspended flag manifold"
- Poster at IMPANGA15, Banach center at Bedlewo, Poland, 13 Apr 2015.

- External products in equivariant homology
- Torus Actions in Geometry, Topology, and Applications, Skoltech, Moscow, 17 Feb, 2015.

- Mod p decompositions of the loop spaces of flag manifolds
- International Mathematics Conference, in honour of the 70th Birthday of Professor S. A. Ilori, Ibadan, 12-13 Jan. 2015.
- Let $G$ be a simply-connected, simple, complex Lie group and $P$ be its parabolic subgroup. The homogeneous space $G/P$ is called the flag manifold. Topological invariants of $G/P$ such as the cohomology ring, K-theory, and its equivariant variants have been extensively studied under the name of Schubert calculus. However, much less is known about the homotopy groups. In this talk, we outline an attempt to give a better understanding of the homotopy groups of $G/P$ using standard techniques from algebraic topology; localisation at a prime and homotopy decomposition. We give a homotopy decomposition of the based loop space of $G/P$ into a product of irreducible spaces ( which are fairy simple ) when localised at a quasi-regular (that is, not too small) prime of $G$. In particular, we give a bound for the p-primary exponent of the homotopy groups of $G/P$. This is a joint work with A. Ohsita and S. Theriault.

- International Mathematics Conference, in honour of the 70th Birthday of Professor S. A. Ilori, Ibadan, 12-13 Jan. 2015.
- Torus equivariant cohomology of flag manifolds
- Topology Seminar at University of Ibadan 15. Jan, 2015
- Let $G$ be a simply-connected, simple, complex Lie group and $P$ be its parabolic subgroup. The homogeneous space $G/P$ is called the flag manifold. Fix a maximal torus $T$ of $G$ which is contained in $P$. It acts on $G/P$ by the group multiplication. One of the main goals of the "equivariant Schubert calculus" is to study the $T$-equivariant (Borel) cohomology $H^*_T(G/P)$. There are three presentations known for $H^*_T(G/P)$: (1) Chevalley gave a free $H^*(BT)$-module basis of it consisting of sub-varieties called the Schubert varieties. (2) Borel computed it with the rational coefficients as the “double coinvariant ring of the Weyl group." (3) Using Goresky-Kottwitz-MacPherson’s theory, it is given a combinatorial presentation called the GKM-presentation. Each presentation has both advantages and disadvantages; for example, in (1) geometric meaning of a class is clear but it is hard to compute the product of two classes while in (2) elements are just polynomials and one can easily multiply them. Therefore, it is preferable to know how to convert an element in one presentation to another. In this talk, I will review those three presentations and investigate how they are related to each other.

- Topology Seminar at University of Ibadan 15. Jan, 2015
- Shape deformation in Computer graphics
- Poster at Discrete, Computational and Algebraic Topology, University of Copenhagen, Nov. 10-14 2014.

- A new algorithm of Polar decomposition
- Poster at Symposium MEIS2014: Mathematical Progress in Expressive Image Synthesis 2014, Nishijin Plaza Kyushu University, Nov. 12-14 2014.

- N-way morphing plugin for Maya [video]
- Poster at Symposium MEIS2014: Mathematical Progress in Expressive Image Synthesis 2014, Nishijin Plaza Kyushu University, Nov. 12-14 2014.

- Mod-p decompositions of the loop spaces of compact symmetric spaces
- 29th British Topology Meeting, Southampton, UK, Sep. 8. 2014.
- We give homotopy decompositions of the based loop spaces of compact symmetric spaces after they are localised at large primes. The factors are fairly simple; namely spheres, sphere bundles over spheres, and their loop spaces. As an application, upper bounds for the homotopy exponents are determined. This is a joint work with A. Ohsita and S. Theriault.

- 29th British Topology Meeting, Southampton, UK, Sep. 8. 2014.
- A product in equivariant homology for compact Lie group actions
- Topology Seminar, CRM, Barcelona, Spain, July 18. 2014.
- The Tate cohomology for a finite group integrates group homology and cohomology into one theory. It is equipped with a cup product, which coincides with the usual one on cohomology and gives a ring structure on homology. A few attempts have been made to generalise this product structure on homology. We follow the line of Kreck and Tene. Kreck defined a product on H_*(BG;Z) for a compact Lie group G based on his geometric homology theory and Tene showed it coincides with the cup product on the Tate cohomology when G is finite. We will generalise this product to one on the equivariant homology of a manifold with a nice action of a Lie group. Our construction is simple and purely homotopy theoretical. This is a joint work with Haggai Tene.

- Topology Seminar, CRM, Barcelona, Spain, July 18. 2014.
- A topological algorithm for Blending Shapes
- Directed Algebraic Topology and Concurrency, Université Lyon 1, France, Jan 30th 2014.

- Lie group action on a GKM manifold
- Manifold Atlas Writing Seminar, Max Planck Institute for Mathematics, Bonn, Apr 28th 2014.
- Topology Seminar, University of Southampton, UK, Dec 9th 2013.
- A GKM manifold is a manifold with a "nice" torus action. Many interesting spaces are examples of GKM manifold including toric manifolds and flag manifolds. Goresky-Kottwitz-MacPherson (for which GKM is an abbreviation) showed that a certain kind of graph, called the GKM graph, is associated to a GKM manifold, from which we can read off the torus equivariant cohomology of the manifold purely combinatorially. I will discuss the case when the torus action extends to a Lie group action, and show how the action is reflected on the combinatorial structure of the GKM graph.

- An Application of Lie theory to Computer Graphics
- Applied Topology, Banach center in Bedlewo, Poland, July 25th 2013.
- In computer graphics, various mathematics is used such as the Navie-Stokes equations for generating clouds and fires, reproducing kernel Hilbert space for mixing different facial expressions, and piecewise linear map (PL-map, in short) for morphing shapes. I will discuss an algorithm to blend/deform shapes based on PL-map and an elementary Lie theory. A shape is represented by a polyhedron and its deformation by a PL-map. The idea is to find a suitable PL-map which minimizes a certain energy functional defined on the space of $3$-dimensional PL-maps.

- Applied Topology, Banach center in Bedlewo, Poland, July 25th 2013.
- Cohomology of a GKM graph with symmetry
- Toric Topology 2012 in Osaka, Osaka city university, Nov. 18th 2012.
- 「代数トポロジーと組み合わせ論の相互作用」, 兵庫教育大学神戸サテライト, Feb. 16th 2013.
- Haifa university colloquium, Israel, Mar. 19th 2013.

- Ordinary and Equivariant Schubert classes
- Characteristic classes and intersection theory seminar, Higher School of Economics, Moscow, Sep. 20th 2012.

- 数学がつなぐカタチ - 幾何学的な形状補間法 -
- ワークショップ「数理科学と情報科学の周辺」, Shinshu University, Feb. 14th 2013.
- CEDEC2012, Pacifico Yokohama, Aug. 20th 2012.
- 少数のキーフレーム(物体の形状データ)から、それらを補間する連続的なフレームデータを生成する技術は、フレーム補間とよばれ、アニメーション作成やインタラクティブな物体変形に使われています。ここでは、特に局所的な形状をなるべく保ったままフレーム補間を行うアルゴリズムを、背後にある数学に焦点を当てながら紹介し ます。技術自体の即効性よりも、アルゴリズムの発見法や数学者の物の見方について主にお話ししたいと思います。

- "Ordinary vs Double Schubert polynomials"
- Poster session at MSJ-SI2012 Schubert calculus, Osaka city university, July 26th 2012.

- "An invitation to Schubert calculus"
- Combinatorics Seminar in Kyushu University, Nishijin plaza, Kyushu university, July 14th 2012.
- Postnikov Seminar, Moscow State University, Sep. 18th 2012.

- "The equivariant cohomology of a manifold with a G-action"
- Topology of Mapping space and around, Okinawa Senin Kaikan May 14th 2012.

- "Frame interpolation for character animation - Overview of the problem and our progress"
- IMI Short-term Joint Research Project "New Animation Interpolation and Proposal of its Evaluation Indicators", Institute of Math-for-Industry, Kyushu University, Mar. 5th. 2012.
- The interpolation technique, which produces continuous images from two or more keyframes, is important in the video production fields since it drastically reduces tedious work of creating animations. Among many suggested methods, Alexa et. al. introduced so-called "As-rigid-as-possible interpolation" (ARAP, in short) in 2000. Since then, improvements and extensions have been studied mostly from CG side, and the way of comparison and evaluation for different techniques is aesthetic. Mathematically, ARAP can be formulated as the problem of finding an appropriate path connecting given polyhedral shapes. Our aim is to establish rigorous framework to study it. In this talk, I will: (1) give a brief survey of the original method, (2) introduce mathematical evaluation indicators for "rigidity", (3) provide possible improvements and extensions.

- IMI Short-term Joint Research Project "New Animation Interpolation and Proposal of its Evaluation Indicators", Institute of Math-for-Industry, Kyushu University, Mar. 5th. 2012.
**"Schubert calculus for G-manifolds"**- International Conference Toric Topology and Automorphic Functions, Pacific National University, Khabarovsk, Russia, Sep. 9th. 2011.
- Toric Topology in Osaka 2011, Osaka city university, Osaka, Nov. 28th. 2011.
- Let G be a Lie group and T be its maximal torus. The homogeneous spaces G/T is known to be a smooth variety and called the flag variety. It admits the action of T by the group multiplication, and the equivariant cohomology H^*_T(G/T) brings up an interesting subject of study called equivariant Schubert calculus. Among remarkable properties of H^*_T(G/T) is the fact that there is a distinguished geometric basis consisting of the Schubert classes, which fits into a hierarchy posed by the divided difference operators. We try to extend this to a litter wider class of G-manifolds, by using so-called ``localization to fixed points'' developed in [GKM].

**"Intersection of topology and combinatorics in Schubert calculus"**- StudioPhones Seminar, Studio Phones, Osaka, Aug. 23rd. 2011.
- Ajou university Colloquim, Suwon, Korea, Nov. 25th. 2011.
- "How many lines are there in the three space which meet all the four given lines ?" In 19th century, H.Schubert considered this problem in an insightful but not rigorous way. He invented a symbolic ``calculation'' for the conditions on lines as follows: [intersecting a given line]^{¥cap 4} = [lying on a given line] ¥cup [lying on a given line],and obtained the answer two. In fact, the ``algebra'' of the conditions on lines is isomorphic to the ring of the symmetric polynomials called Schur polynomials. D.Hilbert asked for a rigorous foundation for the above calculus as the 15th problem in his 1900 lecture and now Schubert’s quiz can be rephrased in terms of cohomology, or equivalently, intersection theory of a Grassmaniann manifold. In this talk, I will briefly review the basics of Schubert calculus with a focus on the correspondence of several algebras occurring in this subject including the above one.

**"A rational homotopy model of quasitoric manifolds"**,- Toric Topology in Himeji 2011, Egret Himeji, Apr. 3rd. 2011.

**"グレブナー基底を用いると、解けそうで解けない少し難しい代数トポロジーの問題"**,- JST CREST Groebner basis young researchers meeting, Yamaguchi Univ. Feb. 17th. 2011.

**"Equivariant cohomology of flag manifolds"**,- トポロジーの多様性, Kinosaki, Dec. 2nd, 2010.
- 京大微分トポロジーセミナー, Kyoto Univ. Dec. 7th.

**"Equivariant Schubert calculus of Coxeter group I_2(m)"**,- The International Conference "GEOMETRY, TOPOLOGY, ALGEBRA and NUMBER THEORY, APPLICATIONS" dedicated to the 120th anniversary of Boris Delone, Moscow, Aug. 19th, 2010.
- International Conference Japan-Mexico on Topology and its Applications, Colima, Mexico, September 27 - October 1, 2010.
- Let $G$ be a Lie group and $T$ be its maximal torus. The homogeneous spaces $G/T$ is known to be a smooth variety and called the
*flag variety*of type $G$. Its cohomology group has a distinguished basis consisting of Schubert classes, which arise from a certain family of sub-varieties. The ring structure of $H^*(G/T)$ with respect to this basis reveals interesting interactions between topology, algebraic geometry, representation theory, and combinatorics, and has been studied under the name of Schubert calculus. One way to study $H^*(G/T)$ is to identify it with the coinvariant ring of the Weyl group $W$ of $G$, i.e. the polynomial ring divided by the ideal generated by the invariant polynomials of $W$. From this point of view, the problem can be rephrased purely in terms of $W$ and extended to any Coxeter group including non-crystallographic ones. In fact, H. Hiller pursued this way in his book ``The geometry of Coxeter groups'' and gave a characterization of a ``Schubert class'' in the coinvariant ring. On the other hand, $G/T$ has the canonical action of $T$ and we can consider the equivariant topology with respect to this action. A similar story goes for the equivariant cohomology $H^*_T(G/T)$ and we can consider equivariant Schubert calculus for Coxeter groups. This time we consider a double version of a coinvariant ring. Along this line, the first difficulty is how to find polynomials in it representing Schubert classes. By several people, such polynomial representatives have been found for type $A_n, B_n, C_n, D_n$. Here we give polynomial representatives for the non-crystallographic group of type $I_2(m)$. The main ingredients is the localization technique, a powerful machinery of equivariant topology.

- Let $G$ be a Lie group and $T$ be its maximal torus. The homogeneous spaces $G/T$ is known to be a smooth variety and called the

**"Schubert calculus, seen from torus equivariant topology"**,- KAIST Toric Topology Workshop 2010, Feb. 25th. 2010.
- I will give two talks on Schubert calculus, with emphasis on the torus action. Schubert calculus is a study of the geometry of a flag variety, which is defined as the homogeneous space of a compact Lie group $G$ divided by its subgroup $P$ containing a maximal torus $T$. Since flag varieties are so basic objects in various areas of mathematics, there are a lot of ways to explore this fertile land. Here we'll take an inclination for a view from torus equivariant topology. A flag variety has an ideal action of the maximal torus $T$, which is hamiltonian with isolated fixed points corresponding to the elements of the Weyl group. Hence, the localization technique, widely known as "GKM theory," offers a powerful machinery to deal with topological invariants such as the equivariant cohomology of flag varieties combinatorially. Fortunately enough for those who have a liking for computation, this is actually applicable to calculations, through which we can see a concrete aspect of the subject. I will start with brief history of the subject and then review basic notions, playing with the most fundamental example of Grassmannian manifolds. Then I will introduce two descriptions of the torus-equivariant cohomology of flag varieties, one by the GKM graph, the other by the polynomial ring with two series of indeterminants, and discuss the interaction between them. With these preparations, we'll set out for a somewhat outskirts region of the study, namely a concrete calculation of the torus equivariant cohomology of the flag varieties associated to exceptional Lie groups.

- KAIST Toric Topology Workshop 2010, Feb. 25th. 2010.
**"Torus equivariant cohomology of flag varieties"****,**- Fukuoka Seminar on Homotopy Theory, Jan. 9th. 2010.
- The Third East Asia Conference on Algebraic Topology, Vietnam National University, Hanoi, Dec. 17th, 2009.
- Let $G$ be a simple complex Lie group and $B$ its Borel subgroup. Then the (right) homogeneous space $G/B$ is called the flag variety. One of its rich structures is that $G/B$ has a stratification given by the
*Schubert varieties*, which is a class of subvarieties indexed by the Weyl group of $G$, and form a module basis of $H^*(G/B;Z)$. Determining the ring structure of $H^*(G/B;Z)$ with respect to this particular basis is a classical but still active theme in*Schubert calculus*. There are a lot of variant of this problem and here we consider the equivariant version. Let $T$ be the maximal torus of $G$. Then $T$ acts on $G/B$ from the left. Since the Schubert varieties are $T$-invariant, the story immediately fits into the $T$-equivariant setting. In this talk we observe a strategy for the concrete calculation of the torus equivariant cohomology $H^*_T(G/B;Z)$, and apply it to the specific case of $G=G_2$, the rank two exceptional group.

- Let $G$ be a simple complex Lie group and $B$ its Borel subgroup. Then the (right) homogeneous space $G/B$ is called the flag variety. One of its rich structures is that $G/B$ has a stratification given by the

**"Schubert calculus of exceptional types"****, [Abstract]**- The 56th Topology Symposium, Hokkaido University, Aug. 11th, 2009.
- "How many lines in the space meet four general lines ?" Starting with the inevitable introduction of
*Schubert calculus*, we make a whirlwind trip through this actively studied area. After introducing the three main characters, the*flag variety*, the*Weyl group*, and the*Schubert variety*, brought together on the*Hasse(GKM) diagram*, we focus on the cases of exceptional types. We explain how one can obtain the explicit description of integral cohomology of flag varieties of exceptional types as quotient rings of polynomial rings, and how to find representatives of Schubert classes there.

- "How many lines in the space meet four general lines ?" Starting with the inevitable introduction of

- The 56th Topology Symposium, Hokkaido University, Aug. 11th, 2009.
**"Divided difference operator and equivariant cohomology"**,- Shinshu Topology Seminar, Shinshu University, Mar. 18th, 2009.

**"Bott-Samelson cycle in Schubert calculus"**,**"An algebraic topological approach toward concrete Schubert calculus"**,- The 2nd East Asia Conference on Algebraic Topology, National University of Singapore, Dec. 19th, 2008.
- Topology Friday Seminar, Kyushu University, Jun. 20th, 2008. Reference list
- Let G be a connected simple complex Lie group and P be a parabolic subgroup. The homogeneous space G/P is known to be a projective variety called the generalized flag variety. The Chow ring A(G/P) of G/P ( ,which is isomorphic to the ordinary cohomology with integral coefficients) is known to have a good Z-module basis consisting of so called Schubert varieties. Schubert calculus in a narrow sense is a study of the ring structure of A(G/P), or more precisely, a study of the structure constants for the intersection products of two arbitrary Schubert varieties. In this talk, we give an algebraic topological method to present A(G/P) explicitly in the form of the quotient of a polynomial algebra, especially when G is an exceptional group. Main tools and ingredients are the divided difference operators introduced by Berstein-Gelfand-Gelfand, and the Borel presentations of the cohomology rings of homogeneous spaces, which are laboriously calculated by Japanese algebraic topologists. This is a joint work with M.Nakagawa.

**"Computers work exceptionally on Lie groups"**,- First Global COE seminar on Mathematical Research Using Computers, Kyoto University, Oct 24, 2008.
- The well known classification theorem states that there are only nine types of simple compact Lie groups; the four classical infinite families and the five exceptional ones. Although topological invariants of classical Lie groups are often given in simple unified forms, those of exceptional groups are usually of ugly looks, which insist on handling by a case by case analysis; this is where computers would do themselves justice. By presenting an example of the computation of the Chow rings of complex Lie groups, I will demonstrate the possibility of using a computer in such a kind of research where tedious and technical labor is required.

- First Global COE seminar on Mathematical Research Using Computers, Kyoto University, Oct 24, 2008.
**"Mod 2 cohomology of some low rank 2-local finite groups"**,- International Conference on Algebraic Topology, Korea University, Oct. 5th, 2008.
- The cohomology of finite groups have been studied successfully by homotopy theoretical approaches. The mod p cohomology of some finite groups of Lie type can be obtained from that of the homotopy fixed points of unstable Adams operations, which have similar properties as the free loop spaces over the classifying spaces of Lie groups. By inspecting this similarity, recently Kishimoto and Kono developed a method to compute the mod p cohomology ring over the Steenrod algebra of the finite groups. We will discuss the actual computation of some low rank cases for p=2, and show that the mod 2 cohomology of the groups and the free loop spaces are isomorphic as rings over the Steenrod algebra in those cases.

- International Conference on Algebraic Topology, Korea University, Oct. 5th, 2008.
**"Chow rings of Complex Algebraic Groups"**, (in Japanese, hand written)- 1. COE tea time, Kyoto University, Jan. 17th, 2008.

**"Chow rings of Complex Algebraic Groups"**,- Workshop on Schubert calculus 2008, Kansai Seminar House, Mar. 20, 2008. [Slide]
- The 4th COE Conference for Young Researchers, Hokkaido Univ, Feb. 13, 2008. [Slide]
- Topology Seminar, Kyoto University, Jan. 29, 2008.
- International Conference on Topology and its Applications, Kyoto University, Dec. 4, 2007.
- Symplectic Geometry Seminar, University of Toronto, Nov. 26, 2007.
- Topology Seminar, Johns Hopkins University, Nov. 13, 2007.
- This work should be viewed as a completion of a series of computations of Chow rings for simply-connected Lie groups by Chevalley, Grothendieck, and most recently by Marlin (1970s), who computed the Chow rings for G = Spin(n), G2, and F4. His method did not cover the cases of the exceptional Lie types G=E6, E7, and E8. On the other hand, these Chow rings A(G) can be determined from the cohomology of the corresponding flag varieties, the latter of which were computed by Borel, Toda, Watanabe, and Nakagawa. We use the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure to obtain efficient presentations of the Chow rings of G=E6, E7, and E8 via geometric generators coming from Schubert classes on G/B. This is a joint work with M.Nakagawa.

**"Homotopy Nilpotency in $p$-compact groups"**,- Geometry & Topology Seminar, McMaster University, Nov. 22, 2007.
- Homotopy nilpotency is a homotopical analogue of the ordinary nilpotency for a group. This is an invariant for a H-space which measures how far its product is from being homotopy commutative. We determine the homotopy nilpotency of compact Lie groups when localized at large primes. This work is a continuation of that of McGibbon (1984) which determined the homotopy commutativity of localized compact Lie groups. By the same methods, we also obtained a similar result for all the p-compact groups, which is a homotopy theoretical generalization of compact Lie groups. This is a joint work with D.Kishimoto.

- Geometry & Topology Seminar, McMaster University, Nov. 22, 2007.
**"Certain $p$-local $H$-space structure on the classifying spaces of gauge groups"**, Takamatsu seminar on homotopy theory, Takamatsu National College of Technology, March 2007.**"Homotopy nilpotency in localized Lie groups"**, The 3rd COE Conference for Young Researchers, Hokkaido University, Feb. 13th. 2007.**"Homotopy nilpotency in localized groups"**, Homotopy Symposium 2006, Ehime University, Nov. 28th. 2006.**"Cohomology of the classifying spaces of gauge groups over 3-manifolds in low dimensions"**, Homotopy Symposium 2004, Okinawa Sen-in Kaikan Hotel, November. 2004.**"Introduction to rational homotopy theory"**, The 1st COE Kinosaki Young Researchers' Seminar, Kinosaki Town Hall, February 2004.**"On the rational $H$-spaces"**, Algebraic and Geometric Models for Spaces and Around, Okayama University, September 2003.